This is SimplicialModules.info, produced by Macaulay2, version 1.26.05

SimplicialModules -- A package for creating and computing objects in the category of Simplicial R-Modules
*********************************************************************************************************

Description
===========

A simplicial R-module is a presheaf on the so-called Simplex Category, with
values in the category of R-modules. Concretely, such objects can be viewed as
nonnegatively graded R-modules equipped with certain face and degeneracy
operators satisfying the simplicial identities. As an example, every R-module M
can be converted into a simplicial R-module whose degree n piece is equal to M
for all n, and with face/degeneracy operators simply given by the identity.

+-----------------------------------------------------------------------------+
|i1 : S = simplicialModule(ZZ^2,3,Degeneracy => true) --the integer 3         |
|                                                                             |
|       2       2       2       2                                             |
|o1 = ZZ  <-- ZZ  <-- ZZ  <-- ZZ <-- ...                                      |
|                                                                             |
|     0       1       2       3                                               |
|                                                                             |
|o1 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|specifies a top degree, the option Degeneracy specifies whether or not to    |
|-----------------------------------------------------------------------------|
|compute degeneracy maps                                                      |
+-----------------------------------------------------------------------------+

The output string for a simplicial module is meant to indicate that this object
is infinite in general, and the user can only compute a finite snapshot of the
object. The face/degeneracy maps can be accessed using the keys dd and ss,
respectively. In order to verify that the resulting face/degeneracy maps
satisfy the simplicial identities, one can use the isSimplicialModule,
command.

+-------------------------------------------+
|i2 : S.dd                                  |
|                                           |
|                2                2         |
|o2 = (0, 0) : ZZ  <----------- ZZ  : (1, 0)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|                2                2         |
|     (0, 1) : ZZ  <----------- ZZ  : (1, 1)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|                2                2         |
|     (1, 0) : ZZ  <----------- ZZ  : (2, 0)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|                2                2         |
|     (1, 1) : ZZ  <----------- ZZ  : (2, 1)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|                2                2         |
|     (1, 2) : ZZ  <----------- ZZ  : (2, 2)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|                2                2         |
|     (2, 0) : ZZ  <----------- ZZ  : (3, 0)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|                2                2         |
|     (2, 1) : ZZ  <----------- ZZ  : (3, 1)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|                2                2         |
|     (2, 2) : ZZ  <----------- ZZ  : (3, 2)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|                2                2         |
|     (2, 3) : ZZ  <----------- ZZ  : (3, 3)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|o2 : SimplicialModuleMap                   |
+-------------------------------------------+
|i3 : S.ss                                  |
|                                           |
|                2                2         |
|o3 = (1, 0) : ZZ  <----------- ZZ  : (0, 0)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|                2                2         |
|     (2, 0) : ZZ  <----------- ZZ  : (1, 0)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|                2                2         |
|     (2, 1) : ZZ  <----------- ZZ  : (1, 1)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|                2                2         |
|     (3, 0) : ZZ  <----------- ZZ  : (2, 0)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|                2                2         |
|     (3, 1) : ZZ  <----------- ZZ  : (2, 1)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|                2                2         |
|     (3, 2) : ZZ  <----------- ZZ  : (2, 2)|
|                     | 1 0 |               |
|                     | 0 1 |               |
|                                           |
|o3 : SimplicialModuleMap                   |
+-------------------------------------------+
|i4 : assert isSimplicialModule S           |
+-------------------------------------------+

In general, simplicial objects are infinite objects. Because of this, the user
can specify a top degree for the resulting simplicial object. If no top degree
is specified, then the default top degree is determined by the input object.

The Dold-Kan Correspondence
===========================

The category of simplicial R-modules is equivalent to the category of
nonnegatively graded chain complexes via an equivalence known as the Dold-Kan
correspondence.

This means that there is a functor that converts a chain complex into a
simplicial object, known as the Dold-Kan functor. In practice, the image of the
Dold-Kan functor is highly nontrivial to compute by hand. However, this package
uses an algorithm of Kock-Sakuranath to compute Dold-Kan images quite
efficiently by using the simplicialModule, command:

+-----------------------------------------------------------------------------+
|i5 : R = ZZ/101[x_1..x_3];                                                   |
+-----------------------------------------------------------------------------+
|i6 : K = koszulComplex vars R                                                |
|                                                                             |
|      1      3      3      1                                                 |
|o6 = R  <-- R  <-- R  <-- R                                                  |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o6 : Complex                                                                 |
+-----------------------------------------------------------------------------+
|i7 : simplicialModule(K) --defaults to top degree 3                          |
|                                                                             |
|      1      4      10      20                                               |
|o7 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o7 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i8 : elapsedTime simplicialModule(K,6) --specify top degree 6                |
| -- .0839178s elapsed                                                        |
|                                                                             |
|      1      4      10      20      35      56      84                       |
|o8 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R  <-- ...                |
|                                                                             |
|     0      1      2       3       4       5       6                         |
|                                                                             |
|o8 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i9 : elapsedTime S' = simplicialModule(K,6, Degeneracy => true)              |
| -- .203081s elapsed                                                         |
|                                                                             |
|      1      4      10      20      35      56      84                       |
|o9 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R  <-- ...                |
|                                                                             |
|     0      1      2       3       4       5       6                         |
|                                                                             |
|o9 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i10 : isWellDefined S'  --this method checks the simplicial identities as    |
|                                                                             |
|o10 = true                                                                   |
|-----------------------------------------------------------------------------|
|well as many other basic checks                                              |
+-----------------------------------------------------------------------------+

The other piece of the Dold-Kan correspondence is the functor that converts a
simplicial module into a chain complex. This functor is often referred to as
the normalization functor, and is also implemented as the command
normalize (SchurFunctors). If a simplicial R-module is computed as
the Dold-Kan image of some complex, then this complex is cached and the
normalize (SchurFunctors), command returns this cached complex. If
the user wants the normalization to be computed without accessing this cached
value, one can use the CheckComplex => false option. By default, the
normalization function does not prune the output, so the user should use
prune (Macaulay2Doc), to get a nicer looking output.

+-----------------------------------------------------------------------------+
|i11 : R = ZZ/101[x_1..x_3];                                                  |
+-----------------------------------------------------------------------------+
|i12 : K = koszulComplex vars R                                               |
|                                                                             |
|       1      3      3      1                                                |
|o12 = R  <-- R  <-- R  <-- R                                                 |
|                                                                             |
|      0      1      2      3                                                 |
|                                                                             |
|o12 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i13 : Kn = normalize(simplicialModule(K,4), CheckComplex => false)           |
|                                                                             |
|       1                                                                     |
|o13 = R  <-- image {0} | 0 0 0 | <-- image {0} | 0 0 0 | <-- image {0} | 0 | |
|                   {1} | 1 0 0 |           {1} | 0 0 0 |           {1} | 0 | |
|      0            {1} | 0 1 0 |           {1} | 0 0 0 |           {1} | 0 | |
|                   {1} | 0 0 1 |           {1} | 0 0 0 |           {1} | 0 | |
|                                           {1} | 0 0 0 |           {1} | 0 | |
|             1                             {1} | 0 0 0 |           {1} | 0 | |
|                                           {1} | 0 0 0 |           {1} | 0 | |
|                                           {2} | 1 0 0 |           {1} | 0 | |
|                                           {2} | 0 1 0 |           {1} | 0 | |
|                                           {2} | 0 0 1 |           {1} | 0 | |
|                                                                   {2} | 0 | |
|                                     2                             {2} | 0 | |
|                                                                   {2} | 0 | |
|                                                                   {2} | 0 | |
|                                                                   {2} | 0 | |
|                                                                   {2} | 0 | |
|                                                                   {2} | 0 | |
|                                                                   {2} | 0 | |
|                                                                   {2} | 0 | |
|                                                                   {3} | 1 | |
|                                                                             |
|                                                             3               |
|                                                                             |
|o13 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i14 : Kn.dd                                                                  |
|                                                                             |
|           1                                                                 |
|o14 = 0 : R  <------------------- image {0} | 0 0 0 | : 1                    |
|                | x_1 x_2 x_3 |         {1} | 1 0 0 |                        |
|                                        {1} | 0 1 0 |                        |
|                                        {1} | 0 0 1 |                        |
|                                                                             |
|      1 : image {0} | 0 0 0 | <-------------------------- image {0} | 0 0 0 ||
|                {1} | 1 0 0 |    {1} | -x_2 -x_3 0    |         {1} | 0 0 0 ||
|                {1} | 0 1 0 |    {1} | x_1  0    -x_3 |         {1} | 0 0 0 ||
|                {1} | 0 0 1 |    {1} | 0    x_1  x_2  |         {1} | 0 0 0 ||
|                                                                {1} | 0 0 0 ||
|                                                                {1} | 0 0 0 ||
|                                                                {1} | 0 0 0 ||
|                                                                {2} | 1 0 0 ||
|                                                                {2} | 0 1 0 ||
|                                                                {2} | 0 0 1 ||
|                                                                             |
|      2 : image {0} | 0 0 0 | <---------------- image {0} | 0 | : 3          |
|                {1} | 0 0 0 |    {2} | x_3  |         {1} | 0 |              |
|                {1} | 0 0 0 |    {2} | -x_2 |         {1} | 0 |              |
|                {1} | 0 0 0 |    {2} | x_1  |         {1} | 0 |              |
|                {1} | 0 0 0 |                         {1} | 0 |              |
|                {1} | 0 0 0 |                         {1} | 0 |              |
|                {1} | 0 0 0 |                         {1} | 0 |              |
|                {2} | 1 0 0 |                         {1} | 0 |              |
|                {2} | 0 1 0 |                         {1} | 0 |              |
|                {2} | 0 0 1 |                         {1} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {3} | 1 |              |
|                                                                             |
|o14 : ComplexMap                                                             |
|-----------------------------------------------------------------------------|
|: 2                                                                          |
+-----------------------------------------------------------------------------+
|i15 : K == prune Kn                                                          |
|                                                                             |
|o15 = true                                                                   |
+-----------------------------------------------------------------------------+

As the above example shows, applying the normalization to the Dold-Kan image
recovers the original complex.

Canonical Extensions of Functors to the Category of Chain Complexes
===================================================================

One of the main utilities of the Dold-Kan correspondence from the perspective
of a commutative algebraist is as a method of canonically extending any
endofunctor of R-modules to an endofunctor of nonnegatively chain complexes.
This is because endofunctors of R-modules naturally extend to endofunctors of
simplicial R-modules by just applying the functor degree-wise. Since any
nonnegatively graded chain complex can be converted into a simplicial R-module
(and vice versa) this yields a canonical way to extend endofunctors of
R-modules to chain complexes in a way that preserves homotopy equivalence.

This extension is often called the Dold-Puppe extension, and was introduced in
pioneering work on the theory of derived nonlinear functors. In general, the
Dold-Puppe extension of a functor looks quite different from the classically
defined functor, and in most cases may not be quasi-isomorphic over arbitrary
base rings!

+-------------------------------------------+
|i16 : Q = ZZ/101[x_1,x_2];                 |
+-------------------------------------------+
|i17 : K1 = complex {matrix{{x_1}}};        |
+-------------------------------------------+
|i18 : K2 = complex {matrix{{x_2}}};        |
+-------------------------------------------+
|i19 : T1 = K1**K2                          |
|                                           |
|       1      2      1                     |
|o19 = Q  <-- Q  <-- Q                      |
|                                           |
|      0      1      2                      |
|                                           |
|o19 : Complex                              |
+-------------------------------------------+
|i20 : T1.dd                                |
|                                           |
|           1                   2           |
|o20 = 0 : Q  <--------------- Q  : 1       |
|                | x_2 x_1 |                |
|                                           |
|           2                    1          |
|      1 : Q  <---------------- Q  : 2      |
|                {1} | x_1  |               |
|                {1} | -x_2 |               |
|                                           |
|o20 : ComplexMap                           |
+-------------------------------------------+
|i21 : T2 = prune simplicialTensor({K1,K2}) |
|                                           |
|       1      3      2                     |
|o21 = Q  <-- Q  <-- Q                      |
|                                           |
|      0      1      2                      |
|                                           |
|o21 : Complex                              |
+-------------------------------------------+
|i22 : T2.dd                                |
|                                           |
|           1                          3    |
|o22 = 0 : Q  <---------------------- Q  : 1|
|                | x_2 x_1 x_1x_2 |         |
|                                           |
|           3                         2     |
|      1 : Q  <--------------------- Q  : 2 |
|                {1} | -x_1 -x_1 |          |
|                {1} | x_2  0    |          |
|                {2} | 0    1    |          |
|                                           |
|o22 : ComplexMap                           |
+-------------------------------------------+
|i23 : phi1 = extend(T1,T2,id_(T1_0))       |
|                                           |
|           1             1                 |
|o23 = 0 : Q  <--------- Q  : 0             |
|                | 1 |                      |
|                                           |
|           2                       3       |
|      1 : Q  <------------------- Q  : 1   |
|                {1} | 1 0 0   |            |
|                {1} | 0 1 x_2 |            |
|                                           |
|           1                     2         |
|      2 : Q  <----------------- Q  : 2     |
|                {2} | -1 -1 |              |
|                                           |
|o23 : ComplexMap                           |
+-------------------------------------------+
|i24 : phi2 = extend(T2,T1,id_(T1_0))       |
|                                           |
|           1             1                 |
|o24 = 0 : Q  <--------- Q  : 0             |
|                | 1 |                      |
|                                           |
|           3                   2           |
|      1 : Q  <--------------- Q  : 1       |
|                {1} | 1 0 |                |
|                {1} | 0 1 |                |
|                {2} | 0 0 |                |
|                                           |
|           2                  1            |
|      2 : Q  <-------------- Q  : 2        |
|                {2} | -1 |                 |
|                {2} | 0  |                 |
|                                           |
|o24 : ComplexMap                           |
+-------------------------------------------+
|i25 : phi1*phi2 == id_T1                   |
|                                           |
|o25 = true                                 |
+-------------------------------------------+
|i26 : isNullHomotopic(phi2*phi1 - id_T2)   |
|                                           |
|o26 = true                                 |
+-------------------------------------------+

The above example allows us to directly verify that the classically defined
tensor product is homotopy equivalent to the Dold-Kan version. This is true
over arbitrary base rings, but there are other functors such as the
symmetric/exterior power functors that have classical definitions for chain
complexes that are not necessarily homotopy equivalent to the Dold-Puppe
extensions:

+-----------------------------------------------------------------------------+
|i27 : needsPackage "ChainComplexOperations"                                  |
|                                                                             |
|o27 = ChainComplexOperations                                                 |
|                                                                             |
|o27 : Package                                                                |
+-----------------------------------------------------------------------------+
|i28 : Q = ZZ/2[x_1..x_3];                                                    |
+-----------------------------------------------------------------------------+
|i29 : K = koszulComplex vars Q;                                              |
+-----------------------------------------------------------------------------+
|i30 : W1 = wedge2 K --the classical "naively" defined exterior power functor |
|                                                                             |
|             3      6      10      6      3                                  |
|o30 = 0 <-- Q  <-- Q  <-- Q   <-- Q  <-- Q  <-- 0                            |
|                                                                             |
|      0     1      2      3       4      5      6                            |
|                                                                             |
|o30 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i31 : W2 = prune extPower(2,K) --the simplicial version of the exterior power|
|                                                                             |
|       6      33      73      81      45      10                             |
|o31 = Q  <-- Q   <-- Q   <-- Q   <-- Q   <-- Q                               |
|                                                                             |
|      1      2       3       4       5       6                               |
|                                                                             |
|o31 : Complex                                                                |
|-----------------------------------------------------------------------------|
|functor                                                                      |
+-----------------------------------------------------------------------------+
|i32 : prune HH W1                                                            |
|                                                                             |
|                                                                             |
|o32 = cokernel {1} | x_2 0   x_3 | <-- cokernel {2} | x_2 x_1 0   x_3 0   0  |
|               {1} | x_1 x_3 0   |              {2} | 0   0   x_3 x_2 x_1 0  |
|               {1} | 0   x_2 x_1 |              {2} | 0   0   0   0   0   x_3|
|                                                                             |
|      1                                2                                     |
|                                                                             |
|                                                                             |
|o32 : Complex                                                                |
|-----------------------------------------------------------------------------|
|                                                            1                |
|0   x_3 | <-- cokernel {3} | x_3 x_2 x_1 0     | <-- 0 <-- Q                 |
|0   0   |              {4} | 0   0   0   x_3^2 |                             |
|x_2 x_1 |              {4} | 0   0   0   x_2^2 |     4     5                 |
|                       {4} | 0   0   0   x_1^2 |                             |
|                                                                             |
|              3                                                              |
+-----------------------------------------------------------------------------+
|i33 : prune HH W2                                                            |
|                                                                             |
|o33 = cokernel {1} | x_1 0   x_2 0   0   x_3 | <-- cokernel {2} | x_2 x_1 0  |
|               {1} | 0   x_2 x_1 0   x_3 0   |              {2} | 0   0   x_3|
|               {1} | 0   0   0   x_3 x_2 x_1 |              {2} | 0   0   0  |
|                                                                             |
|      1                                            2                         |
|                                                                             |
|o33 : Complex                                                                |
|-----------------------------------------------------------------------------|
|x_3 0   0   0   x_3 | <-- cokernel {3} | x_3 x_2 x_1 |                       |
|x_2 x_1 0   0   0   |                                                        |
|0   0   x_3 x_2 x_1 |     3                                                  |
+-----------------------------------------------------------------------------+

Notice that the two definitions of exterior power yield complexes that are not
even quasi-isomorphic, and hence certainly not homotopy equivalent. Moreover,
the "naive" definition yields a complex that does not even have finite length
homology, while the Dold-Puppe extension applied to a complex with finite
length homology is guaranteed to have finite length homology. This preservation
of "good behavior" for Dold-Puppe extensions of functors was the main
motivation of the authors of CITE for using simplicial techniques to prove the
Total Rank Conjecture in characteristic 2. The extPower,
command along with Schur functors in general have also been implemented in a
functorial way and can take morphisms of chain complexes as inputs:

+-----------------------------------------------------------------------------+
|i34 : Q=ZZ/101[x_1,x_2];                                                     |
+-----------------------------------------------------------------------------+
|i35 : K = koszulComplex vars Q;                                              |
+-----------------------------------------------------------------------------+
|i36 : F = complex res ((ideal vars Q)^2);                                    |
+-----------------------------------------------------------------------------+
|i37 : phi = extend(K,F,id_(K_0))                                             |
|                                                                             |
|           1             1                                                   |
|o37 = 0 : Q  <--------- Q  : 0                                               |
|                | 1 |                                                        |
|                                                                             |
|           2                           3                                     |
|      1 : Q  <----------------------- Q  : 1                                 |
|                {1} | x_1 x_2 0   |                                          |
|                {1} | 0   0   x_2 |                                          |
|                                                                             |
|           1                     2                                           |
|      2 : Q  <----------------- Q  : 2                                       |
|                {2} | 0 x_2 |                                                |
|                                                                             |
|o37 : ComplexMap                                                             |
+-----------------------------------------------------------------------------+
|i38 : e2phi = prune extPower(2,phi) --induced map on the exterior powers of t|
|                                                                             |
|           3                                          6                      |
|o38 = 1 : Q  <-------------------------------------- Q  : 1                  |
|                {1} | x_1 x_2 0 0   0      0     |                           |
|                {1} | 0   0   0 x_2 0      0     |                           |
|                {2} | 0   0   0 0   x_1x_2 x_2^2 |                           |
|                                                                             |
|           9                                                                 |
|      2 : Q  <---------------------------------------------------------------|
|                {2} | 0 0   x_1^2 2x_1x_2 x_2^2 x_1x_2 0      0     0     0  |
|                {2} | 0 0   0     0       0     0      x_1x_2 x_2^2 0     0  |
|                {2} | 0 0   0     0       0     0      0      0     x_2^2 0  |
|                {2} | 0 0   0     0       0     0      0      0     0     x_1|
|                {2} | 0 x_2 0     0       0     0      0      0     0     0  |
|                {3} | 0 0   0     0       0     0      0      0     0     0  |
|                {3} | 0 0   0     0       0     0      0      0     0     0  |
|                {3} | 0 0   0     0       0     0      0      0     0     0  |
|                {3} | 0 0   0     0       0     0      0      0     0     0  |
|                                                                             |
|           9                                                                 |
|      3 : Q  <---------------------------------------------------------------|
|                {3} | 0 0 0 0 0 0 0 0 0 x_1x_2 x_2^2 0     0      0     0    |
|                {3} | 0 0 0 0 0 0 0 0 0 0      0     x_2^2 0      0     0    |
|                {3} | 0 0 0 0 0 0 0 0 0 0      0     0     x_1x_2 x_2^2 0    |
|                {3} | 0 0 0 0 0 0 0 0 0 0      0     0     0      0     x_2^2|
|                {3} | 0 0 0 0 0 0 0 0 0 0      0     0     0      0     0    |
|                {3} | 0 0 0 0 0 0 0 0 0 0      0     0     0      0     0    |
|                {4} | 0 0 0 0 0 0 0 0 0 0      0     0     0      0     0    |
|                {4} | 0 0 0 0 0 0 0 0 0 0      0     0     0      0     0    |
|                {4} | 0 0 0 0 0 0 0 0 0 0      0     0     0      0     0    |
|                                                                             |
|           3                                                   12            |
|      4 : Q  <----------------------------------------------- Q   : 4        |
|                {4} | 0 0 0 0 0 0 0 x_2^2 0 0     0 0     |                  |
|                {4} | 0 0 0 0 0 0 0 0     0 x_2^2 0 0     |                  |
|                {4} | 0 0 0 0 0 0 0 0     0 0     0 x_2^2 |                  |
|                                                                             |
|o38 : ComplexMap                                                             |
|-----------------------------------------------------------------------------|
|hese complexes                                                               |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                   24        |
|----------------------------------------------------------------- Q   : 2    |
|    0     0 0 0 0 0 0 0      0     0     0      0     0     0 |              |
|    0     0 0 0 0 0 0 0      0     0     0      0     0     0 |              |
|    0     0 0 0 0 0 0 0      0     0     0      0     0     0 |              |
|x_2 x_2^2 0 0 0 0 0 0 0      0     0     0      0     0     0 |              |
|    0     0 0 0 0 0 0 0      0     0     0      0     0     0 |              |
|    0     0 0 0 0 0 0 x_1x_2 x_2^2 0     0      0     0     0 |              |
|    0     0 0 0 0 0 0 0      0     x_2^2 0      0     0     0 |              |
|    0     0 0 0 0 0 0 0      0     0     x_1x_2 x_2^2 0     0 |              |
|    0     0 0 0 0 0 0 0      0     0     0      0     x_2^2 0 |              |
|                                                                             |
|                                                             30              |
|----------------------------------------------------------- Q   : 3          |
| 0      0     0     0 0 0     0 0 0 0 0 0     0 0     0 |                    |
| 0      0     0     0 0 0     0 0 0 0 0 0     0 0     0 |                    |
| 0      0     0     0 0 0     0 0 0 0 0 0     0 0     0 |                    |
| 0      0     0     0 0 0     0 0 0 0 0 0     0 0     0 |                    |
| x_1x_2 x_2^2 0     0 0 0     0 0 0 0 0 0     0 0     0 |                    |
| 0      0     x_2^2 0 0 0     0 0 0 0 0 0     0 0     0 |                    |
| 0      0     0     0 0 x_2^2 0 0 0 0 0 0     0 0     0 |                    |
| 0      0     0     0 0 0     0 0 0 0 0 x_2^2 0 0     0 |                    |
| 0      0     0     0 0 0     0 0 0 0 0 0     0 x_2^2 0 |                    |
+-----------------------------------------------------------------------------+
|i39 : assert isCommutative e2phi                                             |
+-----------------------------------------------------------------------------+
|i40 : ephi = prune extPower(3,phi,TopDegree => 4); --the TopDegree option    |
|-----------------------------------------------------------------------------|
|specifies how many homological degrees to compute                            |
+-----------------------------------------------------------------------------+
|i41 : assert isCommutative ephi;                                             |
+-----------------------------------------------------------------------------+
|i42 : ephi_1                                                                 |
|                                                                             |
|o42 = {2} | 0 x_1x_2 x_2^2 0 |                                               |
|                                                                             |
|              1      4                                                       |
|o42 : Matrix Q  <-- Q                                                        |
+-----------------------------------------------------------------------------+
|i43 : ephi_2;                                                                |
|                                                                             |
|              18      76                                                     |
|o43 : Matrix Q   <-- Q                                                       |
+-----------------------------------------------------------------------------+
|i44 : prune schurMap({2},phi) --the second symmetric power                   |
|                                                                             |
|           1             1                                                   |
|o44 = 0 : Q  <--------- Q  : 0                                               |
|                | 1 |                                                        |
|                                                                             |
|           5                                                                 |
|      1 : Q  <------------------------------------------------------------- Q|
|                {1} | x_1 x_2 0   0     0      0      0     0     0     |    |
|                {1} | 0   0   x_2 0     0      0      0     0     0     |    |
|                {2} | 0   0   0   x_1^2 x_1x_2 0      x_2^2 0     0     |    |
|                {2} | 0   0   0   0     0      x_1x_2 0     x_2^2 0     |    |
|                {2} | 0   0   0   0     0      0      0     0     x_2^2 |    |
|                                                                             |
|           10                                                                |
|      2 : Q   <--------------------------------------------------------------|
|                 {2} | 0 x_2 0 0      0     0     0      0      0     0     0|
|                 {2} | 0 0   0 x_1x_2 x_2^2 0     0      0      0     0     0|
|                 {2} | 0 0   0 0      0     x_1^2 x_1x_2 0      x_2^2 0     0|
|                 {2} | 0 0   0 0      0     0     0      x_1x_2 0     x_2^2 0|
|                 {2} | 0 0   0 0      0     0     0      0      0     0     x|
|                 {3} | 0 0   0 0      0     0     0      0      0     0     0|
|                 {3} | 0 0   0 0      0     0     0      0      0     0     0|
|                 {3} | 0 0   0 0      0     0     0      0      0     0     0|
|                 {3} | 0 0   0 0      0     0     0      0      0     0     0|
|                 {4} | 0 0   0 0      0     0     0      0      0     0     0|
|                                                                             |
|           9                                                                 |
|      3 : Q  <---------------------------------------------------------------|
|                {3} | 0 x_1x_2 0 x_2^2 0 0     0 0      0 0      0 0     0 0 |
|                {3} | 0 0      0 0     0 x_2^2 0 0      0 0      0 0     0 0 |
|                {3} | 0 0      0 0     0 0     0 x_1x_2 0 0      0 x_2^2 0 0 |
|                {3} | 0 0      0 0     0 0     0 0      0 x_1x_2 0 0     0 x_|
|                {3} | 0 0      0 0     0 0     0 0      0 0      0 0     0 0 |
|                {3} | 0 0      0 0     0 0     0 0      0 0      0 0     0 0 |
|                {4} | 0 0      0 0     0 0     0 0      0 0      0 0     0 0 |
|                {4} | 0 0      0 0     0 0     0 0      0 0      0 0     0 0 |
|                {4} | 0 0      0 0     0 0     0 0      0 0      0 0     0 0 |
|                                                                             |
|           3                                                   12            |
|      4 : Q  <----------------------------------------------- Q   : 4        |
|                {4} | 0 0 x_2^2 0 0 0 0 0     0 0 0 0     |                  |
|                {4} | 0 0 0     0 0 0 0 x_2^2 0 0 0 0     |                  |
|                {4} | 0 0 0     0 0 0 0 0     0 0 0 x_2^2 |                  |
|                                                                             |
|o44 : ComplexMap                                                             |
|-----------------------------------------------------------------------------|
|9                                                                            |
|  : 1                                                                        |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                      26     |
|-------------------------------------------------------------------- Q   : 2 |
|     0 0      0 0     0 0     0 0      0 0     0 0     0 0 0     |           |
|     0 0      0 0     0 0     0 0      0 0     0 0     0 0 0     |           |
|     0 0      0 0     0 0     0 0      0 0     0 0     0 0 0     |           |
|     0 0      0 0     0 0     0 0      0 0     0 0     0 0 0     |           |
|_2^2 0 0      0 0     0 0     0 0      0 0     0 0     0 0 0     |           |
|     0 x_1x_2 0 x_2^2 0 0     0 0      0 0     0 0     0 0 0     |           |
|     0 0      0 0     0 x_2^2 0 0      0 0     0 0     0 0 0     |           |
|     0 0      0 0     0 0     0 x_1x_2 0 x_2^2 0 0     0 0 0     |           |
|     0 0      0 0     0 0     0 0      0 0     0 x_2^2 0 0 0     |           |
|     0 0      0 0     0 0     0 0      0 0     0 0     0 0 x_2^2 |           |
|                                                                             |
|                                                             30              |
|----------------------------------------------------------- Q   : 3          |
|    0 0     0 0     0 0 0 0 0     0 0 0 0     0 0 0     |                    |
|    0 0     0 0     0 0 0 0 0     0 0 0 0     0 0 0     |                    |
|    0 0     0 0     0 0 0 0 0     0 0 0 0     0 0 0     |                    |
|2^2 0 0     0 0     0 0 0 0 0     0 0 0 0     0 0 0     |                    |
|    0 x_2^2 0 0     0 0 0 0 0     0 0 0 0     0 0 0     |                    |
|    0 0     0 x_2^2 0 0 0 0 0     0 0 0 0     0 0 0     |                    |
|    0 0     0 0     0 0 0 0 x_2^2 0 0 0 0     0 0 0     |                    |
|    0 0     0 0     0 0 0 0 0     0 0 0 x_2^2 0 0 0     |                    |
|    0 0     0 0     0 0 0 0 0     0 0 0 0     0 0 x_2^2 |                    |
+-----------------------------------------------------------------------------+
|i45 : sphi = schurMap({2,1},phi,TopDegree => 3);  --schur functor            |
|-----------------------------------------------------------------------------|
|corresponding to partition {2,1} applied to phi                              |
+-----------------------------------------------------------------------------+
|i46 : sphi_1                                                                 |
|                                                                             |
|o46 = {1} | x_1 0     0      0      x_2 0      0     0     0 0 0 0   0       |
|      {2} | 0   x_1^2 x_1x_2 0      0   x_1x_2 x_2^2 0     0 0 0 0   0       |
|      {2} | 0   0     0      x_1x_2 0   0      0     x_2^2 0 0 0 0   0       |
|      {1} | 0   0     0      0      0   0      0     0     0 0 0 x_2 0       |
|      {2} | 0   0     0      0      0   0      0     0     0 0 0 0   x_1x_2  |
|      {2} | 0   0     0      0      0   0      0     0     0 0 0 0   0       |
|      {3} | 0   0     0      0      0   0      0     0     0 0 0 0   0       |
|      {3} | 0   0     0      0      0   0      0     0     0 0 0 0   0       |
|      -----------------------------------------------------------------------|
|      0     0     0        0        0        0     0     |                   |
|      0     0     0        0        0        0     0     |                   |
|      0     0     0        0        0        0     0     |                   |
|      0     0     0        0        0        0     0     |                   |
|      x_2^2 0     0        0        0        0     0     |                   |
|      0     x_2^2 0        0        0        0     0     |                   |
|      0     0     x_1^2x_2 x_1x_2^2 0        x_2^3 0     |                   |
|      0     0     0        0        x_1x_2^2 0     x_2^3 |                   |
|                                                                             |
|              8      20                                                      |
|o46 : Matrix Q  <-- Q                                                        |
+-----------------------------------------------------------------------------+

This package thus lays the groundwork for computing with Dold-Puppe extensions
of nonlinear functors, deriving nonlinear functors, and also allows any
endofunctor of R-modules implemented in Macaulay2 to immediately be canonically
extended to an endofunctor at the level of chain complexes. Moreover, almost
all methods are implemented with an eye toward user accessibility and
comprehension of the outputs. This is illustrated in many of the documentation
nodes below.

Getting started:
================

  * The Dold-Kan Correspondence in Macaulay2
    -- compute the image of a non-negatively graded complex under
    the Dold-Kan correspondence
  * Making simplicial modules -- information
    about the basic constructors
  * Making maps between simplicial modules
    -- information about the basic constructors
  * Basic invariants and properties, --
    information about accessing basic features
  * face/degeneracy maps of a simplicial module
    -- access the face and degeneracy maps of a simplicial
    module

Contributors
============

The following people have generously contributed code or improved existing
code:

Acknowledgments
===============

This package began its life during the 2023 Macaulay2 workshop in Minneapolis.
Many documentation nodes for the basic constructors not specific to the
SimplicialModule type have been repurposed from existing documentation coming
from the Complexes, package.

Authors
=======

  * Keller VandeBogert (see https://sites.google.com/view/kellervandebogert/
    home ) <kvandebo@nd.edu>
  * Michael DeBellevue <michael.debellevue@gmail.com>

Version
=======

This documentation describes version 0.1 of SimplicialModules, released April
27, 2026.

Citation
========

If you have used this package in your research, please cite it as follows:

+-----------------------------------------------------------------------------+
|@misc{SimplicialModulesSource,                                               |
|  title = {{SimplicialModules: methods for working in the category           |
|  author = {Keller VandeBogert and Michael DeBellevue},                      |
|  howpublished = {A \emph{Macaulay2} package available at                    |
|    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/           |
|}                                                                            |
|-----------------------------------------------------------------------------|
| of simplicial modules. Version~0.1}},                                       |
|                                                                             |
|                                                                             |
|packages}}                                                                   |
+-----------------------------------------------------------------------------+

Exports
=======

  * Types
      * SimplicialModule -- the class of all
        simplicial modules
      * SimplicialModuleMap -- the class of all
        maps between simplicial modules
  * Functions and commands
      * exteriorInclusion -- computes the image of
        the 2nd exterior power into the tensor product
      * extPower -- compute the Dold-Puppe extension of the
        exterior power functor to simplicial modules
      * "forgetComplex" -- see forgetComplex(SimplicialModule),
        -- forget the underlying complex
        data of a simplicial module obtained as a Dold-Kan image
      * "forgetDegeneracy" -- see forgetDegeneracy(SimplicialModule),
        -- forget the data of
        degeneracy maps of a simplicial object
      * "isSimplicialModule" -- see isSimplicialModule(SimplicialModule),
        -- check if the simplicial
        identities hold for a simplicial object with degeneracy map keys
      * "isSimplicialMorphism" -- see
        isSimplicialMorphism(SimplicialModuleMap),
        -- whether a simplicial
        module map is a morphism of simplicial modules
      * "naiveNorm" -- see naiveNorm(SimplicialModule,ZZ),
        -- compute the naive
        normalization of a simplicial object
      * "randomSimplicialMap" -- see
        randomSimplicialMap(SimplicialModule,SimplicialModule),
        -- a
        random map of simplicial modules
      * schurMap -- construct the Dold-Puppe extension of the
        Schur functor
      * simplicialModule -- make a simplicial module
      * simplicialTensor -- compute the simplicial
        tensor product and cache direct sum indices for easy access
      * symmetricQuotient -- computes the image of
        the surjection from the simplicial tensor product onto the second
        symmetric power of a simplicial module
      * tensorLES -- computes the long exact sequence of
        homology induced by the canonical short exact sequence of complexes
      * tensorwithComponents -- compute the tensor
        product of direct summands, caching the components for easy access
      * "summandSurjection" -- see The Dold-Kan Correspondence in Macaulay2,
        -- compute the
        image of a non-negatively graded complex under the Dold-Kan
        correspondence
      * topDegree -- top degree of a simplicial module
  * Methods
      * "- SimplicialModuleMap" -- see arithmetic with simplicial module maps,
        -- perform arithmetic
        operations on simplicial module maps
      * "Number * SimplicialModuleMap" -- see arithmetic with simplicial module maps,
        -- perform
        arithmetic operations on simplicial module maps
      * "Number + SimplicialModuleMap" -- see arithmetic with simplicial module maps,
        -- perform
        arithmetic operations on simplicial module maps
      * "Number - SimplicialModuleMap" -- see arithmetic with simplicial module maps,
        -- perform
        arithmetic operations on simplicial module maps
      * "RingElement * SimplicialModuleMap" -- see arithmetic with simplicial module maps,
        --
        perform arithmetic operations on simplicial module maps
      * "RingElement + SimplicialModuleMap" -- see arithmetic with simplicial module maps,
        --
        perform arithmetic operations on simplicial module maps
      * "RingElement - SimplicialModuleMap" -- see arithmetic with simplicial module maps,
        --
        perform arithmetic operations on simplicial module maps
      * "SimplicialModuleMap * Number" -- see arithmetic with simplicial module maps,
        -- perform
        arithmetic operations on simplicial module maps
      * "SimplicialModuleMap * RingElement" -- see arithmetic with simplicial module maps,
        --
        perform arithmetic operations on simplicial module maps
      * "SimplicialModuleMap + Number" -- see arithmetic with simplicial module maps,
        -- perform
        arithmetic operations on simplicial module maps
      * "SimplicialModuleMap + RingElement" -- see arithmetic with simplicial module maps,
        --
        perform arithmetic operations on simplicial module maps
      * "SimplicialModuleMap + SimplicialModuleMap" -- see arithmetic with simplicial module maps,
        --
        perform arithmetic operations on simplicial module maps
      * "SimplicialModuleMap - Number" -- see arithmetic with simplicial module maps,
        -- perform
        arithmetic operations on simplicial module maps
      * "SimplicialModuleMap - RingElement" -- see arithmetic with simplicial module maps,
        --
        perform arithmetic operations on simplicial module maps
      * "SimplicialModuleMap - SimplicialModuleMap" -- see arithmetic with simplicial module maps,
        --
        perform arithmetic operations on simplicial module maps
      * basis(List,SimplicialModule) (missing documentation)
      * basis(List,SimplicialModuleMap) (missing documentation)
      * basis(ZZ,SimplicialModule) (missing documentation)
      * basis(ZZ,SimplicialModuleMap) (missing documentation)
      * coimage(SimplicialModuleMap),
        -- make the coimage of a map of simplicial modules
      * cokernel(SimplicialModuleMap),
        -- make the cokernel of a map of simplicial modules
      * components(SimplicialModule),
        -- list the components of a direct sum
      * components(SimplicialModuleMap)
        -- list the components of a direct
        sum
      * degree(SimplicialModuleMap), --
        get the degree of a map of simplicial modules
      * directSum(SimplicialModule), --
        direct sum of simplicial modules
      * "SimplicialModule ++ SimplicialModule" -- see
        directSum(SimplicialModule), -- direct
        sum of simplicial modules
      * expression(SimplicialModuleMap) (missing documentation)
      * "exteriorInclusion(Complex)" -- see exteriorInclusion,
        -- computes the image of the 2nd exterior power into
        the tensor product
      * "exteriorInclusion(Complex,ZZ)" -- see exteriorInclusion,
        -- computes the image of the 2nd exterior power into
        the tensor product
      * "exteriorInclusion(Module)" -- see exteriorInclusion,
        -- computes the image of the 2nd exterior power into
        the tensor product
      * "exteriorInclusion(SimplicialModule)" -- see exteriorInclusion,
        -- computes the image of the 2nd exterior power into
        the tensor product
      * "extPower(ZZ,Complex)" -- see extPower, -- compute the
        Dold-Puppe extension of the exterior power functor to simplicial
        modules
      * "extPower(ZZ,ComplexMap)" -- see extPower, -- compute
        the Dold-Puppe extension of the exterior power functor to simplicial
        modules
      * "extPower(ZZ,SimplicialModule)" -- see extPower, --
        compute the Dold-Puppe extension of the exterior power functor to
        simplicial modules
      * "extPower(ZZ,SimplicialModuleMap)" -- see extPower, --
        compute the Dold-Puppe extension of the exterior power functor to
        simplicial modules
      * "Symbol ^ SimplicialModule" -- see face/degeneracy maps of a simplicial module,
        --
        access the face and degeneracy maps of a simplicial module
      * forgetComplex(SimplicialModule)
        -- forget the underlying complex
        data of a simplicial module obtained as a Dold-Kan image
      * forgetDegeneracy(SimplicialModule)
        -- forget the data of
        degeneracy maps of a simplicial object
      * HH SimplicialModule (missing documentation)
      * HH SimplicialModuleMap (missing documentation)
      * image(SimplicialModuleMap), --
        make the image of a map of simplicial modules
      * inducedMap(SimplicialModule,SimplicialModule)
        -- make the map of
        simplicial modules induced at each term by the identity map
      * isCommutative(SimplicialModuleMap)
        -- whether a simplicial module
        map commutes with the face/degeneracy maps
      * isHomogeneous(SimplicialModuleMap)
        -- whether a map of simplicial
        modules is homogeneous
      * isShortExactSequence(SimplicialModuleMap,SimplicialModuleMap),
        --
        whether a pair of simplicial module maps forms a short exact sequence
      * isSimplicialModule(SimplicialModule)
        -- check if the simplicial
        identities hold for a simplicial object with degeneracy map keys
      * isSimplicialMorphism(SimplicialModuleMap)
        -- whether a simplicial
        module map is a morphism of simplicial modules
      * isWellDefined(SimplicialModule)
        -- whether a simplicial module is
        well-defined
      * isWellDefined(SimplicialModuleMap)
        -- whether a map of simplicial
        modules is well-defined
      * kernel(SimplicialModuleMap), --
        make the kernel of a map of simplicial modules
      * map(SimplicialModule,SimplicialModule,HashTable)
        -- make a map
        of simplicial modules
      * map(SimplicialModule,SimplicialModule,SimplicialModuleMap),
        --
        make a new map of simplicial modules, induced from an existing one
      * map(SimplicialModule,SimplicialModule,ZZ)
        -- make the zero map
        or identity between simplicial modules
      * minimalPresentation(SimplicialModule)
        -- minimal presentation of
        all terms in a simplicial module
      * "minimalPresentation(SimplicialModuleMap)" -- see
        minimalPresentation(SimplicialModule),
        -- minimal presentation of
        all terms in a simplicial module
      * "prune(SimplicialModule)" -- see
        minimalPresentation(SimplicialModule),
        -- minimal presentation of
        all terms in a simplicial module
      * "prune(SimplicialModuleMap)" -- see
        minimalPresentation(SimplicialModule),
        -- minimal presentation of
        all terms in a simplicial module
      * "naiveNorm(Complex,ZZ)" -- see naiveNorm(SimplicialModule,ZZ),
        -- compute the naive
        normalization of a simplicial object
      * "naiveNorm(SimplicialModule)" -- see
        naiveNorm(SimplicialModule,ZZ),
        -- compute the naive normalization of a simplicial object
      * naiveNorm(SimplicialModule,ZZ)
        -- compute the naive
        normalization of a simplicial object
      * net(SimplicialModule) (missing documentation)
      * net(SimplicialModuleMap) (missing documentation)
      * "normalize(SimplicialModule)" -- see
        normalize(SimplicialModule,ZZ),
        -- normalization functor from simplicial modules to
        nonnegatively-graded chain complexes
      * normalize(SimplicialModule,ZZ)
        -- normalization functor from
        simplicial modules to nonnegatively-graded chain complexes
      * "normalize(SimplicialModuleMap)" -- see
        normalize(SimplicialModule,ZZ),
        -- normalization functor from simplicial modules to
        nonnegatively-graded chain complexes
      * "normalize(SimplicialModuleMap,ZZ)" -- see
        normalize(SimplicialModule,ZZ),
        -- normalization functor from simplicial modules to
        nonnegatively-graded chain complexes
      * randomSimplicialMap(SimplicialModule,SimplicialModule)
        -- a
        random map of simplicial modules
      * ring(SimplicialModule) -- access the
        ring of a simplicial module or a simplicial module map
      * "ring(SimplicialModuleMap)" -- see ring(SimplicialModule),
        -- access the ring of a simplicial module
        or a simplicial module map
      * "Ring ** SimplicialModule" -- see RingMap ** SimplicialModule,
        -- tensor a simplicial module along a
        ring map
      * RingMap ** SimplicialModule, --
        tensor a simplicial module along a ring map
      * "SimplicialModule ** Ring" -- see RingMap ** SimplicialModule,
        -- tensor a simplicial module along a
        ring map
      * "SimplicialModule ** RingMap" -- see RingMap ** SimplicialModule,
        -- tensor a simplicial module along a
        ring map
      * "tensor(RingMap,SimplicialModule)" -- see RingMap ** SimplicialModule,
        -- tensor a
        simplicial module along a ring map
      * "tensor(SimplicialModule,RingMap)" -- see RingMap ** SimplicialModule,
        -- tensor a
        simplicial module along a ring map
      * "Ring ** SimplicialModuleMap" -- see RingMap ** SimplicialModuleMap,
        -- tensor a
        map of simplicial modules along a ring map
      * RingMap ** SimplicialModuleMap
        -- tensor a map of simplicial modules along a ring
        map
      * "SimplicialModuleMap ** Ring" -- see RingMap ** SimplicialModuleMap,
        -- tensor a
        map of simplicial modules along a ring map
      * "SimplicialModuleMap ** RingMap" -- see RingMap ** SimplicialModuleMap,
        -- tensor a
        map of simplicial modules along a ring map
      * "tensor(RingMap,SimplicialModuleMap)" -- see RingMap ** SimplicialModuleMap,
        -- tensor a
        map of simplicial modules along a ring map
      * "tensor(SimplicialModuleMap,RingMap)" -- see RingMap ** SimplicialModuleMap,
        -- tensor a
        map of simplicial modules along a ring map
      * RingMap SimplicialModule -- apply a
        ring map to a simplicial module
      * RingMap SimplicialModuleMap, --
        apply a ring map to a map of simplicial modules
      * "schurMap(List,Complex)" -- see schurMap, -- construct
        the Dold-Puppe extension of the Schur functor
      * "schurMap(List,ComplexMap)" -- see schurMap, --
        construct the Dold-Puppe extension of the Schur functor
      * "schurMap(List,SimplicialModule)" -- see schurMap, --
        construct the Dold-Puppe extension of the Schur functor
      * "schurMap(List,SimplicialModuleMap)" -- see schurMap,
        -- construct the Dold-Puppe extension of the Schur functor
      * "simplicialModule(Complex,HashTable,HashTable,ZZ)" -- see
        simplicialModule, -- make a simplicial module
      * "simplicialModule(Complex,HashTable,ZZ)" -- see simplicialModule,
        -- make a simplicial module
      * "simplicialModule(HashTable,HashTable,HashTable,ZZ)" -- see
        simplicialModule, -- make a simplicial module
      * "simplicialModule(HashTable,HashTable,ZZ)" -- see
        simplicialModule, -- make a simplicial module
      * "Matrix ** SimplicialModule" -- see SimplicialModule ** Matrix,
        -- create the tensor product of a
        simplicial module and a map of modules
      * SimplicialModule ** Matrix, --
        create the tensor product of a simplicial module and a map of modules
      * "Complex ** SimplicialModule" -- see SimplicialModule ** SimplicialModule,
        -- tensor
        product of simplicial modules
      * "Module ** SimplicialModule" -- see SimplicialModule ** SimplicialModule,
        -- tensor
        product of simplicial modules
      * "SimplicialModule ** Complex" -- see SimplicialModule ** SimplicialModule,
        -- tensor
        product of simplicial modules
      * "SimplicialModule ** Module" -- see SimplicialModule ** SimplicialModule,
        -- tensor
        product of simplicial modules
      * SimplicialModule ** SimplicialModule
        -- tensor product of simplicial modules
      * SimplicialModule == SimplicialModule
        -- whether two simplicial modules are equal
      * "SimplicialModule == ZZ" -- see SimplicialModule == SimplicialModule,
        -- whether two
        simplicial modules are equal
      * "ZZ == SimplicialModule" -- see SimplicialModule == SimplicialModule,
        -- whether two
        simplicial modules are equal
      * "SimplicialModule ^ Array" -- see SimplicialModule _ Array,
        -- the canonical inclusion or projection
        map of a direct sum
      * SimplicialModule _ Array -- the
        canonical inclusion or projection map of a direct sum
      * "SimplicialModule _ Sequence" -- see SimplicialModule _ ZZ,
        -- access individual objects of a simplicial
        module
      * SimplicialModule _ ZZ -- access
        individual objects of a simplicial module
      * SimplicialModule Array -- shift a
        simplicial module or simplicial module map
      * "SimplicialModuleMap Array" -- see SimplicialModule Array,
        -- shift a simplicial module or simplicial
        module map
      * simplicialModule(ComplexMap) (missing documentation)
      * simplicialModule(ComplexMap,ZZ) (missing documentation)
      * "simplicialModule(Ideal,ZZ)" -- see simplicialModule(Module,ZZ),
        -- make a simplicial module
        associated to a complex concentrated in degree 0
      * simplicialModule(Module,ZZ),
        -- make a simplicial module associated to a complex concentrated in
        degree 0
      * "simplicialModule(Ring,ZZ)" -- see simplicialModule(Module,ZZ),
        -- make a simplicial module
        associated to a complex concentrated in degree 0
      * SimplicialModuleMap * SimplicialModuleMap
        -- composition of homomorphisms of simplicial
        modules
      * "Complex ** SimplicialModuleMap" -- see SimplicialModuleMap ** SimplicialModuleMap,
        --
        the map of simplicial modules between tensor simplicial modules
      * "ComplexMap ** SimplicialModuleMap" -- see SimplicialModuleMap ** SimplicialModuleMap,
        --
        the map of simplicial modules between tensor simplicial modules
      * "Module ** SimplicialModuleMap" -- see SimplicialModuleMap ** SimplicialModuleMap,
        --
        the map of simplicial modules between tensor simplicial modules
      * "SimplicialModule ** SimplicialModuleMap" -- see
        SimplicialModuleMap ** SimplicialModuleMap,
        -- the map of simplicial modules between tensor
        simplicial modules
      * "SimplicialModuleMap ** Complex" -- see SimplicialModuleMap ** SimplicialModuleMap,
        --
        the map of simplicial modules between tensor simplicial modules
      * "SimplicialModuleMap ** ComplexMap" -- see SimplicialModuleMap ** SimplicialModuleMap,
        --
        the map of simplicial modules between tensor simplicial modules
      * "SimplicialModuleMap ** Module" -- see SimplicialModuleMap ** SimplicialModuleMap,
        --
        the map of simplicial modules between tensor simplicial modules
      * "SimplicialModuleMap ** SimplicialModule" -- see
        SimplicialModuleMap ** SimplicialModuleMap,
        -- the map of simplicial modules between tensor
        simplicial modules
      * SimplicialModuleMap ** SimplicialModuleMap
        -- the map of simplicial modules between
        tensor simplicial modules
      * "tensor(SimplicialModuleMap,SimplicialModuleMap)" -- see
        SimplicialModuleMap ** SimplicialModuleMap,
        -- the map of simplicial modules between tensor
        simplicial modules
      * "directSum(SimplicialModuleMap)" -- see SimplicialModuleMap ++ SimplicialModuleMap,
        --
        direct sum of simplicial module maps
      * SimplicialModuleMap ++ SimplicialModuleMap
        -- direct sum of simplicial module maps
      * SimplicialModuleMap == SimplicialModuleMap
        -- whether two simplicial module maps are equal
      * "SimplicialModuleMap == ZZ" -- see SimplicialModuleMap == SimplicialModuleMap,
        --
        whether two simplicial module maps are equal
      * "ZZ == SimplicialModuleMap" -- see SimplicialModuleMap == SimplicialModuleMap,
        --
        whether two simplicial module maps are equal
      * SimplicialModuleMap ^ Array -- the
        composition with the canonical inclusion or projection map
      * "SimplicialModuleMap _ Array" -- see SimplicialModuleMap ^ Array,
        -- the composition with the canonical
        inclusion or projection map
      * SimplicialModuleMap ^ ZZ -- the n-fold
        composition
      * "SimplicialModuleMap _ Sequence" -- see SimplicialModuleMap _ ZZ,
        -- access individual matrices in a
        simplicial module map
      * SimplicialModuleMap _ ZZ -- access
        individual matrices in a simplicial module map
      * SimplicialModuleMap | SimplicialModuleMap
        -- join or concatenate maps horizontally
      * SimplicialModuleMap || SimplicialModuleMap
        -- join or concatenate maps vertically
      * "simplicialTensor(Complex,Complex)" -- see simplicialTensor,
        -- compute the simplicial tensor product and cache
        direct sum indices for easy access
      * "simplicialTensor(ComplexMap,ComplexMap)" -- see
        simplicialTensor, -- compute the simplicial tensor
        product and cache direct sum indices for easy access
      * "simplicialTensor(List)" -- see simplicialTensor,
        -- compute the simplicial tensor product and cache
        direct sum indices for easy access
      * "simplicialTensor(ZZ,Complex)" -- see simplicialTensor,
        -- compute the simplicial tensor product and cache
        direct sum indices for easy access
      * "simplicialTensor(ZZ,SimplicialModule)" -- see simplicialTensor,
        -- compute the simplicial tensor product and cache
        direct sum indices for easy access
      * simplicialTensor(SimplicialModule,SimplicialModule) (missing
        documentation)
      * source(SimplicialModuleMap), --
        get the source of a map of simplicial modules
      * summandSurjection(ZZ,ZZ) (missing documentation)
      * "symmetricQuotient(Complex)" -- see symmetricQuotient,
        -- computes the image of the surjection from the
        simplicial tensor product onto the second symmetric power of a
        simplicial module
      * "symmetricQuotient(Complex,ZZ)" -- see symmetricQuotient,
        -- computes the image of the surjection from the
        simplicial tensor product onto the second symmetric power of a
        simplicial module
      * "symmetricQuotient(Module)" -- see symmetricQuotient,
        -- computes the image of the surjection from the
        simplicial tensor product onto the second symmetric power of a
        simplicial module
      * "symmetricQuotient(SimplicialModule)" -- see symmetricQuotient,
        -- computes the image of the surjection from the
        simplicial tensor product onto the second symmetric power of a
        simplicial module
      * target(SimplicialModuleMap), --
        get the target of a map of simplicial modules
      * "tensorLES(Complex,ZZ)" -- see tensorLES, -- computes
        the long exact sequence of homology induced by the canonical short
        exact sequence of complexes
      * "tensorwithComponents(List)" -- see tensorwithComponents,
        -- compute the tensor product of direct summands,
        caching the components for easy access
      * "tensorwithComponents(Matrix,Matrix)" -- see
        tensorwithComponents, -- compute the tensor
        product of direct summands, caching the components for easy access
      * "tensorwithComponents(Module,Module)" -- see
        tensorwithComponents, -- compute the tensor
        product of direct summands, caching the components for easy access
      * "simplicialModule(Complex)" -- see The Dold-Kan Correspondence in Macaulay2,
        -- compute the
        image of a non-negatively graded complex under the Dold-Kan
        correspondence
      * "simplicialModule(Complex,ZZ)" -- see The Dold-Kan Correspondence in Macaulay2,
        -- compute the
        image of a non-negatively graded complex under the Dold-Kan
        correspondence
      * "topDegree(SimplicialModule)" -- see topDegree, -- top
        degree of a simplicial module
      * "topDegree(SimplicialModuleMap)" -- see topDegree, --
        top degree of a simplicial module
      * truncate(List,SimplicialModule)
        -- truncation of a simplicial
        module at a specified degree or set of degrees
      * "truncate(ZZ,SimplicialModule)" -- see
        truncate(List,SimplicialModule),
        -- truncation of a simplicial module at a specified degree or set of
        degrees
      * truncate(List,SimplicialModuleMap)
        -- truncation of a simplicial
        module map at a specified degree or set of degrees
      * "truncate(ZZ,SimplicialModuleMap)" -- see
        truncate(List,SimplicialModuleMap),
        -- truncation of a simplicial
        module map at a specified degree or set of degrees
  * Symbols
      * "ss" -- see face/degeneracy maps of a simplicial module,
        -- access the face and
        degeneracy maps of a simplicial module
      * "RememberSummands" -- see forgetComplex(SimplicialModule),
        -- forget the underlying complex
        data of a simplicial module obtained as a Dold-Kan image
      * "CheckComplex" -- see normalize(SimplicialModule,ZZ),
        -- normalization functor from
        simplicial modules to nonnegatively-graded chain complexes
      * "CheckSum" -- see normalize(SimplicialModule,ZZ),
        -- normalization functor from
        simplicial modules to nonnegatively-graded chain complexes
      * "TopDegree" -- see schurMap, -- construct the
        Dold-Puppe extension of the Schur functor
      * "complexLength" -- see SimplicialModule, -- the
        class of all simplicial modules
      * "Degeneracy" -- see SimplicialModule, -- the
        class of all simplicial modules
      * "complexMap" -- see SimplicialModuleMap, --
        the class of all maps between simplicial modules

For the programmer
==================

The object SimplicialModules, is a package (Macaulay2Doc),
defined in SimplicialModules.m2, with auxiliary files
in SimplicialModules/.

-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:134:0.

arithmetic with simplicial module maps -- perform arithmetic operations on simplicial module maps
*************************************************************************************************

  * Usage:
        f + g
        a + f
        f + a
        -f
        f - g
        a - f
        f - a
        a * f
  * Inputs:
      * f, a Map of Simplicial Modules,
      * g, a Map of Simplicial Modules,
      * a, a ring element (Macaulay2Doc), that is, an
        element in the underlying ring or a number
  * Outputs:
      * a Map of Simplicial Modules,

The set of simplicial module maps forms a module over the underlying
ring. These methods implement the basic operations
of addition, subtraction, and scalar multiplication.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[a..d];                                                       |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution coker matrix{{a*b, a*c^2, b*c*d^3,  |
|                                                                             |
|      1      5      13      26                                               |
|o2 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|a^3}}, Degeneracy => true)                                                   |
+-----------------------------------------------------------------------------+
|i3 : D = simplicialModule(freeResolution coker matrix{{a*b, a*c^2, b*c*d^3,  |
|                                                                             |
|      1      6      18      41                                               |
|o3 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o3 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|a^3, a*c*d}}, 3, Degeneracy => true)                                         |
+-----------------------------------------------------------------------------+
|i4 : f = randomSimplicialMap(D, C, Cycle => true);                           |
+-----------------------------------------------------------------------------+
|i5 : prune normalize f                                                       |
|                                                                             |
|          1               1                                                  |
|o5 = 0 : R  <----------- R  : 0                                              |
|               | -47 |                                                       |
|                                                                             |
|          5                                                                  |
|     1 : R  <----------------------------------------------------------------|
|               {2} | -47 0   0   24a3-36a2b-30a2c+19ac2+19bc2-10c3-29a2d-8acd|
|               {3} | 0   -47 0   -24ab+36b2+30bc+18c2+29bd+15cd              |
|               {3} | 0   0   -47 -18a2-19ab-19b2+10bc-8ad-5bd-19cd+47d2      |
|               {3} | 0   0   0   -15a2+8ab+22b2+8ac-38bc+19c2+24bd-47cd      |
|               {5} | 0   0   0   -47                                         |
|                                                                             |
|          7                                                                  |
|     2 : R  <----------------------------------------------------------------|
|               {4} | -47 0   0    -24a2+36ab+30ac+39c2+29ad+43cd           | |
|               {4} | 0   -47 -28d 43a2-19ab+10ac+29ad-38bd-16cd+39d2       | |
|               {4} | 0   0   28c  -35a2+22ab+29ac+38bc+16c2+24ad-39cd-47d2 | |
|               {4} | 0   0   -28b 21a2+34ab-38b2+19ac-16bc-47ad+39bd       | |
|               {5} | 0   0   -47  -18a+39b+13d                             | |
|               {5} | 0   0   0    -15a+43b-13c                             | |
|               {6} | 0   0   0    -47                                      | |
|                                                                             |
|          4                    1                                             |
|     3 : R  <---------------- R  : 3                                         |
|               {5} | -28b |                                                  |
|               {6} | -47  |                                                  |
|               {6} | 0    |                                                  |
|               {6} | 0    |                                                  |
|                                                                             |
|o5 : ComplexMap                                                              |
|-----------------------------------------------------------------------------|
|                        4                                                    |
|---------------------- R  : 1                                                |
|-22bcd+43c2d-24cd2 |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                                                                             |
|   4                                                                         |
|- R  : 2                                                                     |
+-----------------------------------------------------------------------------+
|i6 : g = randomSimplicialMap(D, C, Boundary => true);                        |
+-----------------------------------------------------------------------------+
|i7 : prune normalize g                                                       |
|                                                                             |
|          5                                                                  |
|o7 = 1 : R  <----------------------------------------------------------------|
|               {2} | 0 0 0 -38a3-2a2b-16a2c-45ac2+34bc2+48c3-22a2d-47acd-19bc|
|               {3} | 0 0 0 38ab+2b2+16bc+17c2+22bd+11cd                      |
|               {3} | 0 0 0 -17a2+45ab-34b2-48bc-15ad-24bd-39cd-43d2          |
|               {3} | 0 0 0 -11a2+47ab+19b2+15ac-39bc+39c2+7bd+43cd           |
|               {5} | 0 0 0 0                                                 |
|                                                                             |
|          7                                                            4     |
|     2 : R  <-------------------------------------------------------- R  : 2 |
|               {4} | 0 0 0    38a2+2ab+16ac-38c2+22ad+33cd        |          |
|               {4} | 0 0 48d  -18a2-34ab-48ac-47ad+36bd+35cd+11d2 |          |
|               {4} | 0 0 -48c 14a2+19ab-16ac-36bc-35c2+7ad-11cd   |          |
|               {4} | 0 0 48b  -25a2-23ab+36b2+39ac+35bc+43ad+11bd |          |
|               {5} | 0 0 0    -17a-38b-40d                        |          |
|               {5} | 0 0 0    -11a+33b+40c                        |          |
|               {6} | 0 0 0    0                                   |          |
|                                                                             |
|          4                   1                                              |
|     3 : R  <--------------- R  : 3                                          |
|               {5} | 48b |                                                   |
|               {6} | 0   |                                                   |
|               {6} | 0   |                                                   |
|               {6} | 0   |                                                   |
|                                                                             |
|o7 : ComplexMap                                                              |
|-----------------------------------------------------------------------------|
|                  4                                                          |
|---------------- R  : 1                                                      |
|d-38c2d-7cd2 |                                                               |
|             |                                                               |
|             |                                                               |
|             |                                                               |
|             |                                                               |
+-----------------------------------------------------------------------------+
+---------------------------------------+
|i8 : f+g;                              |
+---------------------------------------+
|i9 : isSimplicialMorphism oo           |
|                                       |
|o9 = true                              |
+---------------------------------------+
|i10 : f-g;                             |
+---------------------------------------+
|i11 : isSimplicialMorphism oo          |
|                                       |
|o11 = true                             |
+---------------------------------------+
|i12 : -f;                              |
+---------------------------------------+
|i13 : 3*f;                             |
+---------------------------------------+
|i14 : 0*f                              |
|                                       |
|o14 = 0                                |
|                                       |
|o14 : SimplicialModuleMap              |
+---------------------------------------+
|i15 : a*f;                             |
+---------------------------------------+
|i16 : assert(0*f == 0)                 |
+---------------------------------------+
|i17 : assert(1*f == f)                 |
+---------------------------------------+
|i18 : assert((-1)*f == -f)             |
+---------------------------------------+
|i19 : assert(-(f-g) == g-f)            |
+---------------------------------------+
|i20 : assert((a+b)*f == a*f + b*f)     |
+---------------------------------------+
|i21 : assert(a*(f+g) == a*f + a*g)     |
+---------------------------------------+
|i22 : assert isSimplicialMorphism (f+g)|
+---------------------------------------+

Adding or subtracting a scalar is the same as adding or subtracting the scalar
multiple of the identity.  In particular, the source and target must be equal.

+-----------------------------------------------------------------------------+
|i23 : h = randomSimplicialMap(C, C);                                         |
+-----------------------------------------------------------------------------+
|i24 : prune normalize h                                                      |
|                                                                             |
|           1              1                                                  |
|o24 = 0 : R  <---------- R  : 0                                              |
|                | 11 |                                                       |
|                                                                             |
|           4                                                                 |
|      1 : R  <---------------------------------------------------------------|
|                {2} | 46 -28a+b-3c+22d -7a+2b+29c-47d -13a3-10a2b+39ab2-20b3+|
|                {3} | 0  -47           15             -19a2+17ab-39b2-20ac+36|
|                {3} | 0  -23           -37            -26a2+22ab-8b2-49ac+43b|
|                {5} | 0  0             0              -30                    |
|                                                                             |
|           4                                                                 |
|      2 : R  <---------------------------------------------------------------|
|                {4} | 41 -28 35a-9b-35c+6d  -41a2-49ab+30b2-13ac-47bc-40c2+4a|
|                {4} | 16 -6  40a+3b-31c+25d -31a2-39ab-29b2-31ac-48bc-37c2-48|
|                {5} | 0  0   -2             28a-18b+46c+d                    |
|                {6} | 0  0   0              40                               |
|                                                                             |
|           1                   1                                             |
|      3 : R  <--------------- R  : 3                                         |
|                {6} | -22 |                                                  |
|                                                                             |
|o24 : ComplexMap                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|30a2c+27abc+24b2c+32ac2-48bc2-18a2d-22abd-30b2d-9acd-15bcd+33c2d-32ad2+39bd2-|
|bc-39c2+44ad+9bd+4cd+13d2                                                    |
|c+36c2-11ad-8bd-3cd-22d2                                                     |
|                                                                             |
|                                                                             |
|                       4                                                     |
|--------------------- R  : 2                                                 |
|d+27bd+37cd-35d2  |                                                          |
|ad+30bd+47cd-49d2 |                                                          |
|                  |                                                          |
|                  |                                                          |
|-----------------------------------------------------------------------------|
|                4                                                            |
|-------------- R  : 1                                                        |
|49cd2-33d3 |                                                                 |
|           |                                                                 |
|           |                                                                 |
|           |                                                                 |
+-----------------------------------------------------------------------------+
|i25 : prune normalize(h+1)                                                   |
|                                                                             |
|           1              1                                                  |
|o25 = 0 : R  <---------- R  : 0                                              |
|                | 12 |                                                       |
|                                                                             |
|           4                                                                 |
|      1 : R  <---------------------------------------------------------------|
|                {2} | 47 -28a+b-3c+22d -7a+2b+29c-47d -13a3-10a2b+39ab2-20b3+|
|                {3} | 0  -46           15             -19a2+17ab-39b2-20ac+36|
|                {3} | 0  -23           -36            -26a2+22ab-8b2-49ac+43b|
|                {5} | 0  0             0              -29                    |
|                                                                             |
|           4                                                                 |
|      2 : R  <---------------------------------------------------------------|
|                {4} | 42 -28 35a-9b-35c+6d  -41a2-49ab+30b2-13ac-47bc-40c2+4a|
|                {4} | 16 -5  40a+3b-31c+25d -31a2-39ab-29b2-31ac-48bc-37c2-48|
|                {5} | 0  0   -1             28a-18b+46c+d                    |
|                {6} | 0  0   0              41                               |
|                                                                             |
|           1                   1                                             |
|      3 : R  <--------------- R  : 3                                         |
|                {6} | -21 |                                                  |
|                                                                             |
|o25 : ComplexMap                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|30a2c+27abc+24b2c+32ac2-48bc2-18a2d-22abd-30b2d-9acd-15bcd+33c2d-32ad2+39bd2-|
|bc-39c2+44ad+9bd+4cd+13d2                                                    |
|c+36c2-11ad-8bd-3cd-22d2                                                     |
|                                                                             |
|                                                                             |
|                       4                                                     |
|--------------------- R  : 2                                                 |
|d+27bd+37cd-35d2  |                                                          |
|ad+30bd+47cd-49d2 |                                                          |
|                  |                                                          |
|                  |                                                          |
|-----------------------------------------------------------------------------|
|                4                                                            |
|-------------- R  : 1                                                        |
|49cd2-33d3 |                                                                 |
|           |                                                                 |
|           |                                                                 |
|           |                                                                 |
+-----------------------------------------------------------------------------+
|i26 : assert(h+1 == h + id_C)                                                |
+-----------------------------------------------------------------------------+
|i27 : assert(h+a == h + a*id_C)                                              |
+-----------------------------------------------------------------------------+
|i28 : assert(1-h == id_C - h)                                                |
+-----------------------------------------------------------------------------+
|i29 : assert(b-c*h == -c*h + b*id_C)                                         |
+-----------------------------------------------------------------------------+
|i30 : assert(b-h*c == -h*c + id_C*b)                                         |
+-----------------------------------------------------------------------------+

See also
========

  * Making maps between simplicial modules
    -- information about the basic constructors
  * randomSimplicialMap
    -- a random
    map of simplicial modules
  * map(SimplicialModule,SimplicialModule,SimplicialModuleMap)
    -- make
    a new map of simplicial modules, induced from an existing one
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2590:0.

Basic invariants and properties -- information about accessing basic features
*****************************************************************************

Predicates for simplicial modules and simplicial module maps
============================================================

  * isWellDefined(SimplicialModule),
    -- whether a simplicial module is well-defined
  * isWellDefined(SimplicialModuleMap)
    -- whether a map of simplicial
    modules is well-defined
  * isCommutative(SimplicialModuleMap)
    -- whether a simplicial module map
    commutes with the face/degeneracy maps
  * isSimplicialMorphism(SimplicialModuleMap)
    -- whether a simplicial
    module map is a morphism of simplicial modules
  * isShortExactSequence(SimplicialModuleMap,SimplicialModuleMap),
    --
    whether a pair of simplicial module maps forms a short exact sequence

Other invariants for simplicial modules
=======================================

  * ring(SimplicialModule) -- access the
    ring of a simplicial module or a simplicial module map
  * topDegree(SimplicialModule) -- top degree of a simplicial
    module
  * components(SimplicialModule), --
    list the components of a direct sum

Other invariants for simplicial module maps
===========================================

  * source(SimplicialModuleMap) -- get
    the source of a map of simplicial modules
  * target(SimplicialModuleMap) -- get
    the target of a map of simplicial modules
  * degree(SimplicialModuleMap) -- get
    the degree of a map of simplicial modules
  * ring(SimplicialModuleMap) -- access the
    ring of a simplicial module or a simplicial module map
  * isHomogeneous(SimplicialModuleMap)
    -- whether a map of simplicial
    modules is homogeneous
  * components(SimplicialModuleMap),
    -- list the components of a direct sum

See also
========

  * Making simplicial modules -- information
    about the basic constructors
  * Making maps between simplicial modules
    -- information about the basic constructors
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:332:0.

coimage(SimplicialModuleMap) -- make the coimage of a map of simplicial modules
*******************************************************************************

  * Function: coimage (Macaulay2Doc),
  * Usage:
        coimage f
  * Inputs:
      * f, a Map of Simplicial Modules,
  * Outputs:
      * a Simplicial Module,

Description
===========

The coimage of a simplicial module map $f : C \to D$ is the simplicial module
$E$ whose $i$-th term is $coimage(f_i)$, and whose face/degeneracy map is
induced from the face/degeneracy map on the source.

In the following example, we first construct a random simplicial morphism $f :
C \to D$.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a,b,c,d];                                                    |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution ideal(b^2-a*c, b*c-a*d, c^2-b*d), 3,|
|                                                                             |
|      1      4      9      16                                                |
|o2 = S  <-- S  <-- S  <-- S  <-- ...                                         |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i3 : D = simplicialModule(freeResolution ideal(a,b,c), Degeneracy => true)   |
|                                                                             |
|      1      4      10      20                                               |
|o3 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : f = randomSimplicialMap(D, C, Cycle => true, InternalDegree => 0)       |
|                                                                             |
|          1               1                                                  |
|o4 = 0 : S  <----------- S  : 0                                              |
|               | -22 |                                                       |
|                                                                             |
|          4                                                        4         |
|     1 : S  <---------------------------------------------------- S  : 1     |
|               {0} | -22 0            0           0           |              |
|               {1} | 0   36b+3c       30b-19c+22d -29b-10c    |              |
|               {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d |              |
|               {1} | 0   19a-29b      19a-8b      10a-29b-22c |              |
|                                                                             |
|          10                                                                 |
|     2 : S   <---------------------------------------------------------------|
|                {0} | -22 0            0           0           0            0|
|                {1} | 0   36b+3c       30b-19c+22d -29b-10c    0            0|
|                {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d 0            0|
|                {1} | 0   19a-29b      19a-8b      10a-29b-22c 0            0|
|                {1} | 0   0            0           0           36b+3c       3|
|                {1} | 0   0            0           0           -36a-22b+29c -|
|                {1} | 0   0            0           0           19a-29b      1|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                                                                             |
|          20                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {0} | -22 0            0           0           0            0|
|                {1} | 0   36b+3c       30b-19c+22d -29b-10c    0            0|
|                {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d 0            0|
|                {1} | 0   19a-29b      19a-8b      10a-29b-22c 0            0|
|                {1} | 0   0            0           0           36b+3c       3|
|                {1} | 0   0            0           0           -36a-22b+29c -|
|                {1} | 0   0            0           0           19a-29b      1|
|                {1} | 0   0            0           0           0            0|
|                {1} | 0   0            0           0           0            0|
|                {1} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {3} | 0   0            0           0           0            0|
|                                                                             |
|o4 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                                              9              |
|------------------------------------------------------------ S  : 2          |
|           0           0                0                |                   |
|           0           0                0                |                   |
|           0           0                0                |                   |
|           0           0                0                |                   |
|0b-19c+22d -29b-10c    0                0                |                   |
|30a-14c    29a+29c+22d 0                0                |                   |
|9a-8b      10a-29b-22c 0                0                |                   |
|           0           -29a-30b+31c-22d 29b+6c-36d       |                   |
|           0           -10a+24b+3c      34b-19c+19d      |                   |
|           0           24a-8b+29c       -24a-29b-14c-29d |                   |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|0b-19c+22d -29b-10c    0            0           0           0                |
|30a-14c    29a+29c+22d 0            0           0           0                |
|9a-8b      10a-29b-22c 0            0           0           0                |
|           0           36b+3c       30b-19c+22d -29b-10c    0                |
|           0           -36a-22b+29c -30a-14c    29a+29c+22d 0                |
|           0           19a-29b      19a-8b      10a-29b-22c 0                |
|           0           0            0           0           -29a-30b+31c-22d |
|           0           0            0           0           -10a+24b+3c      |
|           0           0            0           0           24a-8b+29c       |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|29b+6c-36d       0                0                0                0        |
|34b-19c+19d      0                0                0                0        |
|-24a-29b-14c-29d 0                0                0                0        |
|0                -29a-30b+31c-22d 29b+6c-36d       0                0        |
|0                -10a+24b+3c      34b-19c+19d      0                0        |
|0                24a-8b+29c       -24a-29b-14c-29d 0                0        |
|0                0                0                -29a-30b+31c-22d 29b+6c-36|
|0                0                0                -10a+24b+3c      34b-19c+1|
|0                0                0                24a-8b+29c       -24a-29b-|
|0                0                0                0                0        |
|-----------------------------------------------------------------------------|
|             16                                                              |
|----------- S   : 3                                                          |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|d       |                                                                    |
|9d      |                                                                    |
|14c-29d |                                                                    |
|        |                                                                    |
+-----------------------------------------------------------------------------+
|i5 : g1 = inducedMap(coimage f, source f)                                    |
|                                                                             |
|          1             1                                                    |
|o5 = 0 : S  <--------- S  : 0                                                |
|               | 1 |                                                         |
|                                                                             |
|          4                       4                                          |
|     1 : S  <------------------- S  : 1                                      |
|               {0} | 1 0 0 0 |                                               |
|               {2} | 0 1 0 0 |                                               |
|               {2} | 0 0 1 0 |                                               |
|               {2} | 0 0 0 1 |                                               |
|                                                                             |
|          9                                 9                                |
|     2 : S  <----------------------------- S  : 2                            |
|               {0} | 1 0 0 0 0 0 0 0 0 |                                     |
|               {2} | 0 1 0 0 0 0 0 0 0 |                                     |
|               {2} | 0 0 1 0 0 0 0 0 0 |                                     |
|               {2} | 0 0 0 1 0 0 0 0 0 |                                     |
|               {2} | 0 0 0 0 1 0 0 0 0 |                                     |
|               {2} | 0 0 0 0 0 1 0 0 0 |                                     |
|               {2} | 0 0 0 0 0 0 1 0 0 |                                     |
|               {3} | 0 0 0 0 0 0 0 1 0 |                                     |
|               {3} | 0 0 0 0 0 0 0 0 1 |                                     |
|                                                                             |
|          16                                               16                |
|     3 : S   <------------------------------------------- S   : 3            |
|                {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                      |
|                                                                             |
|o5 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+
|i6 : coimage g1 == coimage f                                                 |
|                                                                             |
|o6 = true                                                                    |
+-----------------------------------------------------------------------------+
|i7 : coker g1 == 0                                                           |
|                                                                             |
|o7 = true                                                                    |
+-----------------------------------------------------------------------------+

Caveat
======

The coimage is more computationally intensive than
image(SimplicialModuleMap), because, unlike
image, it computes kernels of maps of modules.

See also
========

  * Making simplicial modules -- information
    about the basic constructors
  * Making maps between simplicial modules
    -- information about the basic constructors
  * coimage (Macaulay2Doc) -- coimage of a map
  * image(SimplicialModuleMap) -- make
    the image of a map of simplicial modules
  * kernel(SimplicialModuleMap) -- make
    the kernel of a map of simplicial modules
  * cokernel(SimplicialModuleMap), --
    make the cokernel of a map of simplicial modules

Ways to use this method:
========================

  * coimage(SimplicialModuleMap), --
    make the coimage of a map of simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2815:0.

cokernel(SimplicialModuleMap) -- make the cokernel of a map of simplicial modules
*********************************************************************************

  * Function: cokernel (Macaulay2Doc),
  * Usage:
        cokernel f
        coker f
  * Inputs:
      * f, a Map of Simplicial Modules,
  * Outputs:
      * a Simplicial Module,

Description
===========

If $f : C \to D$ is a map of simplicial modules of degree $d$, then the
cokernel is the simplicial module $E$ whose $i$-th term is $cokernel(f_{i-d})$,
and whose face/degeneracy map is induced from the face/degeneracy map on the
target.

In the following example, we first construct a random simplicial morphism $f :
C \to D$.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a,b,c,d];                                                    |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution ideal(b^2-a*c, b*c-a*d, c^2-b*d), 3,|
|                                                                             |
|      1      4      9      16                                                |
|o2 = S  <-- S  <-- S  <-- S  <-- ...                                         |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i3 : D = simplicialModule(freeResolution ideal(a,b,c), Degeneracy => true)   |
|                                                                             |
|      1      4      10      20                                               |
|o3 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : f = randomSimplicialMap(D, C, Cycle => true, InternalDegree => 0)       |
|                                                                             |
|          1               1                                                  |
|o4 = 0 : S  <----------- S  : 0                                              |
|               | -22 |                                                       |
|                                                                             |
|          4                                                        4         |
|     1 : S  <---------------------------------------------------- S  : 1     |
|               {0} | -22 0            0           0           |              |
|               {1} | 0   36b+3c       30b-19c+22d -29b-10c    |              |
|               {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d |              |
|               {1} | 0   19a-29b      19a-8b      10a-29b-22c |              |
|                                                                             |
|          10                                                                 |
|     2 : S   <---------------------------------------------------------------|
|                {0} | -22 0            0           0           0            0|
|                {1} | 0   36b+3c       30b-19c+22d -29b-10c    0            0|
|                {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d 0            0|
|                {1} | 0   19a-29b      19a-8b      10a-29b-22c 0            0|
|                {1} | 0   0            0           0           36b+3c       3|
|                {1} | 0   0            0           0           -36a-22b+29c -|
|                {1} | 0   0            0           0           19a-29b      1|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                                                                             |
|          20                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {0} | -22 0            0           0           0            0|
|                {1} | 0   36b+3c       30b-19c+22d -29b-10c    0            0|
|                {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d 0            0|
|                {1} | 0   19a-29b      19a-8b      10a-29b-22c 0            0|
|                {1} | 0   0            0           0           36b+3c       3|
|                {1} | 0   0            0           0           -36a-22b+29c -|
|                {1} | 0   0            0           0           19a-29b      1|
|                {1} | 0   0            0           0           0            0|
|                {1} | 0   0            0           0           0            0|
|                {1} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {3} | 0   0            0           0           0            0|
|                                                                             |
|o4 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                                              9              |
|------------------------------------------------------------ S  : 2          |
|           0           0                0                |                   |
|           0           0                0                |                   |
|           0           0                0                |                   |
|           0           0                0                |                   |
|0b-19c+22d -29b-10c    0                0                |                   |
|30a-14c    29a+29c+22d 0                0                |                   |
|9a-8b      10a-29b-22c 0                0                |                   |
|           0           -29a-30b+31c-22d 29b+6c-36d       |                   |
|           0           -10a+24b+3c      34b-19c+19d      |                   |
|           0           24a-8b+29c       -24a-29b-14c-29d |                   |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|0b-19c+22d -29b-10c    0            0           0           0                |
|30a-14c    29a+29c+22d 0            0           0           0                |
|9a-8b      10a-29b-22c 0            0           0           0                |
|           0           36b+3c       30b-19c+22d -29b-10c    0                |
|           0           -36a-22b+29c -30a-14c    29a+29c+22d 0                |
|           0           19a-29b      19a-8b      10a-29b-22c 0                |
|           0           0            0           0           -29a-30b+31c-22d |
|           0           0            0           0           -10a+24b+3c      |
|           0           0            0           0           24a-8b+29c       |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|29b+6c-36d       0                0                0                0        |
|34b-19c+19d      0                0                0                0        |
|-24a-29b-14c-29d 0                0                0                0        |
|0                -29a-30b+31c-22d 29b+6c-36d       0                0        |
|0                -10a+24b+3c      34b-19c+19d      0                0        |
|0                24a-8b+29c       -24a-29b-14c-29d 0                0        |
|0                0                0                -29a-30b+31c-22d 29b+6c-36|
|0                0                0                -10a+24b+3c      34b-19c+1|
|0                0                0                24a-8b+29c       -24a-29b-|
|0                0                0                0                0        |
|-----------------------------------------------------------------------------|
|             16                                                              |
|----------- S   : 3                                                          |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|d       |                                                                    |
|9d      |                                                                    |
|14c-29d |                                                                    |
|        |                                                                    |
+-----------------------------------------------------------------------------+
|i5 : prune coker f                                                           |
|                                                                             |
|o5 = 0 <-- cokernel {1} | -b-43c-36d    -8c+45d    50c+43d     | <-- cokernel|
|                    {1} | a+43b+36c+39d 8b-46c-40d 50b-43c-35d |             |
|     0              {1} | -39c          b+40c      a+35c       |             |
|                                                                             |
|           1                                                                 |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                     2       |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o5 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|{1} | -b-43c-36d    -8c+45d    50c+43d     0             0                   |
|{1} | a+43b+36c+39d 8b-46c-40d 50b-43c-35d 0             0                   |
|{1} | -39c          b+40c      a+35c       0             0                   |
|{1} | 0             0          0           -b-43c-36d    -8c+45d             |
|{1} | 0             0          0           a+43b+36c+39d 8b-46c-40d          |
|{1} | 0             0          0           -39c          b+40c               |
|{2} | 0             0          0           0             0                   |
|{2} | 0             0          0           0             0                   |
|{2} | 0             0          0           0             0                   |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|0           0               0          | <-- cokernel {1} | -b-43c-36d       |
|0           0               0          |              {1} | a+43b+36c+39d    |
|0           0               0          |              {1} | -39c             |
|50c+43d     0               0          |              {1} | 0                |
|50b-43c-35d 0               0          |              {1} | 0                |
|a+35c       0               0          |              {1} | 0                |
|0           13a-10b-34c+26d 3b+25c-49d |              {1} | 0                |
|0           a-26b+42c-12d   7b+5c-5d   |              {1} | 0                |
|0           34b+49c+13d     a-3b+9c-3d |              {1} | 0                |
|                                                      {2} | 0                |
|                                                      {2} | 0                |
|                                                      {2} | 0                |
|                                                      {2} | 0                |
|                                                      {2} | 0                |
|                                                      {2} | 0                |
|                                                      {2} | 0                |
|                                                      {2} | 0                |
|                                                      {2} | 0                |
|                                                      {3} | 0                |
|                                                                             |
|                                             3                               |
|-----------------------------------------------------------------------------|
|-8c+45d    50c+43d     0             0          0           0                |
|8b-46c-40d 50b-43c-35d 0             0          0           0                |
|b+40c      a+35c       0             0          0           0                |
|0          0           -b-43c-36d    -8c+45d    50c+43d     0                |
|0          0           a+43b+36c+39d 8b-46c-40d 50b-43c-35d 0                |
|0          0           -39c          b+40c      a+35c       0                |
|0          0           0             0          0           -b-43c-36d       |
|0          0           0             0          0           a+43b+36c+39d    |
|0          0           0             0          0           -39c             |
|0          0           0             0          0           0                |
|0          0           0             0          0           0                |
|0          0           0             0          0           0                |
|0          0           0             0          0           0                |
|0          0           0             0          0           0                |
|0          0           0             0          0           0                |
|0          0           0             0          0           0                |
|0          0           0             0          0           0                |
|0          0           0             0          0           0                |
|0          0           0             0          0           0                |
|-----------------------------------------------------------------------------|
|0          0           0               0          0               0          |
|0          0           0               0          0               0          |
|0          0           0               0          0               0          |
|0          0           0               0          0               0          |
|0          0           0               0          0               0          |
|0          0           0               0          0               0          |
|-8c+45d    50c+43d     0               0          0               0          |
|8b-46c-40d 50b-43c-35d 0               0          0               0          |
|b+40c      a+35c       0               0          0               0          |
|0          0           13a-10b-34c+26d 3b+25c-49d 0               0          |
|0          0           a-26b+42c-12d   7b+5c-5d   0               0          |
|0          0           34b+49c+13d     a-3b+9c-3d 0               0          |
|0          0           0               0          13a-10b-34c+26d 3b+25c-49d |
|0          0           0               0          a-26b+42c-12d   7b+5c-5d   |
|0          0           0               0          34b+49c+13d     a-3b+9c-3d |
|0          0           0               0          0               0          |
|0          0           0               0          0               0          |
|0          0           0               0          0               0          |
|0          0           0               0          0               0          |
|-----------------------------------------------------------------------------|
|0               0          |<-- ...                                          |
|0               0          |                                                 |
|0               0          |                                                 |
|0               0          |                                                 |
|0               0          |                                                 |
|0               0          |                                                 |
|0               0          |                                                 |
|0               0          |                                                 |
|0               0          |                                                 |
|0               0          |                                                 |
|0               0          |                                                 |
|0               0          |                                                 |
|0               0          |                                                 |
|0               0          |                                                 |
|0               0          |                                                 |
|13a-10b-34c+26d 3b+25c-49d |                                                 |
|a-26b+42c-12d   7b+5c-5d   |                                                 |
|34b+49c+13d     a-3b+9c-3d |                                                 |
|0               0          |                                                 |
+-----------------------------------------------------------------------------+
|i6 : prune normalize oo                                                      |
|                                                                             |
|                                                                             |
|o6 = cokernel {1} | -b-43c-36d    -8c+45d    50c+43d     | <-- cokernel {2} ||
|              {1} | a+43b+36c+39d 8b-46c-40d 50b-43c-35d |              {2} ||
|              {1} | -39c          b+40c      a+35c       |              {2} ||
|                                                                             |
|     1                                                         2             |
|                                                                             |
|o6 : Complex                                                                 |
|-----------------------------------------------------------------------------|
|                                  1                                          |
|13a-10b-34c+26d 3b+25c-49d | <-- S                                           |
|a-26b+42c-12d   7b+5c-5d   |                                                 |
|34b+49c+13d     a-3b+9c-3d |     3                                           |
+-----------------------------------------------------------------------------+
|i7 : prune HH coker f                                                        |
|                                                                             |
|o7 = 0 <-- cokernel {1} | -d -c -b 0  -d -c | <-- cokernel {1} | -d -c -b 0  |
|                    {1} | c  b  a  -d 0  0  |              {1} | c  b  a  -d |
|     0              {1} | 0  0  0  c  b  a  |              {1} | 0  0  0  c  |
|                                                           {1} | 0  0  0  0  |
|           1                                               {1} | 0  0  0  0  |
|                                                           {1} | 0  0  0  0  |
|                                                                             |
|                                                  2                          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o7 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|-d -c 0  0  0  0  0  0  | <-- cokernel {1} | -d -c -b 0  -d -c 0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {1} | c  b  a  -d 0  0  0  0  0  0  0 |
|b  a  0  0  0  0  0  0  |              {1} | 0  0  0  c  b  a  0  0  0  0  0 |
|0  0  -d -c -b 0  -d -c |              {1} | 0  0  0  0  0  0  -d -c -b 0  -d|
|0  0  c  b  a  -d 0  0  |              {1} | 0  0  0  0  0  0  c  b  a  -d 0 |
|0  0  0  0  0  c  b  a  |              {1} | 0  0  0  0  0  0  0  0  0  c  b |
|                                       {1} | 0  0  0  0  0  0  0  0  0  0  0 |
|                                       {1} | 0  0  0  0  0  0  0  0  0  0  0 |
|                                       {1} | 0  0  0  0  0  0  0  0  0  0  0 |
|                                                                             |
|                              3                                              |
|-----------------------------------------------------------------------------|
|0  0  0  0  0  0  0  |<-- ...                                                |
|0  0  0  0  0  0  0  |                                                       |
|0  0  0  0  0  0  0  |                                                       |
|-c 0  0  0  0  0  0  |                                                       |
|0  0  0  0  0  0  0  |                                                       |
|a  0  0  0  0  0  0  |                                                       |
|0  -d -c -b 0  -d -c |                                                       |
|0  c  b  a  -d 0  0  |                                                       |
|0  0  0  0  c  b  a  |                                                       |
+-----------------------------------------------------------------------------+
|i8 : g1 = inducedMap(coker f, target f)                                      |
|                                                                             |
|                                  1                                          |
|o8 = 0 : cokernel | -22 | <----- S  : 0                                      |
|                             0                                               |
|                                                                             |
|                                                                             |
|     1 : cokernel {0} | -22 0            0           0                       |
|                  {1} | 0   36b+3c       30b-19c+22d -29b-10c                |
|                  {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d             |
|                  {1} | 0   19a-29b      19a-8b      10a-29b-22c             |
|                                                                             |
|                                                                             |
|                                                                             |
|     2 : cokernel {0} | -22 0            0           0                       |
|                  {1} | 0   36b+3c       30b-19c+22d -29b-10c                |
|                  {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d             |
|                  {1} | 0   19a-29b      19a-8b      10a-29b-22c             |
|                  {1} | 0   0            0           0                       |
|                  {1} | 0   0            0           0                       |
|                  {1} | 0   0            0           0                       |
|                  {2} | 0   0            0           0                       |
|                  {2} | 0   0            0           0                       |
|                  {2} | 0   0            0           0                       |
|                                                                             |
|                                                                             |
|                                                                             |
|     3 : cokernel {0} | -22 0            0           0                       |
|                  {1} | 0   36b+3c       30b-19c+22d -29b-10c                |
|                  {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d             |
|                  {1} | 0   19a-29b      19a-8b      10a-29b-22c             |
|                  {1} | 0   0            0           0                       |
|                  {1} | 0   0            0           0                       |
|                  {1} | 0   0            0           0                       |
|                  {1} | 0   0            0           0                       |
|                  {1} | 0   0            0           0                       |
|                  {1} | 0   0            0           0                       |
|                  {2} | 0   0            0           0                       |
|                  {2} | 0   0            0           0                       |
|                  {2} | 0   0            0           0                       |
|                  {2} | 0   0            0           0                       |
|                  {2} | 0   0            0           0                       |
|                  {2} | 0   0            0           0                       |
|                  {2} | 0   0            0           0                       |
|                  {2} | 0   0            0           0                       |
|                  {2} | 0   0            0           0                       |
|                  {3} | 0   0            0           0                       |
|                                                                             |
|                                                                             |
|o8 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                        4                                                    |
|| <------------------- S  : 1                                                |
||    {0} | 0 0 0 0 |                                                         |
||    {1} | 0 1 0 0 |                                                         |
||    {1} | 0 0 1 0 |                                                         |
|     {1} | 0 0 0 1 |                                                         |
|                                                                             |
|                                                                             |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|36b+3c       30b-19c+22d -29b-10c                                            |
|-36a-22b+29c -30a-14c    29a+29c+22d                                         |
|19a-29b      19a-8b      10a-29b-22c                                         |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|                                                                             |
|                                                                             |
|                                                                             |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|36b+3c       30b-19c+22d -29b-10c                                            |
|-36a-22b+29c -30a-14c    29a+29c+22d                                         |
|19a-29b      19a-8b      10a-29b-22c                                         |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|0            0           0                                                   |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                      10     |
|0                0                | <------------------------------- S   :   |
|0                0                |    {0} | 0 0 0 0 0 0 0 0 0 0 |           |
|0                0                |    {1} | 0 1 0 0 0 0 0 0 0 0 |           |
|0                0                |    {1} | 0 0 1 0 0 0 0 0 0 0 |           |
|0                0                |    {1} | 0 0 0 1 0 0 0 0 0 0 |           |
|0                0                |    {1} | 0 0 0 0 1 0 0 0 0 0 |           |
|0                0                |    {1} | 0 0 0 0 0 1 0 0 0 0 |           |
|-29a-30b+31c-22d 29b+6c-36d       |    {1} | 0 0 0 0 0 0 1 0 0 0 |           |
|-10a+24b+3c      34b-19c+19d      |    {2} | 0 0 0 0 0 0 0 1 0 0 |           |
|24a-8b+29c       -24a-29b-14c-29d |    {2} | 0 0 0 0 0 0 0 0 1 0 |           |
|                                       {2} | 0 0 0 0 0 0 0 0 0 1 |           |
|                                                                             |
|                                                                             |
|0            0           0           0                0                0     |
|0            0           0           0                0                0     |
|0            0           0           0                0                0     |
|0            0           0           0                0                0     |
|0            0           0           0                0                0     |
|0            0           0           0                0                0     |
|0            0           0           0                0                0     |
|36b+3c       30b-19c+22d -29b-10c    0                0                0     |
|-36a-22b+29c -30a-14c    29a+29c+22d 0                0                0     |
|19a-29b      19a-8b      10a-29b-22c 0                0                0     |
|0            0           0           -29a-30b+31c-22d 29b+6c-36d       0     |
|0            0           0           -10a+24b+3c      34b-19c+19d      0     |
|0            0           0           24a-8b+29c       -24a-29b-14c-29d 0     |
|0            0           0           0                0                -29a  |
|0            0           0           0                0                -10a  |
|0            0           0           0                0                24a-  |
|0            0           0           0                0                0     |
|0            0           0           0                0                0     |
|0            0           0           0                0                0     |
|0            0           0           0                0                0     |
|                                                                             |
|-----------------------------------------------------------------------------|
|2                                                                            |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|             0                0                0                |            |
|             0                0                0                |            |
|             0                0                0                |            |
|             0                0                0                |            |
|             0                0                0                |            |
|             0                0                0                |            |
|             0                0                0                |            |
|             0                0                0                |            |
|             0                0                0                |            |
|             0                0                0                |            |
|             0                0                0                |            |
|             0                0                0                |            |
|             0                0                0                |            |
|-30b+31c-22d 29b+6c-36d       0                0                |            |
|+24b+3c      34b-19c+19d      0                0                |            |
|8b+29c       -24a-29b-14c-29d 0                0                |            |
|             0                -29a-30b+31c-22d 29b+6c-36d       |            |
|             0                -10a+24b+3c      34b-19c+19d      |            |
|             0                24a-8b+29c       -24a-29b-14c-29d |            |
|             0                0                0                |            |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                      20                     |
|<--------------------------------------------------- S   : 3                 |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                           |
|   {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                           |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                           |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                           |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                           |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                           |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                           |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                           |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                           |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                           |
|   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                           |
+-----------------------------------------------------------------------------+
|i9 : coker f == image g1                                                     |
|                                                                             |
|o9 = true                                                                    |
+-----------------------------------------------------------------------------+
|i10 : coker g1 == 0                                                          |
|                                                                             |
|o10 = true                                                                   |
+-----------------------------------------------------------------------------+

See also
========

  * Making simplicial modules -- information
    about the basic constructors
  * Making maps between simplicial modules
    -- information about the basic constructors
  * cokernel (Macaulay2Doc) -- cokernel of a map
  * image(SimplicialModuleMap) -- make
    the image of a map of simplicial modules
  * coimage(SimplicialModuleMap), --
    make the coimage of a map of simplicial modules
  * kernel(SimplicialModuleMap) -- make
    the kernel of a map of simplicial modules

Ways to use this method:
========================

  * cokernel(SimplicialModuleMap), --
    make the cokernel of a map of simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2895:0.

components(SimplicialModule) -- list the components of a direct sum
*******************************************************************

  * Function: components (Macaulay2Doc),
  * Usage:
        components C
  * Inputs:
      * C, a Simplicial Module,
  * Outputs:
      * a list (Macaulay2Doc), the component simplicial modules of
        a direct sum (of simplicial modules)

Description
===========

A simplicial module which has been constructed as a direct sum stores its
component simplicial modules.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a,b,c];                                                      |
+-----------------------------------------------------------------------------+
|i2 : C1 = simplicialModule freeResolution coker vars S                       |
|                                                                             |
|      1      4      10      20                                               |
|o2 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o2 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i3 : C2 = simplicialModule(complex (ideal(a,b,c)), 3)                        |
|                                                                             |
|o3 = image | a b c | <-- image | a b c | <-- image | a b c | <-- image | a b |
|                                                                             |
|     0                   1                   2                   3           |
|                                                                             |
|o3 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|c |<-- ...                                                                   |
+-----------------------------------------------------------------------------+
|i4 : D = C1 ++ C2                                                            |
|                                                                             |
|o4 = image | 1 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 | <-- image {0} | 1 0 0 |
|           | 0 a b c |           {1} | 0 1 0 0 0 0 0 |           {1} | 0 1 0 |
|                                 {1} | 0 0 1 0 0 0 0 |           {1} | 0 0 1 |
|     0                           {1} | 0 0 0 1 0 0 0 |           {1} | 0 0 0 |
|                                 {0} | 0 0 0 0 a b c |           {1} | 0 0 0 |
|                                                                 {1} | 0 0 0 |
|                           1                                     {1} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {0} | 0 0 0 |
|                                                                             |
|                                                           2                 |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o4 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 |           {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|1 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 1 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 1 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 1 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 1 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 1 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 1 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 a b c |           {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|                                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|                                {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|                          3                                                  |
|-----------------------------------------------------------------------------|
|0 0 0 |<-- ...                                                               |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|a b c |                                                                      |
+-----------------------------------------------------------------------------+
|i5 : L = components D                                                        |
|                                                                             |
|       1      4      10      20                                              |
|o5 = {S  <-- S  <-- S   <-- S  <-- ..., image | a b c | <-- image | a b c |  |
|                                                                             |
|      0      1      2       3           0                   1                |
|     ------------------------------------------------------------------------|
|     <-- image | a b c | <-- image | a b c |<-- ...}                         |
|                                                                             |
|         2                   3                                               |
|                                                                             |
|o5 : List                                                                    |
+-----------------------------------------------------------------------------+
|i6 : L_0 === C1                                                              |
|                                                                             |
|o6 = true                                                                    |
+-----------------------------------------------------------------------------+
|i7 : L_1 === C2                                                              |
|                                                                             |
|o7 = true                                                                    |
+-----------------------------------------------------------------------------+
|i8 : E = (peanut => C1) ++ (butter => C2)                                    |
|                                                                             |
|o8 = image | 1 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 | <-- image {0} | 1 0 0 |
|           | 0 a b c |           {1} | 0 1 0 0 0 0 0 |           {1} | 0 1 0 |
|                                 {1} | 0 0 1 0 0 0 0 |           {1} | 0 0 1 |
|     0                           {1} | 0 0 0 1 0 0 0 |           {1} | 0 0 0 |
|                                 {0} | 0 0 0 0 a b c |           {1} | 0 0 0 |
|                                                                 {1} | 0 0 0 |
|                           1                                     {1} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {0} | 0 0 0 |
|                                                                             |
|                                                           2                 |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o8 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 |           {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|1 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 1 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 1 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 1 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 1 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 1 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 1 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 a b c |           {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|                                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|                                {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|                          3                                                  |
|-----------------------------------------------------------------------------|
|0 0 0 |<-- ...                                                               |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|a b c |                                                                      |
+-----------------------------------------------------------------------------+
|i9 : components E                                                            |
|                                                                             |
|       1      4      10      20                                              |
|o9 = {S  <-- S  <-- S   <-- S  <-- ..., image | a b c | <-- image | a b c |  |
|                                                                             |
|      0      1      2       3           0                   1                |
|     ------------------------------------------------------------------------|
|     <-- image | a b c | <-- image | a b c |<-- ...}                         |
|                                                                             |
|         2                   3                                               |
|                                                                             |
|o9 : List                                                                    |
+-----------------------------------------------------------------------------+

The names of the component simplicial modules are called indices, and are used
to access the relevant inclusion and projection maps.

+-----------------------------------------------------------------------------+
|i10 : indices D                                                              |
|                                                                             |
|o10 = {0, 1}                                                                 |
|                                                                             |
|o10 : List                                                                   |
+-----------------------------------------------------------------------------+
|i11 : D^[0]                                                                  |
|                                                                             |
|           1                                                                 |
|o11 = 0 : S  <--------------- image | 1 0 0 0 | : 0                          |
|                | 1 0 0 0 |         | 0 a b c |                              |
|                                                                             |
|           4                                                                 |
|      1 : S  <------------------------- image {0} | 1 0 0 0 0 0 0 | : 1      |
|                {0} | 1 0 0 0 0 0 0 |         {1} | 0 1 0 0 0 0 0 |          |
|                {1} | 0 1 0 0 0 0 0 |         {1} | 0 0 1 0 0 0 0 |          |
|                {1} | 0 0 1 0 0 0 0 |         {1} | 0 0 0 1 0 0 0 |          |
|                {1} | 0 0 0 1 0 0 0 |         {0} | 0 0 0 0 a b c |          |
|                                                                             |
|           10                                                                |
|      2 : S   <------------------------------------- image {0} | 1 0 0 0     |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 1 0 0     |
|                 {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 1 0     |
|                 {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 1     |
|                 {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 0     |
|                 {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 0     |
|                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 |         {1} | 0 0 0 0     |
|                 {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 |         {2} | 0 0 0 0     |
|                 {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 |         {2} | 0 0 0 0     |
|                 {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 |         {2} | 0 0 0 0     |
|                 {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 |         {0} | 0 0 0 0     |
|                                                                             |
|           20                                                                |
|      3 : S   <---------------------------------------------------------     |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |       |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |       |
|                                                                             |
|o11 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 | : 2                                                      |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|1 0 0 0 0 0 0 0 0 |                                                          |
|0 1 0 0 0 0 0 0 0 |                                                          |
|0 0 1 0 0 0 0 0 0 |                                                          |
|0 0 0 1 0 0 0 0 0 |                                                          |
|0 0 0 0 1 0 0 0 0 |                                                          |
|0 0 0 0 0 1 0 0 0 |                                                          |
|0 0 0 0 0 0 a b c |                                                          |
|                                                                             |
|                                                                             |
|image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | : 3              |
|      {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                  |
|      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                  |
|      {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |                  |
+-----------------------------------------------------------------------------+
|i12 : indices E                                                              |
|                                                                             |
|o12 = {peanut, butter}                                                       |
|                                                                             |
|o12 : List                                                                   |
+-----------------------------------------------------------------------------+
|i13 : E_[butter]                                                             |
|                                                                             |
|o13 = 0 : image | 1 0 0 0 |                                                  |
|                | 0 a b c |                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      1 : image {0} | 1 0 0                                                  |
|                {1} | 0 1 0                                                  |
|                {1} | 0 0 1                                                  |
|                {1} | 0 0 0                                                  |
|                {0} | 0 0 0                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      2 : image {0} | 1 0 0                                                  |
|                {1} | 0 1 0                                                  |
|                {1} | 0 0 1                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {0} | 0 0 0                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      3 : image {0} | 1 0 0                                                  |
|                {1} | 0 1 0                                                  |
|                {1} | 0 0 1                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {0} | 0 0 0                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o13 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|<----------------- image | a b c | : 0                                       |
|   {0} | 0 0 0 |                                                             |
|   {1} | 1 0 0 |                                                             |
|   {1} | 0 1 0 |                                                             |
|   {1} | 0 0 1 |                                                             |
|                                                                             |
|0 0 0 0 | <----------------- image | a b c | : 1                             |
|0 0 0 0 |    {0} | 0 0 0 |                                                   |
|0 0 0 0 |    {1} | 0 0 0 |                                                   |
|1 0 0 0 |    {1} | 0 0 0 |                                                   |
|0 a b c |    {1} | 0 0 0 |                                                   |
|             {1} | 1 0 0 |                                                   |
|             {1} | 0 1 0 |                                                   |
|             {1} | 0 0 1 |                                                   |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 | <----------------- image | a b c | : 2                 |
|0 0 0 0 0 0 0 0 0 0 |    {0} | 0 0 0 |                                       |
|0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|1 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 1 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 0 1 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 0 0 1 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 0 0 0 1 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 0 0 0 0 1 0 0 0 0 |    {2} | 0 0 0 |                                       |
|0 0 0 0 0 0 1 0 0 0 |    {2} | 0 0 0 |                                       |
|0 0 0 0 0 0 0 a b c |    {2} | 0 0 0 |                                       |
|                         {1} | 1 0 0 |                                       |
|                         {1} | 0 1 0 |                                       |
|                         {1} | 0 0 1 |                                       |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <----------------- image | a b c | |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {0} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |    {3} | 0 0 0 |                   |
|                                             {1} | 1 0 0 |                   |
|                                             {1} | 0 1 0 |                   |
|                                             {1} | 0 0 1 |                   |
|-----------------------------------------------------------------------------|
|: 3                                                                          |
+-----------------------------------------------------------------------------+

See also
========

  * directSum(SimplicialModule), --
    direct sum of simplicial modules
  * indices (Macaulay2Doc) -- indices of a polynomial; also
    components for a direct sum
  * SimplicialModule _ Array -- the
    canonical inclusion or projection map of a direct sum
  * SimplicialModule ^ Array -- the
    canonical inclusion or projection map of a direct sum

Ways to use this method:
========================

  * components(SimplicialModule), --
    list the components of a direct sum
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:913:0.

components(SimplicialModuleMap) -- list the components of a direct sum
**********************************************************************

  * Function: components (Macaulay2Doc),
  * Usage:
        components f
  * Inputs:
      * f, a Map of Simplicial Modules,
  * Outputs:
      * a list (Macaulay2Doc), the component maps of a direct sum
        of maps of simplicial modules

Description
===========

A map of simplicial modules stores its component maps.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a,b,c];                                                      |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution coker vars S, Degeneracy => true)   |
|                                                                             |
|      1      4      10      20                                               |
|o2 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o2 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i3 : g1 = id_C                                                               |
|                                                                             |
|          1             1                                                    |
|o3 = 0 : S  <--------- S  : 0                                                |
|               | 1 |                                                         |
|                                                                             |
|          4                       4                                          |
|     1 : S  <------------------- S  : 1                                      |
|               {0} | 1 0 0 0 |                                               |
|               {1} | 0 1 0 0 |                                               |
|               {1} | 0 0 1 0 |                                               |
|               {1} | 0 0 0 1 |                                               |
|                                                                             |
|          10                                   10                            |
|     2 : S   <------------------------------- S   : 2                        |
|                {0} | 1 0 0 0 0 0 0 0 0 0 |                                  |
|                {1} | 0 1 0 0 0 0 0 0 0 0 |                                  |
|                {1} | 0 0 1 0 0 0 0 0 0 0 |                                  |
|                {1} | 0 0 0 1 0 0 0 0 0 0 |                                  |
|                {1} | 0 0 0 0 1 0 0 0 0 0 |                                  |
|                {1} | 0 0 0 0 0 1 0 0 0 0 |                                  |
|                {1} | 0 0 0 0 0 0 1 0 0 0 |                                  |
|                {2} | 0 0 0 0 0 0 0 1 0 0 |                                  |
|                {2} | 0 0 0 0 0 0 0 0 1 0 |                                  |
|                {2} | 0 0 0 0 0 0 0 0 0 1 |                                  |
|                                                                             |
|          20                                                       20        |
|     3 : S   <--------------------------------------------------- S   : 3    |
|                {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |              |
|                {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |              |
|                {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |              |
|                {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |              |
|                {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |              |
|                {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |              |
|                {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |              |
|                {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |              |
|                {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |              |
|                {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |              |
|                {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |              |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |              |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |              |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |              |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |              |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |              |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |              |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |              |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |              |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |              |
|                                                                             |
|o3 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+
|i4 : g2 = randomSimplicialMap(C[1], C[2], Boundary => true)                  |
|                                                                             |
|          3                                          3                       |
|o4 = 0 : S  <-------------------------------------- S  : 0                   |
|               {1} | 24b-36c  -29b+19c -10b-29c |                            |
|               {1} | -24a-30c 29a+19c  10a-8c   |                            |
|               {1} | 36a+30b  -19a-19b 29a+8b   |                            |
|                                                                             |
|          6                                                       4          |
|     1 : S  <--------------------------------------------------- S  : 1      |
|               {1} | 24b-36c  -29b+19c -10b-29c 0            |               |
|               {1} | -24a-30c 29a+19c  10a-8c   0            |               |
|               {1} | 36a+30b  -19a-19b 29a+8b   0            |               |
|               {2} | 0        0        0        -10a+29b+46c |               |
|               {2} | 0        0        0        -29a-41b-36c |               |
|               {2} | 0        0        0        14a-19b-30c  |               |
|                                                                             |
|          10                                                                 |
|     2 : S   <---------------------------------------------------------------|
|                {1} | 24b-36c  -29b+19c -10b-29c 0            0            | |
|                {1} | -24a-30c 29a+19c  10a-8c   0            0            | |
|                {1} | 36a+30b  -19a-19b 29a+8b   0            0            | |
|                {2} | 0        0        0        -10a+29b+46c 0            | |
|                {2} | 0        0        0        -29a-41b-36c 0            | |
|                {2} | 0        0        0        14a-19b-30c  0            | |
|                {2} | 0        0        0        0            -10a+29b+46c | |
|                {2} | 0        0        0        0            -29a-41b-36c | |
|                {2} | 0        0        0        0            14a-19b-30c  | |
|                {3} | 0        0        0        0            0            | |
|                                                                             |
|          15                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {1} | 24b-36c  -29b+19c -10b-29c 0            0            0 |
|                {1} | -24a-30c 29a+19c  10a-8c   0            0            0 |
|                {1} | 36a+30b  -19a-19b 29a+8b   0            0            0 |
|                {2} | 0        0        0        -10a+29b+46c 0            0 |
|                {2} | 0        0        0        -29a-41b-36c 0            0 |
|                {2} | 0        0        0        14a-19b-30c  0            0 |
|                {2} | 0        0        0        0            -10a+29b+46c 0 |
|                {2} | 0        0        0        0            -29a-41b-36c 0 |
|                {2} | 0        0        0        0            14a-19b-30c  0 |
|                {2} | 0        0        0        0            0            -1|
|                {2} | 0        0        0        0            0            -2|
|                {2} | 0        0        0        0            0            14|
|                {3} | 0        0        0        0            0            0 |
|                {3} | 0        0        0        0            0            0 |
|                {3} | 0        0        0        0            0            0 |
|                                                                             |
|o4 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|   5                                                                         |
|- S  : 2                                                                     |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                6                                                            |
|-------------- S  : 3                                                        |
|           |                                                                 |
|           |                                                                 |
|           |                                                                 |
|           |                                                                 |
|           |                                                                 |
|           |                                                                 |
|           |                                                                 |
|           |                                                                 |
|           |                                                                 |
|0a+29b+46c |                                                                 |
|9a-41b-36c |                                                                 |
|a-19b-30c  |                                                                 |
|           |                                                                 |
|           |                                                                 |
|           |                                                                 |
+-----------------------------------------------------------------------------+
|i5 : f = g1 ++ g2                                                            |
|                                                                             |
|          4                                            4                     |
|o5 = 0 : S  <---------------------------------------- S  : 0                 |
|               {0} | 1 0        0        0        |                          |
|               {1} | 0 24b-36c  -29b+19c -10b-29c |                          |
|               {1} | 0 -24a-30c 29a+19c  10a-8c   |                          |
|               {1} | 0 36a+30b  -19a-19b 29a+8b   |                          |
|                                                                             |
|          10                                                               8 |
|     1 : S   <----------------------------------------------------------- S  |
|                {0} | 1 0 0 0 0        0        0        0            |      |
|                {1} | 0 1 0 0 0        0        0        0            |      |
|                {1} | 0 0 1 0 0        0        0        0            |      |
|                {1} | 0 0 0 1 0        0        0        0            |      |
|                {1} | 0 0 0 0 24b-36c  -29b+19c -10b-29c 0            |      |
|                {1} | 0 0 0 0 -24a-30c 29a+19c  10a-8c   0            |      |
|                {1} | 0 0 0 0 36a+30b  -19a-19b 29a+8b   0            |      |
|                {2} | 0 0 0 0 0        0        0        -10a+29b+46c |      |
|                {2} | 0 0 0 0 0        0        0        -29a-41b-36c |      |
|                {2} | 0 0 0 0 0        0        0        14a-19b-30c  |      |
|                                                                             |
|          20                                                                 |
|     2 : S   <---------------------------------------------------------------|
|                {0} | 1 0 0 0 0 0 0 0 0 0 0        0        0        0       |
|                {1} | 0 1 0 0 0 0 0 0 0 0 0        0        0        0       |
|                {1} | 0 0 1 0 0 0 0 0 0 0 0        0        0        0       |
|                {1} | 0 0 0 1 0 0 0 0 0 0 0        0        0        0       |
|                {1} | 0 0 0 0 1 0 0 0 0 0 0        0        0        0       |
|                {1} | 0 0 0 0 0 1 0 0 0 0 0        0        0        0       |
|                {1} | 0 0 0 0 0 0 1 0 0 0 0        0        0        0       |
|                {2} | 0 0 0 0 0 0 0 1 0 0 0        0        0        0       |
|                {2} | 0 0 0 0 0 0 0 0 1 0 0        0        0        0       |
|                {2} | 0 0 0 0 0 0 0 0 0 1 0        0        0        0       |
|                {1} | 0 0 0 0 0 0 0 0 0 0 24b-36c  -29b+19c -10b-29c 0       |
|                {1} | 0 0 0 0 0 0 0 0 0 0 -24a-30c 29a+19c  10a-8c   0       |
|                {1} | 0 0 0 0 0 0 0 0 0 0 36a+30b  -19a-19b 29a+8b   0       |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0        0        0        -10a+29b|
|                {2} | 0 0 0 0 0 0 0 0 0 0 0        0        0        -29a-41b|
|                {2} | 0 0 0 0 0 0 0 0 0 0 0        0        0        14a-19b-|
|                {2} | 0 0 0 0 0 0 0 0 0 0 0        0        0        0       |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0        0        0        0       |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0        0        0        0       |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0        0        0        0       |
|                                                                             |
|          35                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0        0     |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0        0     |
|                {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24b-36c  -29b+1|
|                {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -24a-30c 29a+19|
|                {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 36a+30b  -19a-1|
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        0     |
|                                                                             |
|o5 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|: 1                                                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                       15                                                    |
|--------------------- S   : 2                                                |
|     0            |                                                          |
|     0            |                                                          |
|     0            |                                                          |
|     0            |                                                          |
|     0            |                                                          |
|     0            |                                                          |
|     0            |                                                          |
|     0            |                                                          |
|     0            |                                                          |
|     0            |                                                          |
|     0            |                                                          |
|     0            |                                                          |
|     0            |                                                          |
|+46c 0            |                                                          |
|-36c 0            |                                                          |
|30c  0            |                                                          |
|     -10a+29b+46c |                                                          |
|     -29a-41b-36c |                                                          |
|     14a-19b-30c  |                                                          |
|     0            |                                                          |
|                                                                             |
|                                                        26                   |
|------------------------------------------------------ S   : 3               |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|9c -10b-29c 0            0            0            |                         |
|c  10a-8c   0            0            0            |                         |
|9b 29a+8b   0            0            0            |                         |
|   0        -10a+29b+46c 0            0            |                         |
|   0        -29a-41b-36c 0            0            |                         |
|   0        14a-19b-30c  0            0            |                         |
|   0        0            -10a+29b+46c 0            |                         |
|   0        0            -29a-41b-36c 0            |                         |
|   0        0            14a-19b-30c  0            |                         |
|   0        0            0            -10a+29b+46c |                         |
|   0        0            0            -29a-41b-36c |                         |
|   0        0            0            14a-19b-30c  |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
|   0        0            0            0            |                         |
+-----------------------------------------------------------------------------+
|i6 : assert isWellDefined f                                                  |
+-----------------------------------------------------------------------------+
|i7 : L = components f                                                        |
|                                                                             |
|           1             1                                                   |
|o7 = {0 : S  <--------- S  : 0                                            , 0|
|                | 1 |                                                        |
|                                                                             |
|           4                       4                                         |
|      1 : S  <------------------- S  : 1                                     |
|                {0} | 1 0 0 0 |                                              |
|                {1} | 0 1 0 0 |                                             1|
|                {1} | 0 0 1 0 |                                              |
|                {1} | 0 0 0 1 |                                              |
|                                                                             |
|           10                                   10                           |
|      2 : S   <------------------------------- S   : 2                       |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 |                                 |
|                 {1} | 0 1 0 0 0 0 0 0 0 0 |                                 |
|                 {1} | 0 0 1 0 0 0 0 0 0 0 |                                 |
|                 {1} | 0 0 0 1 0 0 0 0 0 0 |                                2|
|                 {1} | 0 0 0 0 1 0 0 0 0 0 |                                 |
|                 {1} | 0 0 0 0 0 1 0 0 0 0 |                                 |
|                 {1} | 0 0 0 0 0 0 1 0 0 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 1 0 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 0 1 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 0 0 1 |                                 |
|                                                                             |
|           20                                                       20       |
|      3 : S   <--------------------------------------------------- S   : 3   |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            3|
|                 {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |             |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |             |
|     ------------------------------------------------------------------------|
|        3                                          3                         |
|     : S  <-------------------------------------- S  : 0                     |
|             {1} | 24b-36c  -29b+19c -10b-29c |                              |
|             {1} | -24a-30c 29a+19c  10a-8c   |                              |
|             {1} | 36a+30b  -19a-19b 29a+8b   |                              |
|                                                                             |
|        6                                                       4            |
|     : S  <--------------------------------------------------- S  : 1        |
|             {1} | 24b-36c  -29b+19c -10b-29c 0            |                 |
|             {1} | -24a-30c 29a+19c  10a-8c   0            |                 |
|             {1} | 36a+30b  -19a-19b 29a+8b   0            |                 |
|             {2} | 0        0        0        -10a+29b+46c |                 |
|             {2} | 0        0        0        -29a-41b-36c |                 |
|             {2} | 0        0        0        14a-19b-30c  |                 |
|                                                                             |
|        10                                                                   |
|     : S   <---------------------------------------------------------------- |
|              {1} | 24b-36c  -29b+19c -10b-29c 0            0            |   |
|              {1} | -24a-30c 29a+19c  10a-8c   0            0            |   |
|              {1} | 36a+30b  -19a-19b 29a+8b   0            0            |   |
|              {2} | 0        0        0        -10a+29b+46c 0            |   |
|              {2} | 0        0        0        -29a-41b-36c 0            |   |
|              {2} | 0        0        0        14a-19b-30c  0            |   |
|              {2} | 0        0        0        0            -10a+29b+46c |   |
|              {2} | 0        0        0        0            -29a-41b-36c |   |
|              {2} | 0        0        0        0            14a-19b-30c  |   |
|              {3} | 0        0        0        0            0            |   |
|                                                                             |
|        15                                                                   |
|     : S   <-----------------------------------------------------------------|
|              {1} | 24b-36c  -29b+19c -10b-29c 0            0            0   |
|              {1} | -24a-30c 29a+19c  10a-8c   0            0            0   |
|              {1} | 36a+30b  -19a-19b 29a+8b   0            0            0   |
|              {2} | 0        0        0        -10a+29b+46c 0            0   |
|              {2} | 0        0        0        -29a-41b-36c 0            0   |
|              {2} | 0        0        0        14a-19b-30c  0            0   |
|              {2} | 0        0        0        0            -10a+29b+46c 0   |
|              {2} | 0        0        0        0            -29a-41b-36c 0   |
|              {2} | 0        0        0        0            14a-19b-30c  0   |
|              {2} | 0        0        0        0            0            -10a|
|              {2} | 0        0        0        0            0            -29a|
|              {2} | 0        0        0        0            0            14a-|
|              {3} | 0        0        0        0            0            0   |
|              {3} | 0        0        0        0            0            0   |
|              {3} | 0        0        0        0            0            0   |
|     ------------------------------------------------------------------------|
|                        }                                                    |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      5                                                                      |
|     S  : 2                                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                   6                                                         |
|     ------------ S  : 3                                                     |
|              |                                                              |
|              |                                                              |
|              |                                                              |
|              |                                                              |
|              |                                                              |
|              |                                                              |
|              |                                                              |
|              |                                                              |
|              |                                                              |
|     +29b+46c |                                                              |
|     -41b-36c |                                                              |
|     19b-30c  |                                                              |
|              |                                                              |
|              |                                                              |
|              |                                                              |
|                                                                             |
|o7 : List                                                                    |
+-----------------------------------------------------------------------------+
|i8 : L_0 === g1                                                              |
|                                                                             |
|o8 = true                                                                    |
+-----------------------------------------------------------------------------+
|i9 : L_1 === g2                                                              |
|                                                                             |
|o9 = true                                                                    |
+-----------------------------------------------------------------------------+
|i10 : indices f                                                              |
|                                                                             |
|o10 = {0, 1}                                                                 |
|                                                                             |
|o10 : List                                                                   |
+-----------------------------------------------------------------------------+

The names of the components are called indices, and are used to access the
relevant inclusion and projection maps.

+-----------------------------------------------------------------------------+
|i11 : f^[0]                                                                  |
|                                                                             |
|           1                   4                                             |
|o11 = 0 : S  <--------------- S  : 0                                         |
|                | 1 0 0 0 |                                                  |
|                                                                             |
|           4                               8                                 |
|      1 : S  <--------------------------- S  : 1                             |
|                {0} | 1 0 0 0 0 0 0 0 |                                      |
|                {1} | 0 1 0 0 0 0 0 0 |                                      |
|                {1} | 0 0 1 0 0 0 0 0 |                                      |
|                {1} | 0 0 0 1 0 0 0 0 |                                      |
|                                                                             |
|           10                                             15                 |
|      2 : S   <----------------------------------------- S   : 2             |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                       |
|                 {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                       |
|                 {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                       |
|                 {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                       |
|                 {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                       |
|                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                       |
|                 {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                       |
|                 {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                       |
|                 {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                       |
|                 {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                       |
|                                                                             |
|           20                                                                |
|      3 : S   <--------------------------------------------------------------|
|                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | |
|                                                                             |
|o11 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|   26                                                                        |
|- S   : 3                                                                    |
+-----------------------------------------------------------------------------+
|i12 : f_[0]                                                                  |
|                                                                             |
|           4                 1                                               |
|o12 = 0 : S  <------------- S  : 0                                           |
|                {0} | 1 |                                                    |
|                {1} | 0 |                                                    |
|                {1} | 0 |                                                    |
|                {1} | 0 |                                                    |
|                                                                             |
|           10                       4                                        |
|      1 : S   <------------------- S  : 1                                    |
|                 {0} | 1 0 0 0 |                                             |
|                 {1} | 0 1 0 0 |                                             |
|                 {1} | 0 0 1 0 |                                             |
|                 {1} | 0 0 0 1 |                                             |
|                 {1} | 0 0 0 0 |                                             |
|                 {1} | 0 0 0 0 |                                             |
|                 {1} | 0 0 0 0 |                                             |
|                 {2} | 0 0 0 0 |                                             |
|                 {2} | 0 0 0 0 |                                             |
|                 {2} | 0 0 0 0 |                                             |
|                                                                             |
|           20                                   10                           |
|      2 : S   <------------------------------- S   : 2                       |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 |                                 |
|                 {1} | 0 1 0 0 0 0 0 0 0 0 |                                 |
|                 {1} | 0 0 1 0 0 0 0 0 0 0 |                                 |
|                 {1} | 0 0 0 1 0 0 0 0 0 0 |                                 |
|                 {1} | 0 0 0 0 1 0 0 0 0 0 |                                 |
|                 {1} | 0 0 0 0 0 1 0 0 0 0 |                                 |
|                 {1} | 0 0 0 0 0 0 1 0 0 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 1 0 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 0 1 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 0 0 1 |                                 |
|                 {1} | 0 0 0 0 0 0 0 0 0 0 |                                 |
|                 {1} | 0 0 0 0 0 0 0 0 0 0 |                                 |
|                 {1} | 0 0 0 0 0 0 0 0 0 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 |                                 |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 |                                 |
|                                                                             |
|           35                                                       20       |
|      3 : S   <--------------------------------------------------- S   : 3   |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |             |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |             |
|                 {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                                                                             |
|o12 : SimplicialModuleMap                                                    |
+-----------------------------------------------------------------------------+

See also
========

  * directSum(SimplicialModuleMap)
    -- direct sum of simplicial module maps
  * components(SimplicialModule), --
    list the components of a direct sum
  * indices (Macaulay2Doc) -- indices of a polynomial; also
    components for a direct sum
  * SimplicialModuleMap _ Array -- the
    composition with the canonical inclusion or projection map
  * SimplicialModuleMap ^ Array -- the
    composition with the canonical inclusion or projection map

Ways to use this method:
========================

  * components(SimplicialModuleMap),
    -- list the components of a direct sum
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1941:0.

degree(SimplicialModuleMap) -- get the degree of a map of simplicial modules
****************************************************************************

  * Function: degree (Macaulay2Doc),
  * Usage:
        degree f
  * Inputs:
      * f, a Map of Simplicial Modules,
  * Outputs:
      * an integer (Macaulay2Doc),

Description
===========

A simplicial module map $f : C \to D$ of degree $d$ is a sequence of maps $f_i
: C_i \to D_{i+d}$. This method returns $d$.

The degree of the face/degeneracy map of a simplicial module is -1 or 1,
respectively.

+----------------------------------------------------------------+
|i1 : R = ZZ/101[a..d];                                          |
+----------------------------------------------------------------+
|i2 : I = ideal(a^2, b^2, c^2)                                   |
|                                                                |
|             2   2   2                                          |
|o2 = ideal (a , b , c )                                         |
|                                                                |
|o2 : Ideal of R                                                 |
+----------------------------------------------------------------+
|i3 : FI = simplicialModule(freeResolution I, Degeneracy => true)|
|                                                                |
|      1      4      10      20                                  |
|o3 = R  <-- R  <-- R   <-- R  <-- ...                           |
|                                                                |
|     0      1      2       3                                    |
|                                                                |
|o3 : SimplicialModule                                           |
+----------------------------------------------------------------+
|i4 : assert(degree dd^FI == -1)                                 |
+----------------------------------------------------------------+
|i5 : assert(degree ss^FI == 1)                                  |
+----------------------------------------------------------------+
+-----------------------------------------------------------------------------+
|i6 : FJ = simplicialModule(freeResolution (I + ideal(a*b*c)), Degeneracy =>  |
|                                                                             |
|      1      5      15      34                                               |
|o6 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o6 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|true)                                                                        |
+-----------------------------------------------------------------------------+
|i7 : f = randomSimplicialMap(FJ, FI, Cycle=>true, Degree => -2);             |
+-----------------------------------------------------------------------------+
|i8 : normalize f                                                             |
|                                                                             |
|          1                                                                  |
|o8 = 0 : R  <----------------------------------------------------------------|
|               | 18a4+44a3b-29a2b2+8ab3+6b4+38a3c+50a2bc-48ab2c+50b3c-32a2c2+|
|                                                                             |
|          4                                                                  |
|     1 : R  <----------------------------------------------------------------|
|               {2} | 24a4-36a3b+19a2b2-29ab3-18b4-30a3c+19a2bc-24ab2c-13b3c-2|
|               {2} | 19a4-16a3b-23a2b2+36ab3-3b4+7a3c+39a2bc+35ab2c+22b3c-17a|
|               {2} | 27a4-22a3b-32a2b2+39ab3+36b4+32a3c-20a2bc+9b3c-30a2c2-49|
|               {3} | 40a3-22a2b+30ab2-47b3+10a2c+13abc+18b2c-13ac2-30bc2-48c3|
|                                                                             |
|o8 : ComplexMap                                                              |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|4abc2-24b2c2+17ac3+22bc3+43c4+31a3d+49a2bd+4ab2d-23b3d+12a2cd-3abcd+7b2cd-20a|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|9a2c2-16abc2-15b2c2+34ac3+38bc3+45c4-29a3d-10a2bd-38ab2d-43b3d-8a2cd+39abcd-2|
|2c2-38abc2-23b2c2+11ac3+29bc3-13c4+15a3d+43a2bd+11ab2d-47b3d-11a2cd+33abcd-7b|
|abc2+4b2c2+17ac3+22bc3+43c4-9a3d+24a2bd+33ab2d-39b3d-48a2cd-33abcd+13b2cd-20a|
|+7a2d-17abd-28b2d+3acd+26bcd+30c2d-41ad2-bd2-37cd2                           |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|c2d-49bc2d-8c3d+37a2d2+45abd2+9b2d2+44acd2-11bcd2+36c2d2-39ad3-8bd3-3cd3-22d4|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|8b2cd+19ac2d+2bc2d-34c3d-22a2d2+21abd2-47b2d2-47acd2+16bcd2-48c2d2-39ad3+22bd|
|2cd+46ac2d-47bc2d-10c3d+48a2d2+40abd2+2b2d2-28acd2+15bcd2+30c2d2+ad3-37bd3-18|
|c2d-49bc2d-8c3d-15a2d2-19abd2-26b2d2+44acd2-11bcd2+36c2d2-39ad3-8bd3-3cd3-22d|
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
| -32a4+20a3b+50a2b2-36ab3+3b4+23a3c+22a2bc+12ab2c-22b3c-18a2c2-11abc2-7b2c2-2|
|                                                                             |
|                  1                                                          |
|---------------- R  : 1                                                      |
|3-47cd3+47d4 |                                                               |
|cd3+39d4     |                                                               |
|4            |                                                               |
|             |                                                               |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|9ac3+12bc3-15c4+39a3d-6a2bd-11ab2d+47b3d-20a2cd-5abcd+7b2cd-38bc2d+4c3d+14a2d|
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|2-40abd2-2b2d2+29acd2-15bcd2+5c2d2-ad3+37bd3+18cd3-39d4 24a4-36a3b+6a2b2-25ab|
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|3+9b4-30a3c-42a2bc-16ab2c-23b3c-20a2c2+29abc2-40b2c2+28ac3-45bc3+47c4-29a3d-1|
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|0a2bd+16ab2d-6b3d-8a2cd+46abcd+25b2cd-21ac2d-20bc2d+7c3d-22a2d2+21abd2+15b2d2|
|-----------------------------------------------------------------------------|
|                                                3                            |
|---------------------------------------------- R  : 0                        |
|-47acd2-25bcd2+c2d2-39ad3+22bd3-47cd3+47d4 |                                 |
+-----------------------------------------------------------------------------+

See also
========

  * Basic invariants and properties, --
    information about accessing basic features
  * source(SimplicialModuleMap) -- get
    the source of a map of simplicial modules
  * target(SimplicialModuleMap) -- get
    the target of a map of simplicial modules
  * randomSimplicialMap(SimplicialModule,SimplicialModule)
    -- a random
    map of simplicial modules

Ways to use this method:
========================

  * degree(SimplicialModuleMap) -- get
    the degree of a map of simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1809:0.

directSum(SimplicialModule) -- direct sum of simplicial modules
***************************************************************

  * Function: directSum (Macaulay2Doc),
  * Usage:
        D = C1 ++ C2
        D = directSum(C1,C2,...)
        D = directSum(name1 => C1, name2 => C2, ...)
  * Inputs:
      * Ci, a Simplicial Module,
  * Outputs:
      * D, a Simplicial Module, the direct sum of the
        input simplicial modules

Description
===========

The direct sum of two simplicial modules is another simplicial module.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a,b,c];                                                      |
+-----------------------------------------------------------------------------+
|i2 : C1 = simplicialModule(freeResolution coker vars S, Degeneracy => true)  |
|                                                                             |
|      1      4      10      20                                               |
|o2 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o2 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i3 : D1 = C1 ++ simplicialModule(complex(S^13)[-2], 3, Degeneracy => true)   |
|                                                                             |
|      1      4      23      59                                               |
|o3 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : isWellDefined D1                                                        |
|                                                                             |
|o4 = true                                                                    |
+-----------------------------------------------------------------------------+
|i5 : D1.?ss --knows to cache degeneracy maps if inputs have degeneracy maps  |
|                                                                             |
|o5 = true                                                                    |
+-----------------------------------------------------------------------------+
|i6 : C2 = simplicialModule(ideal(a,b,c), 3, Degeneracy => true)              |
|                                                                             |
|o6 = image | a b c | <-- image | a b c | <-- image | a b c | <-- image | a b |
|                                                                             |
|     0                   1                   2                   3           |
|                                                                             |
|o6 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|c |<-- ...                                                                   |
+-----------------------------------------------------------------------------+
|i7 : C1 ++ C2                                                                |
|                                                                             |
|o7 = image | 1 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 | <-- image {0} | 1 0 0 |
|           | 0 a b c |           {1} | 0 1 0 0 0 0 0 |           {1} | 0 1 0 |
|                                 {1} | 0 0 1 0 0 0 0 |           {1} | 0 0 1 |
|     0                           {1} | 0 0 0 1 0 0 0 |           {1} | 0 0 0 |
|                                 {0} | 0 0 0 0 a b c |           {1} | 0 0 0 |
|                                                                 {1} | 0 0 0 |
|                           1                                     {1} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {0} | 0 0 0 |
|                                                                             |
|                                                           2                 |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o7 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 |           {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|1 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 1 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 1 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 1 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 1 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 1 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 1 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 a b c |           {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|                                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|                                {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|                          3                                                  |
|-----------------------------------------------------------------------------|
|0 0 0 |<-- ...                                                               |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|a b c |                                                                      |
+-----------------------------------------------------------------------------+
|i8 : assert isWellDefined(C1 ++ C2)                                          |
+-----------------------------------------------------------------------------+

The direct sum of a sequence of simplicial modules can be computed as follows.

+-----------------------------------------------------------------------------+
|i9 : C3 = directSum(C1,C2,C2[-2])                                            |
|                                                                             |
|o9 = image | 1 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 | <-- image {0} | 1 0 0 |
|           | 0 a b c |           {1} | 0 1 0 0 0 0 0 |           {1} | 0 1 0 |
|                                 {1} | 0 0 1 0 0 0 0 |           {1} | 0 0 1 |
|     0                           {1} | 0 0 0 1 0 0 0 |           {1} | 0 0 0 |
|                                 {0} | 0 0 0 0 a b c |           {1} | 0 0 0 |
|                                                                 {1} | 0 0 0 |
|                           1                                     {1} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {0} | 0 0 0 |
|                                                                 {0} | 0 0 0 |
|                                                                             |
|                                                           2                 |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o9 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|1 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 1 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 1 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 1 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 1 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 1 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 1 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 a b c 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 a b c |           {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|                                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|                                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
|                                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0|
|                                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|                                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|                                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                      {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                      {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                      {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                      {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|                                3                                            |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |<-- ...                                       |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 a b c 0 0 0 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 a b c 0 0 0 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 a b c 0 0 0 |                                              |
|0 0 0 0 0 0 0 0 0 0 0 0 a b c |                                              |
+-----------------------------------------------------------------------------+
|i10 : assert isWellDefined C3                                                |
+-----------------------------------------------------------------------------+

The direct sum is an n-ary operator with projection and inclusion maps from
each component satisfying appropriate identities.

+-----------------------------------------------------------------------------+
|i11 : C4 = directSum(first => C1, second => C2)                              |
|                                                                             |
|o11 = image | 1 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 | <-- image {0} | 1 0 0|
|            | 0 a b c |           {1} | 0 1 0 0 0 0 0 |           {1} | 0 1 0|
|                                  {1} | 0 0 1 0 0 0 0 |           {1} | 0 0 1|
|      0                           {1} | 0 0 0 1 0 0 0 |           {1} | 0 0 0|
|                                  {0} | 0 0 0 0 a b c |           {1} | 0 0 0|
|                                                                  {1} | 0 0 0|
|                            1                                     {1} | 0 0 0|
|                                                                  {2} | 0 0 0|
|                                                                  {2} | 0 0 0|
|                                                                  {2} | 0 0 0|
|                                                                  {0} | 0 0 0|
|                                                                             |
|                                                            2                |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o11 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 |           {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|1 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 1 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 1 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 1 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 1 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 1 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 1 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 a b c |           {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|                                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|                                {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|                          3                                                  |
|-----------------------------------------------------------------------------|
|0 0 0 |<-- ...                                                               |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|a b c |                                                                      |
+-----------------------------------------------------------------------------+
|i12 : C4_[first] -- inclusion map C1 --> C4                                  |
|                                                                             |
|                                            1                                |
|o12 = 0 : image | 1 0 0 0 | <------------- S  : 0                            |
|                | 0 a b c |    {0} | 1 |                                     |
|                               {1} | 0 |                                     |
|                               {1} | 0 |                                     |
|                               {1} | 0 |                                     |
|                                                                             |
|                                                            4                |
|      1 : image {0} | 1 0 0 0 0 0 0 | <------------------- S  : 1            |
|                {1} | 0 1 0 0 0 0 0 |    {0} | 1 0 0 0 |                     |
|                {1} | 0 0 1 0 0 0 0 |    {1} | 0 1 0 0 |                     |
|                {1} | 0 0 0 1 0 0 0 |    {1} | 0 0 1 0 |                     |
|                {0} | 0 0 0 0 a b c |    {1} | 0 0 0 1 |                     |
|                                         {1} | 0 0 0 0 |                     |
|                                         {1} | 0 0 0 0 |                     |
|                                         {1} | 0 0 0 0 |                     |
|                                                                             |
|                                                                             |
|      2 : image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 | <--------------------------|
|                {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 |    {0} | 1 0 0 0 0 0 0 0 0 |
|                {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 1 0 0 0 0 0 0 0 |
|                {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 1 0 0 0 0 0 0 |
|                {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 1 0 0 0 0 0 |
|                {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 1 0 0 0 0 |
|                {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 |    {1} | 0 0 0 0 0 1 0 0 0 |
|                {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 |    {1} | 0 0 0 0 0 0 1 0 0 |
|                {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 1 0 |
|                {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 1 |
|                {0} | 0 0 0 0 0 0 0 0 0 0 a b c |    {2} | 0 0 0 0 0 0 0 0 0 |
|                                                     {1} | 0 0 0 0 0 0 0 0 0 |
|                                                     {1} | 0 0 0 0 0 0 0 0 0 |
|                                                     {1} | 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|                                                                             |
|      3 : image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <------|
|                {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {0} |
|                {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} |
|                {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} |
|                {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} |
|                {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} |
|                {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} |
|                {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} |
|                {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} |
|                {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} |
|                {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |    {2} |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |    {2} |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |    {2} |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |    {2} |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |    {2} |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |    {2} |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |    {2} |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |    {2} |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |    {2} |
|                {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |    {3} |
|                                                                         {1} |
|                                                                         {1} |
|                                                                         {1} |
|                                                                             |
|o12 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|       10                                                                    |
|----- S   : 2                                                                |
|0 |                                                                          |
|0 |                                                                          |
|0 |                                                                          |
|0 |                                                                          |
|0 |                                                                          |
|0 |                                                                          |
|0 |                                                                          |
|0 |                                                                          |
|0 |                                                                          |
|1 |                                                                          |
|0 |                                                                          |
|0 |                                                                          |
|0 |                                                                          |
|                                                                             |
|                                               20                            |
|--------------------------------------------- S   : 3                        |
|| 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                  |
|| 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                  |
|| 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                  |
|| 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                  |
|| 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                  |
|| 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                  |
|| 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                  |
|| 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                                  |
|| 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                                  |
|| 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                                  |
|| 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                                  |
|| 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                                  |
|| 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                                  |
|| 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                                  |
|| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                                  |
|| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                                  |
|| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                                  |
|| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                                  |
|| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                                  |
|| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                                  |
|| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                  |
|| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                  |
|| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                  |
+-----------------------------------------------------------------------------+
|i13 : C4^[first] -- projection map C4 --> C1                                 |
|                                                                             |
|           1                                                                 |
|o13 = 0 : S  <--------------- image | 1 0 0 0 | : 0                          |
|                | 1 0 0 0 |         | 0 a b c |                              |
|                                                                             |
|           4                                                                 |
|      1 : S  <------------------------- image {0} | 1 0 0 0 0 0 0 | : 1      |
|                {0} | 1 0 0 0 0 0 0 |         {1} | 0 1 0 0 0 0 0 |          |
|                {1} | 0 1 0 0 0 0 0 |         {1} | 0 0 1 0 0 0 0 |          |
|                {1} | 0 0 1 0 0 0 0 |         {1} | 0 0 0 1 0 0 0 |          |
|                {1} | 0 0 0 1 0 0 0 |         {0} | 0 0 0 0 a b c |          |
|                                                                             |
|           10                                                                |
|      2 : S   <------------------------------------- image {0} | 1 0 0 0     |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 1 0 0     |
|                 {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 1 0     |
|                 {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 1     |
|                 {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 0     |
|                 {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 |         {1} | 0 0 0 0     |
|                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 |         {1} | 0 0 0 0     |
|                 {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 |         {2} | 0 0 0 0     |
|                 {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 |         {2} | 0 0 0 0     |
|                 {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 |         {2} | 0 0 0 0     |
|                 {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 |         {0} | 0 0 0 0     |
|                                                                             |
|           20                                                                |
|      3 : S   <---------------------------------------------------------     |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |       |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |       |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |       |
|                                                                             |
|o13 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 | : 2                                                      |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|1 0 0 0 0 0 0 0 0 |                                                          |
|0 1 0 0 0 0 0 0 0 |                                                          |
|0 0 1 0 0 0 0 0 0 |                                                          |
|0 0 0 1 0 0 0 0 0 |                                                          |
|0 0 0 0 1 0 0 0 0 |                                                          |
|0 0 0 0 0 1 0 0 0 |                                                          |
|0 0 0 0 0 0 a b c |                                                          |
|                                                                             |
|                                                                             |
|image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | : 3              |
|      {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                  |
|      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                  |
|      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                  |
|      {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |                  |
+-----------------------------------------------------------------------------+
|i14 : C4^[first] * C4_[first] == 1                                           |
|                                                                             |
|o14 = true                                                                   |
+-----------------------------------------------------------------------------+
|i15 : C4^[second] * C4_[second] == 1                                         |
|                                                                             |
|o15 = true                                                                   |
+-----------------------------------------------------------------------------+
|i16 : C4^[first] * C4_[second] == 0                                          |
|                                                                             |
|o16 = true                                                                   |
+-----------------------------------------------------------------------------+
|i17 : C4^[second] * C4_[first] == 0                                          |
|                                                                             |
|o17 = true                                                                   |
+-----------------------------------------------------------------------------+
|i18 : C4_[first] * C4^[first] + C4_[second] * C4^[second] == 1               |
|                                                                             |
|o18 = true                                                                   |
+-----------------------------------------------------------------------------+

There are two short exact sequences associated to a direct sum.

+---------------------------------------------------+
|i19 : isShortExactSequence(C4^[first], C4_[second])|
|                                                   |
|o19 = true                                         |
+---------------------------------------------------+
|i20 : isShortExactSequence(C4^[second], C4_[first])|
|                                                   |
|o20 = true                                         |
+---------------------------------------------------+

Given a simplicial module which is a direct sum, we obtain the component
simplicial modules and their names (indices) as follows.

+-----------------------------------------------------------------------------+
|i21 : components C3                                                          |
|                                                                             |
|        1      4      10      20                                             |
|o21 = {S  <-- S  <-- S   <-- S  <-- ..., image | a b c | <-- image | a b c | |
|                                                                             |
|       0      1      2       3           0                   1               |
|                                                                             |
|                                                                             |
|      -----------------------------------------------------------------------|
|      <-- image | a b c | <-- image | a b c |<-- ..., 0 <-- 0 <-- image | a b|
|                                                                             |
|          2                   3                       0     1     2          |
|                                                                             |
|                                                                             |
|      -----------------------------------------------------------------------|
|      c | <-- image | a b c 0 0 0 0 0 0 |<-- ...}                            |
|                    | 0 0 0 a b c 0 0 0 |                                    |
|                    | 0 0 0 0 0 0 a b c |                                    |
|                                                                             |
|              3                                                              |
|                                                                             |
|o21 : List                                                                   |
+-----------------------------------------------------------------------------+
|i22 : indices C3                                                             |
|                                                                             |
|o22 = {0, 1, 2}                                                              |
|                                                                             |
|o22 : List                                                                   |
+-----------------------------------------------------------------------------+
|i23 : components C4                                                          |
|                                                                             |
|        1      4      10      20                                             |
|o23 = {S  <-- S  <-- S   <-- S  <-- ..., image | a b c | <-- image | a b c | |
|                                                                             |
|       0      1      2       3           0                   1               |
|      -----------------------------------------------------------------------|
|      <-- image | a b c | <-- image | a b c |<-- ...}                        |
|                                                                             |
|          2                   3                                              |
|                                                                             |
|o23 : List                                                                   |
+-----------------------------------------------------------------------------+
|i24 : indices C4                                                             |
|                                                                             |
|o24 = {first, second}                                                        |
|                                                                             |
|o24 : List                                                                   |
+-----------------------------------------------------------------------------+

See also
========

  * components(SimplicialModule), --
    list the components of a direct sum
  * indices (Macaulay2Doc) -- indices of a polynomial; also
    components for a direct sum
  * SimplicialModule ^ Array -- the
    canonical inclusion or projection map of a direct sum
  * SimplicialModule _ Array -- the
    canonical inclusion or projection map of a direct sum
  * isShortExactSequence(SimplicialModuleMap,SimplicialModuleMap),
    --
    whether a pair of simplicial module maps forms a short exact sequence

Ways to use this method:
========================

  * directSum(SimplicialModule), --
    direct sum of simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3215:0.

exteriorInclusion -- computes the image of the 2nd exterior power into the tensor product
*****************************************************************************************

  * Usage:
        exteriorInclusion(S)
  * Inputs:
      * M, a Simplicial Module, or Complex (Complexes),
        or Module (Macaulay2Doc), optional
        integer argument specifies top degree
  * Outputs:
      * a Map of Simplicial Modules, or
        Matrix (Macaulay2Doc), the map representing the image of the
        2nd exterior power of $S$ into the tensor product $S \otimes S$.

Description
===========

Given a simplicial module $S$ (or a complex), this function computes the map
$$\bigwedge^2 S \to S \otimes S,$$ The cokernel of this map is by definition
the second symmetric power of $S$. This method is mainly used in conjunction
with the tensorLES, command to compute induced maps on
homology for canonical short exact sequences.

+---------------------------------------------------------------------+
|i1 : Q = ZZ/101[a,b,c]                                               |
|                                                                     |
|o1 = Q                                                               |
|                                                                     |
|o1 : PolynomialRing                                                  |
+---------------------------------------------------------------------+
|i2 : K = koszulComplex vars Q                                        |
|                                                                     |
|      1      3      3      1                                         |
|o2 = Q  <-- Q  <-- Q  <-- Q                                          |
|                                                                     |
|     0      1      2      3                                          |
|                                                                     |
|o2 : Complex                                                         |
+---------------------------------------------------------------------+
|i3 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3|
| -- .261123s elapsed                                                 |
+---------------------------------------------------------------------+
|i4 : isWellDefined phi                                               |
|                                                                     |
|o4 = true                                                            |
+---------------------------------------------------------------------+
|i5 : isCommutative phi                                               |
|                                                                     |
|o5 = true                                                            |
+---------------------------------------------------------------------+
|i6 : prune coker phi                                                 |
|                                                                     |
|      1      9      36      74                                       |
|o6 = Q  <-- Q  <-- Q   <-- Q                                         |
|                                                                     |
|     0      1      2       3                                         |
|                                                                     |
|o6 : Complex                                                         |
+---------------------------------------------------------------------+

All of the homology of the second symmetric/exterior powers are guaranteed to
be concentrated in degrees $0$ to $3$ in the above example, so it suffices to
compute only $4$ terms to understand all of the homology.

+-----------------------------------------------------------------------------+
|i7 : for i to 3 list prune HH_i source phi                                   |
|                                                                             |
|                                                 46                          |
|o7 = {0, cokernel {1} | c b a 0 0 0 0 0 0 |, 0, Q  }                         |
|                  {1} | 0 0 0 c b a 0 0 0 |                                  |
|                  {1} | 0 0 0 0 0 0 c b a |                                  |
|                                                                             |
|o7 : List                                                                    |
+-----------------------------------------------------------------------------+
|i8 : for i to 3 list prune HH_i (coker phi)                                  |
|                                                                             |
|o8 = {cokernel | c b a |, 0, cokernel {2} | c b a 0 0 0 0 0 0 |, cokernel {4}|
|                                      {2} | 0 0 0 c b a 0 0 0 |           {4}|
|                                      {2} | 0 0 0 0 0 0 c b a |           {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {4}|
|                                                                          {5}|
|                                                                          {5}|
|                                                                          {5}|
|                                                                          {5}|
|                                                                          {5}|
|                                                                          {5}|
|                                                                          {5}|
|                                                                          {5}|
|                                                                          {5}|
|                                                                          {6}|
|     ------------------------------------------------------------------------|
|     | 2c 0    0    |}                                                       |
|     | 0  0    -b   |                                                        |
|     | 0  a    0    |                                                        |
|     | -b 0    50c  |                                                        |
|     | b  0    -50c |                                                        |
|     | a  50c  0    |                                                        |
|     | 0  -50b -50a |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | -b 0    -50c |                                                        |
|     | -b 0    -50c |                                                        |
|     | 0  c    0    |                                                        |
|     | a  50c  0    |                                                        |
|     | a  -50c 0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  -b   0    |                                                        |
|     | 0  50b  50a  |                                                        |
|     | 0  0    a    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|     | 0  0    0    |                                                        |
|                                                                             |
|o8 : List                                                                    |
+-----------------------------------------------------------------------------+

Notice that since the tensor square splits outside of characteristic 2, the
symmetric power picks up the even degree homology and the exterior square picks
up the odd homology. In characteristic 2 this changes:

+-------------------------------------------------------------------------+
|i9 : Q = ZZ/2[a,b,c]                                                     |
|                                                                         |
|o9 = Q                                                                   |
|                                                                         |
|o9 : PolynomialRing                                                      |
+-------------------------------------------------------------------------+
|i10 : K = koszulComplex vars Q                                           |
|                                                                         |
|       1      3      3      1                                            |
|o10 = Q  <-- Q  <-- Q  <-- Q                                             |
|                                                                         |
|      0      1      2      3                                             |
|                                                                         |
|o10 : Complex                                                            |
+-------------------------------------------------------------------------+
|i11 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3   |
| -- .283736s elapsed                                                     |
+-------------------------------------------------------------------------+
|i12 : isWellDefined phi                                                  |
|                                                                         |
|o12 = true                                                               |
+-------------------------------------------------------------------------+
|i13 : isCommutative phi                                                  |
|                                                                         |
|o13 = true                                                               |
+-------------------------------------------------------------------------+
|i14 : for i to 2 list prune HH_i source phi                              |
|                                                                         |
|o14 = {0, cokernel {1} | a 0 b 0 0 c |, cokernel {2} | b a 0 c 0 0 0 c |}|
|                   {1} | 0 b a 0 c 0 |           {2} | 0 0 c b a 0 0 0 | |
|                   {1} | 0 0 0 c b a |           {2} | 0 0 0 0 0 c b a | |
|                                                                         |
|o14 : List                                                               |
+-------------------------------------------------------------------------+
|i15 : for i to 2 list prune HH_i (coker phi)                             |
|                                                                         |
|o15 = {cokernel | c b a |, 0, cokernel {2} | b a 0 c 0 0 0 c |}          |
|                                       {2} | 0 0 c b a 0 0 0 |           |
|                                       {2} | 0 0 0 0 0 c b a |           |
|                                                                         |
|o15 : List                                                               |
+-------------------------------------------------------------------------+

See also
========

  * symmetricQuotient -- computes the image of the
    surjection from the simplicial tensor product onto the second symmetric
    power of a simplicial module
  * extPower -- compute the Dold-Puppe extension of the
    exterior power functor to simplicial modules
  * simplicialTensor -- compute the simplicial tensor
    product and cache direct sum indices for easy access

Ways to use exteriorInclusion:
==============================

  * "exteriorInclusion(Complex)"
  * "exteriorInclusion(Complex,ZZ)"
  * "exteriorInclusion(Module)"
  * "exteriorInclusion(SimplicialModule)"

For the programmer
==================

The object exteriorInclusion, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3906:0.

extPower -- compute the Dold-Puppe extension of the exterior power functor to simplicial modules
************************************************************************************************

  * Usage:
        extPower(d, S, Options => {Degeneracy => false, TopDegree => null})
  * Inputs:
      * d, an integer (Macaulay2Doc), An integer representing the
        degree of the exterior power.
      * S, a Simplicial Module, A simplicial module to
        which the exterior power functor extension is applied.
  * Optional inputs (Macaulay2Doc),:
      * Degeneracy, => a
        Boolean value (Macaulay2Doc), default value false,
        Default value is false. Indicates whether to include degeneracy maps in
        the construction.
      * TopDegree, => an
        integer (Macaulay2Doc), default value null, Default is null.
        If provided, specifies the top degree for the construction of the
        simplicial module.
  * Outputs:
      * a Simplicial Module, The Dold-Puppe extension
        of the exterior power functor applied to the simplicial module S.

Description
===========

This function computes the Dold-Puppe extension of the exterior power functor
to the category of chain complexes. It can be applied to a simplicial module, a
complex, or maps thereof. This method is equivalent to using schurMap,
for the partition $(1, 1, \dots , 1)$, but is generally much faster.

+-----------------------------------------------------------------------------+
|i1 : Q = ZZ/2[a,b]                                                           |
|                                                                             |
|o1 = Q                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : K = koszulComplex vars Q                                                |
|                                                                             |
|      1      2      1                                                        |
|o2 = Q  <-- Q  <-- Q                                                         |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o2 : Complex                                                                 |
+-----------------------------------------------------------------------------+
|i3 : w3K = elapsedTime prune extPower(3, K)                                  |
| -- 7.14048s elapsed                                                         |
|                                                                             |
|      1      18      63      91      60      15                              |
|o3 = Q  <-- Q   <-- Q   <-- Q   <-- Q   <-- Q                                |
|                                                                             |
|     1      2       3       4       5       6                                |
|                                                                             |
|o3 : Complex                                                                 |
+-----------------------------------------------------------------------------+
|i4 : (minimize w3K).dd                                                       |
|                                                                             |
|          3                              7                                   |
|o4 = 2 : Q  <-------------------------- Q  : 3                               |
|               {2} | a b 0 0 0 0 b2 |                                        |
|               {2} | 0 a b 0 a b ab |                                        |
|               {2} | 0 0 a b 0 0 a2 |                                        |
|                                                                             |
|          7                             6                                    |
|     3 : Q  <------------------------- Q  : 4                                |
|               {3} | b 0 0 0 b2 0  |                                         |
|               {3} | a 0 0 0 0  b2 |                                         |
|               {3} | 0 0 b 0 a2 0  |                                         |
|               {3} | 0 0 a 0 0  a2 |                                         |
|               {3} | a b 0 b 0  0  |                                         |
|               {3} | 0 a b a 0  0  |                                         |
|               {4} | 0 0 0 0 a  b  |                                         |
|                                                                             |
|          6                      3                                           |
|     4 : Q  <------------------ Q  : 5                                       |
|               {4} | 0 0 b2 |                                                |
|               {4} | a b ab |                                                |
|               {4} | 0 0 a2 |                                                |
|               {4} | a b 0  |                                                |
|               {5} | 0 0 b  |                                                |
|               {5} | 0 0 a  |                                                |
|                                                                             |
|          3                 1                                                |
|     5 : Q  <------------- Q  : 6                                            |
|               {5} | b |                                                     |
|               {5} | a |                                                     |
|               {6} | 0 |                                                     |
|                                                                             |
|o4 : ComplexMap                                                              |
+-----------------------------------------------------------------------------+
|i5 : prune HH w3K                                                            |
|                                                                             |
|o5 = cokernel {2} | b a 0 0 0 0 | <-- cokernel {4} | b a | <-- cokernel {4} ||
|              {2} | 0 0 b a 0 0 |                                            |
|              {2} | 0 0 0 0 b a |     3                        4             |
|                                                                             |
|     2                                                                       |
|                                                                             |
|o5 : Complex                                                                 |
|-----------------------------------------------------------------------------|
|b a |                                                                        |
+-----------------------------------------------------------------------------+

This method is also implemented in a functorial way:

+-----------------------------------------------------------------------------+
|i6 : F = freeResolution( (ideal vars Q)^2)                                   |
|                                                                             |
|      1      3      2                                                        |
|o6 = Q  <-- Q  <-- Q                                                         |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o6 : Complex                                                                 |
+-----------------------------------------------------------------------------+
|i7 : phi = extend(K, F, id_(K_0))                                            |
|                                                                             |
|          1             1                                                    |
|o7 = 0 : Q  <--------- Q  : 0                                                |
|               | 1 |                                                         |
|                                                                             |
|          2                     3                                            |
|     1 : Q  <----------------- Q  : 1                                        |
|               {1} | a b 0 |                                                 |
|               {1} | 0 0 b |                                                 |
|                                                                             |
|          1                   2                                              |
|     2 : Q  <--------------- Q  : 2                                          |
|               {2} | 0 b |                                                   |
|                                                                             |
|o7 : ComplexMap                                                              |
+-----------------------------------------------------------------------------+
|i8 : f = elapsedTime prune extPower(2, phi)                                  |
| -- .50202s elapsed                                                          |
|                                                                             |
|          3                             6                                    |
|o8 = 1 : Q  <------------------------- Q  : 1                                |
|               {1} | a b 0 0 0  0  |                                         |
|               {1} | 0 0 0 b 0  0  |                                         |
|               {2} | 0 0 0 0 ab b2 |                                         |
|                                                                             |
|          9                                                                  |
|     2 : Q  <----------------------------------------------------------------|
|               {2} | 0 0 a2 0 b2 ab 0  0  0  0  0  0 0 0 0 0 0 0  0  0  0  0 |
|               {2} | 0 0 0  0 0  0  ab b2 0  0  0  0 0 0 0 0 0 0  0  0  0  0 |
|               {2} | 0 0 0  0 0  0  0  0  b2 0  0  0 0 0 0 0 0 0  0  0  0  0 |
|               {2} | 0 0 0  0 0  0  0  0  0  ab b2 0 0 0 0 0 0 0  0  0  0  0 |
|               {2} | 0 b 0  0 0  0  0  0  0  0  0  0 0 0 0 0 0 0  0  0  0  0 |
|               {3} | 0 0 0  0 0  0  0  0  0  0  0  0 0 0 0 0 0 ab b2 0  0  0 |
|               {3} | 0 0 0  0 0  0  0  0  0  0  0  0 0 0 0 0 0 0  0  b2 0  0 |
|               {3} | 0 0 0  0 0  0  0  0  0  0  0  0 0 0 0 0 0 0  0  0  ab b2|
|               {3} | 0 0 0  0 0  0  0  0  0  0  0  0 0 0 0 0 0 0  0  0  0  0 |
|                                                                             |
|          9                                                                  |
|     3 : Q  <----------------------------------------------------------------|
|               {3} | 0 0 0 0 0 0 0 0 0 ab b2 0  0  0  0  0  0  0  0 0 0  0 0 |
|               {3} | 0 0 0 0 0 0 0 0 0 0  0  b2 0  0  0  0  0  0  0 0 0  0 0 |
|               {3} | 0 0 0 0 0 0 0 0 0 0  0  0  ab b2 0  0  0  0  0 0 0  0 0 |
|               {3} | 0 0 0 0 0 0 0 0 0 0  0  0  0  0  b2 0  0  0  0 0 0  0 0 |
|               {3} | 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  ab b2 0  0 0 0  0 0 |
|               {3} | 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  b2 0 0 0  0 0 |
|               {4} | 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 b2 0 0 |
|               {4} | 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0  0 0 |
|               {4} | 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0  0 0 |
|                                                                             |
|          3                                          12                      |
|     4 : Q  <-------------------------------------- Q   : 4                  |
|               {4} | 0 0 0 0 0 0 0 b2 0 0  0 0  |                            |
|               {4} | 0 0 0 0 0 0 0 0  0 b2 0 0  |                            |
|               {4} | 0 0 0 0 0 0 0 0  0 0  0 b2 |                            |
|                                                                             |
|o8 : ComplexMap                                                              |
|-----------------------------------------------------------------------------|
|           24                                                                |
|--------- Q   : 2                                                            |
| 0  0 |                                                                      |
| 0  0 |                                                                      |
| 0  0 |                                                                      |
| 0  0 |                                                                      |
| 0  0 |                                                                      |
| 0  0 |                                                                      |
| 0  0 |                                                                      |
| 0  0 |                                                                      |
| b2 0 |                                                                      |
|                                                                             |
|                     30                                                      |
|------------------- Q   : 3                                                  |
|0 0 0 0  0 0  0 |                                                            |
|0 0 0 0  0 0  0 |                                                            |
|0 0 0 0  0 0  0 |                                                            |
|0 0 0 0  0 0  0 |                                                            |
|0 0 0 0  0 0  0 |                                                            |
|0 0 0 0  0 0  0 |                                                            |
|0 0 0 0  0 0  0 |                                                            |
|0 0 0 b2 0 0  0 |                                                            |
|0 0 0 0  0 b2 0 |                                                            |
+-----------------------------------------------------------------------------+
|i9 : prune HH f                                                              |
|                                                                             |
|o9 = 1 : cokernel {1} | a 0 b | <----------------- cokernel {2} | b 0 b2 a2  |
|                  {1} | 0 b a |    {1} | 0 b 0 |            {2} | a b 0  0   |
|                                   {1} | 0 0 0 |            {2} | 0 a 0  0   |
|                                                                             |
|o9 : ComplexMap                                                              |
|-----------------------------------------------------------------------------|
|0  | : 1                                                                     |
|0  |                                                                         |
|b2 |                                                                         |
+-----------------------------------------------------------------------------+

Since this method runs significantly faster than using schurMap, we can compute
the full exterior power complex of a longer complex:

+-----------------------------------------------------------------------------+
|i10 : Q = ZZ/2[a..c];                                                        |
+-----------------------------------------------------------------------------+
|i11 : K = koszulComplex vars Q                                               |
|                                                                             |
|       1      3      3      1                                                |
|o11 = Q  <-- Q  <-- Q  <-- Q                                                 |
|                                                                             |
|      0      1      2      3                                                 |
|                                                                             |
|o11 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i12 : w2K = prune extPower(2, K)                                             |
|                                                                             |
|       6      33      73      81      45      10                             |
|o12 = Q  <-- Q   <-- Q   <-- Q   <-- Q   <-- Q                               |
|                                                                             |
|      1      2       3       4       5       6                               |
|                                                                             |
|o12 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i13 : (minimize w2K).dd                                                      |
|                                                                             |
|           3                                 9                               |
|o13 = 1 : Q  <----------------------------- Q  : 2                           |
|                {1} | a b 0 c 0 0 b c 0 |                                    |
|                {1} | 0 a b 0 c 0 a 0 c |                                    |
|                {1} | 0 0 0 a b c 0 a b |                                    |
|                                                                             |
|           9                                            13                   |
|      2 : Q  <---------------------------------------- Q   : 3               |
|                {2} | 0 0 0 0 0 0 0 0 0 0 b2 c2 0  |                         |
|                {2} | a b 0 0 c 0 c 0 0 0 ab 0  0  |                         |
|                {2} | 0 0 0 0 0 0 0 0 0 0 a2 0  c2 |                         |
|                {2} | 0 0 b a 0 c b 0 0 0 0  ac 0  |                         |
|                {2} | 0 0 a 0 a 0 0 b c 0 0  0  bc |                         |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0  a2 b2 |                         |
|                {2} | a b c 0 0 0 0 0 0 c 0  0  0  |                         |
|                {2} | 0 0 0 a b c 0 0 0 b 0  0  0  |                         |
|                {2} | 0 0 0 0 0 0 a b c a 0  0  0  |                         |
|                                                                             |
|           13                                     10                         |
|      3 : Q   <--------------------------------- Q   : 4                     |
|                 {3} | b c 0 0 0 0 c 0 0 bc2 |                               |
|                 {3} | a 0 0 c 0 0 0 c 0 ac2 |                               |
|                 {3} | 0 a 0 b 0 0 0 0 c 0   |                               |
|                 {3} | 0 b c 0 0 0 b 0 0 0   |                               |
|                 {3} | 0 a 0 0 c 0 0 b 0 abc |                               |
|                 {3} | 0 0 a 0 b 0 0 0 b ab2 |                               |
|                 {3} | 0 0 0 b c 0 a 0 0 0   |                               |
|                 {3} | 0 0 0 a 0 c 0 a 0 0   |                               |
|                 {3} | 0 0 0 0 a b 0 0 a 0   |                               |
|                 {3} | 0 0 0 0 0 0 a b c 0   |                               |
|                 {4} | 0 0 0 0 0 0 0 0 0 c2  |                               |
|                 {4} | 0 0 0 0 0 0 0 0 0 b2  |                               |
|                 {4} | 0 0 0 0 0 0 0 0 0 a2  |                               |
|                                                                             |
|           10                        4                                       |
|      4 : Q   <-------------------- Q  : 5                                   |
|                 {4} | 0 0 0 c2 |                                            |
|                 {4} | b c 0 bc |                                            |
|                 {4} | 0 0 0 b2 |                                            |
|                 {4} | a 0 c ac |                                            |
|                 {4} | 0 a b ab |                                            |
|                 {4} | 0 0 0 a2 |                                            |
|                 {4} | b c 0 0  |                                            |
|                 {4} | a 0 c 0  |                                            |
|                 {4} | 0 a b 0  |                                            |
|                 {6} | 0 0 0 0  |                                            |
|                                                                             |
|           4                 1                                               |
|      5 : Q  <------------- Q  : 6                                           |
|                {5} | c |                                                    |
|                {5} | b |                                                    |
|                {5} | a |                                                    |
|                {6} | 0 |                                                    |
|                                                                             |
|o13 : ComplexMap                                                             |
+-----------------------------------------------------------------------------+
|i14 : prune HH w2K --has more interesting homology than                      |
|                                                                             |
|o14 = cokernel {1} | a 0 b 0 0 c | <-- cokernel {2} | b                      |
|               {1} | 0 b a 0 c 0 |              {2} | 0                      |
|               {1} | 0 0 0 c b a |              {2} | 0                      |
|                                                                             |
|      1                                2                                     |
|                                                                             |
|o14 : Complex                                                                |
|-----------------------------------------------------------------------------|
|in the nonmodular case                                                       |
|                                                                             |
|a 0 c 0 0 0 c | <-- cokernel {3} | c b a |                                   |
|0 c b a 0 0 0 |                                                              |
|0 0 0 0 c b a |     3                                                        |
+-----------------------------------------------------------------------------+
|i15 : needsPackage "ChainComplexOperations"                                  |
|                                                                             |
|o15 = ChainComplexOperations                                                 |
|                                                                             |
|o15 : Package                                                                |
+-----------------------------------------------------------------------------+
|i16 : Q = ZZ/3[a..c]                                                         |
|                                                                             |
|o16 = Q                                                                      |
|                                                                             |
|o16 : PolynomialRing                                                         |
+-----------------------------------------------------------------------------+
|i17 : K = koszulComplex vars Q                                               |
|                                                                             |
|       1      3      3      1                                                |
|o17 = Q  <-- Q  <-- Q  <-- Q                                                 |
|                                                                             |
|      0      1      2      3                                                 |
|                                                                             |
|o17 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i18 : prune HH wedge2 K --notice: homology concentrated in odd degrees       |
|                                                                             |
|o18 = cokernel {1} | c b a 0 0 0 0 0 0 | <-- 0 <-- cokernel {3} | c b a |    |
|               {1} | 0 0 0 c b a 0 0 0 |                                     |
|               {1} | 0 0 0 0 0 0 c b a |     2     3                         |
|                                                                             |
|      1                                                                      |
|                                                                             |
|o18 : Complex                                                                |
+-----------------------------------------------------------------------------+

Caveat
======

This method may take a very long time to run if the user inputs a large/long
complex. Use the option TopDegree => d to only run the computation up to a
certain degree.

See also
========

  * schurMap -- construct the Dold-Puppe extension of the
    Schur functor
  * exteriorInclusion -- computes the image of the
    2nd exterior power into the tensor product
  * symmetricQuotient -- computes the image of the
    surjection from the simplicial tensor product onto the second symmetric
    power of a simplicial module
  * simplicialTensor -- compute the simplicial tensor
    product and cache direct sum indices for easy access

Ways to use extPower:
=====================

  * "extPower(ZZ,Complex)"
  * "extPower(ZZ,ComplexMap)"
  * "extPower(ZZ,SimplicialModule)"
  * "extPower(ZZ,SimplicialModuleMap)"

For the programmer
==================

The object extPower, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3612:0.

extPower(...,Degeneracy=>...) -- whether to compute degeneracy maps
*******************************************************************

  * Usage:
        extPower(..., Degeneracy => true)

Description
===========

If Degeneracy => true, the degeneracy maps are also computed alongside the face
maps. By default, degeneracy maps are not computed.

See also
========

  * extPower -- compute the Dold-Puppe extension of the
    exterior power functor to simplicial modules

Functions with optional argument named Degeneracy:
==================================================

  * extPower(...,Degeneracy=>...)
    -- whether to compute
    degeneracy maps
  * schurMap(...,Degeneracy=>...)
    -- whether to compute
    degeneracy maps
  * simplicialModule(...,Degeneracy=>...)
    -- whether to
    compute degeneracy maps
  * simplicialTensor(...,Degeneracy=>...)
    -- whether to
    compute degeneracy maps

Further information
===================

  * Default value: false (Macaulay2Doc),
  * Function: extPower, -- compute the Dold-Puppe extension of
    the exterior power functor to simplicial modules
  * Option key: Degeneracy, -- the class of all
    simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:4040:0.

extPower(...,TopDegree=>...) -- top degree for the construction
***************************************************************

  * Usage:
        extPower(..., TopDegree => n)

Description
===========

Specifies the top simplicial degree for the resulting simplicial module.

See also
========

  * extPower -- compute the Dold-Puppe extension of the
    exterior power functor to simplicial modules

Functions with optional argument named TopDegree:
=================================================

  * extPower(...,TopDegree=>...)
    -- top degree for the
    construction
  * schurMap(...,TopDegree=>...)
    -- top degree for the
    construction
  * simplicialTensor(...,TopDegree=>...)
    -- top degree for
    the construction

Further information
===================

  * Default value: null
  * Function: extPower, -- compute the Dold-Puppe extension of
    the exterior power functor to simplicial modules
  * Option key: TopDegree, -- construct the Dold-Puppe
    extension of the Schur functor
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:4054:0.

face/degeneracy maps of a simplicial module -- access the face and degeneracy maps of a simplicial module
*********************************************************************************************************

  * Usage:
        dd^C
        ss^C
  * Inputs:
      * C, a Simplicial Module,
  * Outputs:
      * a Map of Simplicial Modules, a map of
        degree -1 or 1, respectively

A simplicial module is a sequence of modules connected by homomorphisms, called
face/degeneracy maps, such that these maps satisfy a set of identities known as
the simplicial identities. The face maps are accessed via the symbol dd (the
same symbol used for the differentials of a complex), and the degeneracy maps
via ss, after the conventional letter $s$ (or $\sigma$) used for degeneracy
maps in the literature.

More precisely we should have the following equalities: 1. For face maps: \[
d_j d_i = d_i d_{j-1} \text{ for } 0 \leq i < j \leq n \] 2. For face and
degeneracy maps: \[ d_i s_j = s_{j-1} d_i \text{ for } i < j \] \[ d_j s_j =
\text{id} \] \[ d_{j+1} s_j = \text{id} \] \[ d_k s_j = s_j d_{k-1} \text{ for
} k > j+1 \] 3. For degeneracy maps: \[ s_j s_i = s_i s_{j+1} \text{ for } i
\leq j \]

The face/degeneracy maps are considered as double indexed in this package. This
is because for a fixed integer $n$, the module $S_n$ comes equipped with
$(n+1)$ face and degeneracy maps: $$d_{n,i} : S_n \to S_{n-1}, \quad \text{and}
\quad s_{n,i} : S_n \to S_{n+1}.$$ The above maps can be accessed as follows:

+-----------------------------------------------------------------------------+
|i1 : R = QQ[a..d];                                                           |
+-----------------------------------------------------------------------------+
|i2 : I = ideal(a*d-b*c, b^2-a*c, c^2-b*d);                                   |
|                                                                             |
|o2 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i3 : C = simplicialModule(freeResolution(R^1/I), 4, Degeneracy => true)      |
|                                                                             |
|      1      4      9      16      25                                        |
|o3 = R  <-- R  <-- R  <-- R   <-- R  <-- ...                                 |
|                                                                             |
|     0      1      2      3       4                                          |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : isWellDefined C                                                         |
|                                                                             |
|o4 = true                                                                    |
+-----------------------------------------------------------------------------+
|i5 : dd^C                                                                    |
|                                                                             |
|               1                               4                             |
|o5 = (0, 0) : R  <--------------------------- R  : (1, 0)                    |
|                    | 1 b2-ac bc-ad c2-bd |                                  |
|                                                                             |
|               1                   4                                         |
|     (0, 1) : R  <--------------- R  : (1, 1)                                |
|                    | 1 0 0 0 |                                              |
|                                                                             |
|               4                                               9             |
|     (1, 0) : R  <------------------------------------------- R  : (2, 0)    |
|                    {0} | 1 b2-ac bc-ad c2-bd 0 0 0 0  0  |                  |
|                    {2} | 0 0     0     0     1 0 0 -c d  |                  |
|                    {2} | 0 0     0     0     0 1 0 b  -c |                  |
|                    {2} | 0 0     0     0     0 0 1 -a b  |                  |
|                                                                             |
|               4                                 9                           |
|     (1, 1) : R  <----------------------------- R  : (2, 1)                  |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 1 0 0 1 0 0 0 0 |                                |
|                    {2} | 0 0 1 0 0 1 0 0 0 |                                |
|                    {2} | 0 0 0 1 0 0 1 0 0 |                                |
|                                                                             |
|               4                                 9                           |
|     (1, 2) : R  <----------------------------- R  : (2, 2)                  |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 1 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 1 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 1 0 0 0 0 0 |                                |
|                                                                             |
|               9                                                             |
|     (2, 0) : R  <-----------------------------------------------------------|
|                    {0} | 1 b2-ac bc-ad c2-bd 0 0 0 0 0 0 0  0  0  0  0 0 |  |
|                    {2} | 0 0     0     0     1 0 0 0 0 0 -c d  0  0  0 0 |  |
|                    {2} | 0 0     0     0     0 1 0 0 0 0 b  -c 0  0  0 0 |  |
|                    {2} | 0 0     0     0     0 0 1 0 0 0 -a b  0  0  0 0 |  |
|                    {2} | 0 0     0     0     0 0 0 1 0 0 0  0  -c d  0 0 |  |
|                    {2} | 0 0     0     0     0 0 0 0 1 0 0  0  b  -c 0 0 |  |
|                    {2} | 0 0     0     0     0 0 0 0 0 1 0  0  -a b  0 0 |  |
|                    {3} | 0 0     0     0     0 0 0 0 0 0 0  0  0  0  1 0 |  |
|                    {3} | 0 0     0     0     0 0 0 0 0 0 0  0  0  0  0 1 |  |
|                                                                             |
|               9                                               16            |
|     (2, 1) : R  <------------------------------------------- R   : (3, 1)   |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |                  |
|                                                                             |
|               9                                               16            |
|     (2, 2) : R  <------------------------------------------- R   : (3, 2)   |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |                  |
|                                                                             |
|               9                                               16            |
|     (2, 3) : R  <------------------------------------------- R   : (3, 3)   |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                  |
|                                                                             |
|               16                                                            |
|     (3, 0) : R   <----------------------------------------------------------|
|                     {0} | 1 b2-ac bc-ad c2-bd 0 0 0 0 0 0 0 0 0 0  0  0  0  |
|                     {2} | 0 0     0     0     1 0 0 0 0 0 0 0 0 -c d  0  0  |
|                     {2} | 0 0     0     0     0 1 0 0 0 0 0 0 0 b  -c 0  0  |
|                     {2} | 0 0     0     0     0 0 1 0 0 0 0 0 0 -a b  0  0  |
|                     {2} | 0 0     0     0     0 0 0 1 0 0 0 0 0 0  0  -c d  |
|                     {2} | 0 0     0     0     0 0 0 0 1 0 0 0 0 0  0  b  -c |
|                     {2} | 0 0     0     0     0 0 0 0 0 1 0 0 0 0  0  -a b  |
|                     {2} | 0 0     0     0     0 0 0 0 0 0 1 0 0 0  0  0  0  |
|                     {2} | 0 0     0     0     0 0 0 0 0 0 0 1 0 0  0  0  0  |
|                     {2} | 0 0     0     0     0 0 0 0 0 0 0 0 1 0  0  0  0  |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  |
|                                                                             |
|               16                                                            |
|     (3, 1) : R   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
|                                                                             |
|               16                                                            |
|     (3, 2) : R   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |
|                                                                             |
|               16                                                            |
|     (3, 3) : R   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |
|                                                                             |
|               16                                                            |
|     (3, 4) : R   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
|                                                                             |
|o5 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|  16                                                                         |
| R   : (3, 0)                                                                |
|                                                                             |
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|                                                                             |
|                                                                             |
|                       25                                                    |
|--------------------- R   : (4, 0)                                           |
|0  0  0 0 0 0 0 0 |                                                          |
|0  0  0 0 0 0 0 0 |                                                          |
|0  0  0 0 0 0 0 0 |                                                          |
|0  0  0 0 0 0 0 0 |                                                          |
|0  0  0 0 0 0 0 0 |                                                          |
|0  0  0 0 0 0 0 0 |                                                          |
|0  0  0 0 0 0 0 0 |                                                          |
|-c d  0 0 0 0 0 0 |                                                          |
|b  -c 0 0 0 0 0 0 |                                                          |
|-a b  0 0 0 0 0 0 |                                                          |
|0  0  1 0 0 0 0 0 |                                                          |
|0  0  0 1 0 0 0 0 |                                                          |
|0  0  0 0 1 0 0 0 |                                                          |
|0  0  0 0 0 1 0 0 |                                                          |
|0  0  0 0 0 0 1 0 |                                                          |
|0  0  0 0 0 0 0 1 |                                                          |
|                                                                             |
|     25                                                                      |
|--- R   : (4, 1)                                                             |
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|                                                                             |
|     25                                                                      |
|--- R   : (4, 2)                                                             |
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|                                                                             |
|     25                                                                      |
|--- R   : (4, 3)                                                             |
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|                                                                             |
|     25                                                                      |
|--- R   : (4, 4)                                                             |
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+-----------------------------------------------------------------------------+
|i6 : C.dd                                                                    |
|                                                                             |
|               1                               4                             |
|o6 = (0, 0) : R  <--------------------------- R  : (1, 0)                    |
|                    | 1 b2-ac bc-ad c2-bd |                                  |
|                                                                             |
|               1                   4                                         |
|     (0, 1) : R  <--------------- R  : (1, 1)                                |
|                    | 1 0 0 0 |                                              |
|                                                                             |
|               4                                               9             |
|     (1, 0) : R  <------------------------------------------- R  : (2, 0)    |
|                    {0} | 1 b2-ac bc-ad c2-bd 0 0 0 0  0  |                  |
|                    {2} | 0 0     0     0     1 0 0 -c d  |                  |
|                    {2} | 0 0     0     0     0 1 0 b  -c |                  |
|                    {2} | 0 0     0     0     0 0 1 -a b  |                  |
|                                                                             |
|               4                                 9                           |
|     (1, 1) : R  <----------------------------- R  : (2, 1)                  |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 1 0 0 1 0 0 0 0 |                                |
|                    {2} | 0 0 1 0 0 1 0 0 0 |                                |
|                    {2} | 0 0 0 1 0 0 1 0 0 |                                |
|                                                                             |
|               4                                 9                           |
|     (1, 2) : R  <----------------------------- R  : (2, 2)                  |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 1 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 1 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 1 0 0 0 0 0 |                                |
|                                                                             |
|               9                                                             |
|     (2, 0) : R  <-----------------------------------------------------------|
|                    {0} | 1 b2-ac bc-ad c2-bd 0 0 0 0 0 0 0  0  0  0  0 0 |  |
|                    {2} | 0 0     0     0     1 0 0 0 0 0 -c d  0  0  0 0 |  |
|                    {2} | 0 0     0     0     0 1 0 0 0 0 b  -c 0  0  0 0 |  |
|                    {2} | 0 0     0     0     0 0 1 0 0 0 -a b  0  0  0 0 |  |
|                    {2} | 0 0     0     0     0 0 0 1 0 0 0  0  -c d  0 0 |  |
|                    {2} | 0 0     0     0     0 0 0 0 1 0 0  0  b  -c 0 0 |  |
|                    {2} | 0 0     0     0     0 0 0 0 0 1 0  0  -a b  0 0 |  |
|                    {3} | 0 0     0     0     0 0 0 0 0 0 0  0  0  0  1 0 |  |
|                    {3} | 0 0     0     0     0 0 0 0 0 0 0  0  0  0  0 1 |  |
|                                                                             |
|               9                                               16            |
|     (2, 1) : R  <------------------------------------------- R   : (3, 1)   |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |                  |
|                                                                             |
|               9                                               16            |
|     (2, 2) : R  <------------------------------------------- R   : (3, 2)   |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |                  |
|                                                                             |
|               9                                               16            |
|     (2, 3) : R  <------------------------------------------- R   : (3, 3)   |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                  |
|                                                                             |
|               16                                                            |
|     (3, 0) : R   <----------------------------------------------------------|
|                     {0} | 1 b2-ac bc-ad c2-bd 0 0 0 0 0 0 0 0 0 0  0  0  0  |
|                     {2} | 0 0     0     0     1 0 0 0 0 0 0 0 0 -c d  0  0  |
|                     {2} | 0 0     0     0     0 1 0 0 0 0 0 0 0 b  -c 0  0  |
|                     {2} | 0 0     0     0     0 0 1 0 0 0 0 0 0 -a b  0  0  |
|                     {2} | 0 0     0     0     0 0 0 1 0 0 0 0 0 0  0  -c d  |
|                     {2} | 0 0     0     0     0 0 0 0 1 0 0 0 0 0  0  b  -c |
|                     {2} | 0 0     0     0     0 0 0 0 0 1 0 0 0 0  0  -a b  |
|                     {2} | 0 0     0     0     0 0 0 0 0 0 1 0 0 0  0  0  0  |
|                     {2} | 0 0     0     0     0 0 0 0 0 0 0 1 0 0  0  0  0  |
|                     {2} | 0 0     0     0     0 0 0 0 0 0 0 0 1 0  0  0  0  |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0  0  0  |
|                                                                             |
|               16                                                            |
|     (3, 1) : R   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
|                                                                             |
|               16                                                            |
|     (3, 2) : R   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |
|                                                                             |
|               16                                                            |
|     (3, 3) : R   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |
|                                                                             |
|               16                                                            |
|     (3, 4) : R   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
|                                                                             |
|o6 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|  16                                                                         |
| R   : (3, 0)                                                                |
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|                                                                             |
|                       25                                                    |
|--------------------- R   : (4, 0)                                           |
|0  0  0 0 0 0 0 0 |                                                          |
|0  0  0 0 0 0 0 0 |                                                          |
|0  0  0 0 0 0 0 0 |                                                          |
|0  0  0 0 0 0 0 0 |                                                          |
|0  0  0 0 0 0 0 0 |                                                          |
|0  0  0 0 0 0 0 0 |                                                          |
|0  0  0 0 0 0 0 0 |                                                          |
|-c d  0 0 0 0 0 0 |                                                          |
|b  -c 0 0 0 0 0 0 |                                                          |
|-a b  0 0 0 0 0 0 |                                                          |
|0  0  1 0 0 0 0 0 |                                                          |
|0  0  0 1 0 0 0 0 |                                                          |
|0  0  0 0 1 0 0 0 |                                                          |
|0  0  0 0 0 1 0 0 |                                                          |
|0  0  0 0 0 0 1 0 |                                                          |
|0  0  0 0 0 0 0 1 |                                                          |
|                                                                             |
|     25                                                                      |
|--- R   : (4, 1)                                                             |
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|     25                                                                      |
|--- R   : (4, 2)                                                             |
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|     25                                                                      |
|--- R   : (4, 3)                                                             |
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|     25                                                                      |
|--- R   : (4, 4)                                                             |
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+-----------------------------------------------------------------------------+
|i7 : ss^C                                                                    |
|                                                                             |
|               4                 1                                           |
|o7 = (1, 0) : R  <------------- R  : (0, 0)                                  |
|                    {0} | 1 |                                                |
|                    {2} | 0 |                                                |
|                    {2} | 0 |                                                |
|                    {2} | 0 |                                                |
|                                                                             |
|               9                       4                                     |
|     (2, 0) : R  <------------------- R  : (1, 0)                            |
|                    {0} | 1 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 1 0 0 |                                          |
|                    {2} | 0 0 1 0 |                                          |
|                    {2} | 0 0 0 1 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                                                                             |
|               9                       4                                     |
|     (2, 1) : R  <------------------- R  : (1, 1)                            |
|                    {0} | 1 0 0 0 |                                          |
|                    {2} | 0 1 0 0 |                                          |
|                    {2} | 0 0 1 0 |                                          |
|                    {2} | 0 0 0 1 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                                                                             |
|               16                                 9                          |
|     (3, 0) : R   <----------------------------- R  : (2, 0)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 1 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 1 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 1 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 1 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 1 |                               |
|                                                                             |
|               16                                 9                          |
|     (3, 1) : R   <----------------------------- R  : (2, 1)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 1 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 1 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 1 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 1 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 1 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                                                                             |
|               16                                 9                          |
|     (3, 2) : R   <----------------------------- R  : (2, 2)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 1 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 1 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 1 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 1 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 1 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                                                                             |
|               25                                               16           |
|     (4, 0) : R   <------------------------------------------- R   : (3, 0)  |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                 |
|                                                                             |
|               25                                               16           |
|     (4, 1) : R   <------------------------------------------- R   : (3, 1)  |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                 |
|                                                                             |
|               25                                               16           |
|     (4, 2) : R   <------------------------------------------- R   : (3, 2)  |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                                                                             |
|               25                                               16           |
|     (4, 3) : R   <------------------------------------------- R   : (3, 3)  |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+
|i8 : C.ss                                                                    |
|                                                                             |
|               4                 1                                           |
|o8 = (1, 0) : R  <------------- R  : (0, 0)                                  |
|                    {0} | 1 |                                                |
|                    {2} | 0 |                                                |
|                    {2} | 0 |                                                |
|                    {2} | 0 |                                                |
|                                                                             |
|               9                       4                                     |
|     (2, 0) : R  <------------------- R  : (1, 0)                            |
|                    {0} | 1 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 1 0 0 |                                          |
|                    {2} | 0 0 1 0 |                                          |
|                    {2} | 0 0 0 1 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                                                                             |
|               9                       4                                     |
|     (2, 1) : R  <------------------- R  : (1, 1)                            |
|                    {0} | 1 0 0 0 |                                          |
|                    {2} | 0 1 0 0 |                                          |
|                    {2} | 0 0 1 0 |                                          |
|                    {2} | 0 0 0 1 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                                                                             |
|               16                                 9                          |
|     (3, 0) : R   <----------------------------- R  : (2, 0)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 1 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 1 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 1 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 1 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 1 |                               |
|                                                                             |
|               16                                 9                          |
|     (3, 1) : R   <----------------------------- R  : (2, 1)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 1 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 1 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 1 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 1 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 1 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                                                                             |
|               16                                 9                          |
|     (3, 2) : R   <----------------------------- R  : (2, 2)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 1 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 1 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 1 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 1 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 1 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                                                                             |
|               25                                               16           |
|     (4, 0) : R   <------------------------------------------- R   : (3, 0)  |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                 |
|                                                                             |
|               25                                               16           |
|     (4, 1) : R   <------------------------------------------- R   : (3, 1)  |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                 |
|                                                                             |
|               25                                               16           |
|     (4, 2) : R   <------------------------------------------- R   : (3, 2)  |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                                                                             |
|               25                                               16           |
|     (4, 3) : R   <------------------------------------------- R   : (3, 3)  |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                                                                             |
|o8 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+
|i9 : assert(dd^C === C.dd)                                                   |
+-----------------------------------------------------------------------------+
|i10 : assert(source dd^C === C)                                              |
+-----------------------------------------------------------------------------+
|i11 : assert(target dd^C === C)                                              |
+-----------------------------------------------------------------------------+
|i12 : assert(degree dd^C === -1)                                             |
+-----------------------------------------------------------------------------+
|i13 : assert(source ss^C === C)                                              |
+-----------------------------------------------------------------------------+
|i14 : assert(target ss^C === C)                                              |
+-----------------------------------------------------------------------------+
|i15 : assert(degree ss^C === 1)                                              |
+-----------------------------------------------------------------------------+

The individual maps between terms are indexed by their source.

+---------------------------------------------+
|i16 : dd^C_(2,0)                             |
|                                             |
|o16 = {0} | 1 b2-ac bc-ad c2-bd 0 0 0 0  0  ||
|      {2} | 0 0     0     0     1 0 0 -c d  ||
|      {2} | 0 0     0     0     0 1 0 b  -c ||
|      {2} | 0 0     0     0     0 0 1 -a b  ||
|                                             |
|              4      9                       |
|o16 : Matrix R  <-- R                        |
+---------------------------------------------+
|i17 : assert(source dd^C_2 === C_2)          |
+---------------------------------------------+
|i18 : assert(target dd^C_2 === C_1)          |
+---------------------------------------------+

See also
========

  * Making maps between simplicial modules
    -- information about the basic constructors
  * SimplicialModuleMap _ ZZ -- access
    individual matrices in a simplicial module map
  * SimplicialModule _ ZZ -- access individual
    objects of a simplicial module
  * source(SimplicialModuleMap) -- get
    the source of a map of simplicial modules
  * target(SimplicialModuleMap) -- get
    the target of a map of simplicial modules
  * degree(SimplicialModuleMap) -- get
    the degree of a map of simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:808:0.

forgetComplex(...,RememberSummands=>...) -- whether to remember direct sum structure
************************************************************************************

  * Usage:
        forgetComplex(..., RememberSummands => true)

Description
===========

If RememberSummands => true, the direct sum structure of the terms is
remembered when forgetting the underlying complex.

See also
========

  * forgetComplex -- forget the
    underlying complex data of a simplicial module obtained as a Dold-Kan image

Functions with optional argument named RememberSummands:
========================================================

  * forgetComplex(...,RememberSummands=>...)
    -- whether to
    remember direct sum structure

Further information
===================

  * Default value: true (Macaulay2Doc),
  * Function: forgetComplex, --
    forget the underlying complex data of a simplicial module obtained as a
    Dold-Kan image
  * Option key: RememberSummands, --
    forget the underlying complex data of a simplicial module obtained as a
    Dold-Kan image
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:4025:0.

forgetComplex(SimplicialModule) -- forget the underlying complex data of a simplicial module obtained as a Dold-Kan image
*************************************************************************************************************************

  * Function: forgetComplex,
  * Usage:
        forgetComplex(S)
  * Inputs:
      * S, a Simplicial Module, A simplicial module,
        which is assumed to be obtained as a Dold-Kan image.
  * Optional inputs (Macaulay2Doc),:
      * RememberSummands,
        => a
        Boolean value (Macaulay2Doc), default value true, Default
        value is true. Indicates whether to remember the component summands of
        the simplicial module when creating the new simplicial module.
  * Outputs:
      * a Simplicial Module, A new simplicial module
        that is no longer viewed as a Dold-Kan image of some complex

Description
===========

This function removes the data of the underlying complex from a simplicial
module $S$ that is obtained as a Dold-Kan image. The function checks if the
simplicial module has an associated complex and, if so, it reconstructs the
simplicial module without the complex data while preserving the face and
degeneracy maps.

If the option `RememberSummands` is set to true (the default), the function
will remember the summands of the simplicial module when reconstructing it. The
face and degeneracy maps of the original simplicial module are preserved in the
new simplicial module. This function is good for testing that the normalization
of the Dold-Kan functor recovers the original complex, since the
normalize (SchurFunctors), command by default first checks if a
simplicial module is obtained as a Dold-Kan image before attempting a more
costly computation.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[x_1..x_3];                                                   |
+-----------------------------------------------------------------------------+
|i2 : K = koszulComplex vars R                                                |
|                                                                             |
|      1      3      3      1                                                 |
|o2 = R  <-- R  <-- R  <-- R                                                  |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o2 : Complex                                                                 |
+-----------------------------------------------------------------------------+
|i3 : S = simplicialModule(K,4, Degeneracy => true)                           |
|                                                                             |
|      1      4      10      20      35                                       |
|o3 = R  <-- R  <-- R   <-- R   <-- R  <-- ...                                |
|                                                                             |
|     0      1      2       3       4                                         |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : S.?complex                                                              |
|                                                                             |
|o4 = true                                                                    |
+-----------------------------------------------------------------------------+
|i5 : fS = forgetComplex S                                                    |
|                                                                             |
|      1      4      10      20      35                                       |
|o5 = R  <-- R  <-- R   <-- R   <-- R  <-- ...                                |
|                                                                             |
|     0      1      2       3       4                                         |
|                                                                             |
|o5 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i6 : components fS_3                                                         |
|                                                                             |
|       1   3   3   3   3   3   3   1                                         |
|o6 = {R , R , R , R , R , R , R , R }                                        |
|                                                                             |
|o6 : List                                                                    |
+-----------------------------------------------------------------------------+
|i7 : ffS = forgetComplex(S, RememberSummands => false)                       |
|                                                                             |
|      1      4      10      20      35                                       |
|o7 = R  <-- R  <-- R   <-- R   <-- R  <-- ...                                |
|                                                                             |
|     0      1      2       3       4                                         |
|                                                                             |
|o7 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i8 : components ffS_3                                                        |
|                                                                             |
|       20                                                                    |
|o8 = {R  }                                                                   |
|                                                                             |
|o8 : List                                                                    |
+-----------------------------------------------------------------------------+
|i9 : Kn = normalize fS                                                       |
|                                                                             |
|      1                                                                      |
|o9 = R  <-- image {0} | 0 0 0 | <-- image {0} | 0 0 0 | <-- image {0} | 0 |  |
|                  {1} | 1 0 0 |           {1} | 0 0 0 |           {1} | 0 |  |
|     0            {1} | 0 1 0 |           {1} | 0 0 0 |           {1} | 0 |  |
|                  {1} | 0 0 1 |           {1} | 0 0 0 |           {1} | 0 |  |
|                                          {1} | 0 0 0 |           {1} | 0 |  |
|            1                             {1} | 0 0 0 |           {1} | 0 |  |
|                                          {1} | 0 0 0 |           {1} | 0 |  |
|                                          {2} | 1 0 0 |           {1} | 0 |  |
|                                          {2} | 0 1 0 |           {1} | 0 |  |
|                                          {2} | 0 0 1 |           {1} | 0 |  |
|                                                                  {2} | 0 |  |
|                                    2                             {2} | 0 |  |
|                                                                  {2} | 0 |  |
|                                                                  {2} | 0 |  |
|                                                                  {2} | 0 |  |
|                                                                  {2} | 0 |  |
|                                                                  {2} | 0 |  |
|                                                                  {2} | 0 |  |
|                                                                  {2} | 0 |  |
|                                                                  {3} | 1 |  |
|                                                                             |
|                                                            3                |
|                                                                             |
|o9 : Complex                                                                 |
|-----------------------------------------------------------------------------|
|<-- 0                                                                        |
|                                                                             |
|    4                                                                        |
+-----------------------------------------------------------------------------+
|i10 : Knn = normalize ffS                                                    |
|                                                                             |
|       1                                                                     |
|o10 = R  <-- image {0} | 0 0 0 | <-- image {0} | 0 0 0 | <-- image {0} | 0 | |
|                   {1} | 1 0 0 |           {1} | 0 0 0 |           {1} | 0 | |
|      0            {1} | 0 1 0 |           {1} | 0 0 0 |           {1} | 0 | |
|                   {1} | 0 0 1 |           {1} | 0 0 0 |           {1} | 0 | |
|                                           {1} | 0 0 0 |           {1} | 0 | |
|             1                             {1} | 0 0 0 |           {1} | 0 | |
|                                           {1} | 0 0 0 |           {1} | 0 | |
|                                           {2} | 1 0 0 |           {1} | 0 | |
|                                           {2} | 0 1 0 |           {1} | 0 | |
|                                           {2} | 0 0 1 |           {1} | 0 | |
|                                                                   {2} | 0 | |
|                                     2                             {2} | 0 | |
|                                                                   {2} | 0 | |
|                                                                   {2} | 0 | |
|                                                                   {2} | 0 | |
|                                                                   {2} | 0 | |
|                                                                   {2} | 0 | |
|                                                                   {2} | 0 | |
|                                                                   {2} | 0 | |
|                                                                   {3} | 1 | |
|                                                                             |
|                                                             3               |
|                                                                             |
|o10 : Complex                                                                |
|-----------------------------------------------------------------------------|
|<-- 0                                                                        |
|                                                                             |
|    4                                                                        |
+-----------------------------------------------------------------------------+
|i11 : Kn.dd                                                                  |
|                                                                             |
|           1                                                                 |
|o11 = 0 : R  <------------------- image {0} | 0 0 0 | : 1                    |
|                | x_1 x_2 x_3 |         {1} | 1 0 0 |                        |
|                                        {1} | 0 1 0 |                        |
|                                        {1} | 0 0 1 |                        |
|                                                                             |
|      1 : image {0} | 0 0 0 | <-------------------------- image {0} | 0 0 0 ||
|                {1} | 1 0 0 |    {1} | -x_2 -x_3 0    |         {1} | 0 0 0 ||
|                {1} | 0 1 0 |    {1} | x_1  0    -x_3 |         {1} | 0 0 0 ||
|                {1} | 0 0 1 |    {1} | 0    x_1  x_2  |         {1} | 0 0 0 ||
|                                                                {1} | 0 0 0 ||
|                                                                {1} | 0 0 0 ||
|                                                                {1} | 0 0 0 ||
|                                                                {2} | 1 0 0 ||
|                                                                {2} | 0 1 0 ||
|                                                                {2} | 0 0 1 ||
|                                                                             |
|      2 : image {0} | 0 0 0 | <---------------- image {0} | 0 | : 3          |
|                {1} | 0 0 0 |    {2} | x_3  |         {1} | 0 |              |
|                {1} | 0 0 0 |    {2} | -x_2 |         {1} | 0 |              |
|                {1} | 0 0 0 |    {2} | x_1  |         {1} | 0 |              |
|                {1} | 0 0 0 |                         {1} | 0 |              |
|                {1} | 0 0 0 |                         {1} | 0 |              |
|                {1} | 0 0 0 |                         {1} | 0 |              |
|                {2} | 1 0 0 |                         {1} | 0 |              |
|                {2} | 0 1 0 |                         {1} | 0 |              |
|                {2} | 0 0 1 |                         {1} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {2} | 0 |              |
|                                                      {3} | 1 |              |
|                                                                             |
|      3 : image {0} | 0 | <----- 0 : 4                                       |
|                {1} | 0 |    0                                               |
|                {1} | 0 |                                                    |
|                {1} | 0 |                                                    |
|                {1} | 0 |                                                    |
|                {1} | 0 |                                                    |
|                {1} | 0 |                                                    |
|                {1} | 0 |                                                    |
|                {1} | 0 |                                                    |
|                {1} | 0 |                                                    |
|                {2} | 0 |                                                    |
|                {2} | 0 |                                                    |
|                {2} | 0 |                                                    |
|                {2} | 0 |                                                    |
|                {2} | 0 |                                                    |
|                {2} | 0 |                                                    |
|                {2} | 0 |                                                    |
|                {2} | 0 |                                                    |
|                {2} | 0 |                                                    |
|                {3} | 1 |                                                    |
|                                                                             |
|o11 : ComplexMap                                                             |
|-----------------------------------------------------------------------------|
|: 2                                                                          |
+-----------------------------------------------------------------------------+
|i12 : K == prune Kn                                                          |
|                                                                             |
|o12 = true                                                                   |
+-----------------------------------------------------------------------------+

See also
========

  * normalize (SchurFunctors) -- sort each column of a
    Filling, with Koszul sign
  * forgetDegeneracy -- forget
    the data of degeneracy maps of a simplicial object

Ways to use this method:
========================

  * forgetComplex(SimplicialModule),
    -- forget the underlying complex data of a simplicial module obtained as a
    Dold-Kan image
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3313:0.

forgetDegeneracy(SimplicialModule) -- forget the data of degeneracy maps of a simplicial object
***********************************************************************************************

  * Function: forgetDegeneracy,
  * Usage:
        forgetDegeneracy(S)
  * Inputs:
      * S, a Simplicial Module, A simplicial module
        whose degeneracy maps are to be forgotten.
  * Outputs:
      * a Simplicial Module, The simplicial module S,
        but with no degeneracy maps stored.

Description
===========

This function removes the data of degeneracy maps from a simplicial module `S`.
It is useful when the user wants to ignore degeneracy maps for the purpose of
speeding up computations or simplifying the simplicial object.

+-------------------------------------------------------------------------+
|i1 : Q = ZZ/101[a..d]                                                    |
|                                                                         |
|o1 = Q                                                                   |
|                                                                         |
|o1 : PolynomialRing                                                      |
+-------------------------------------------------------------------------+
|i2 : K = koszulComplex vars Q                                            |
|                                                                         |
|      1      4      6      4      1                                      |
|o2 = Q  <-- Q  <-- Q  <-- Q  <-- Q                                       |
|                                                                         |
|     0      1      2      3      4                                       |
|                                                                         |
|o2 : Complex                                                             |
+-------------------------------------------------------------------------+
|i3 : S = simplicialModule(K, 6, Degeneracy => true)                      |
|                                                                         |
|      1      5      15      35      70      126      210                 |
|o3 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- Q    <-- Q   <-- ...          |
|                                                                         |
|     0      1      2       3       4       5        6                    |
|                                                                         |
|o3 : SimplicialModule                                                    |
+-------------------------------------------------------------------------+
|i4 : elapsedTime S**S                                                    |
| -- .572311s elapsed                                                     |
|                                                                         |
|      1      25      225      1225      4900      15876      44100       |
|o4 = Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q      <-- Q     <-- ...|
|                                                                         |
|     0      1       2        3         4         5          6            |
|                                                                         |
|o4 : SimplicialModule                                                    |
+-------------------------------------------------------------------------+
|i5 : fS = forgetDegeneracy S                                             |
|                                                                         |
|      1      5      15      35      70      126      210                 |
|o5 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- Q    <-- Q   <-- ...          |
|                                                                         |
|     0      1      2       3       4       5        6                    |
|                                                                         |
|o5 : SimplicialModule                                                    |
+-------------------------------------------------------------------------+
|i6 : elapsedTime fS**fS --faster when degeneracy is ignored              |
| -- .39696s elapsed                                                      |
|                                                                         |
|      1      25      225      1225      4900      15876      44100       |
|o6 = Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q      <-- Q     <-- ...|
|                                                                         |
|     0      1       2        3         4         5          6            |
|                                                                         |
|o6 : SimplicialModule                                                    |
+-------------------------------------------------------------------------+

The change in speed becomes much more noticeable as ranks get larger.

See also
========

  * forgetComplex -- forget the
    underlying complex data of a simplicial module obtained as a Dold-Kan image

Ways to use this method:
========================

  * forgetDegeneracy(SimplicialModule)
    -- forget the data of degeneracy
    maps of a simplicial object
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3346:0.

id _ SimplicialModule -- the identity map of a simplicial module
****************************************************************

  * Scripted functor: id (Macaulay2Doc),
  * Usage:
        f = id_C
  * Inputs:
      * C, a Simplicial Module,
  * Outputs:
      * f, a Map of Simplicial Modules, the
        identity map from $C$ to itself

Description
===========

The simplicial modules together with simplicial morphisms forms a category.  In
particular, every simplicial module has an identity map.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[x,y]/(x^3, y^3)                                              |
|                                                                             |
|o1 = R                                                                       |
|                                                                             |
|o1 : QuotientRing                                                            |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution(coker vars R, LengthLimit=>6), 6,   |
|                                                                             |
|      1      3      8      20      48      112      256                      |
|o2 = R  <-- R  <-- R  <-- R   <-- R   <-- R    <-- R   <-- ...               |
|                                                                             |
|     0      1      2      3       4       5        6                         |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i3 : f = id_C;                                                               |
+-----------------------------------------------------------------------------+
|i4 : assert isWellDefined f                                                  |
+-----------------------------------------------------------------------------+
|i5 : assert isSimplicialMorphism f                                           |
+-----------------------------------------------------------------------------+

See also
========

  * map(SimplicialModule,SimplicialModule,ZZ)
    -- make the zero map or
    identity between simplicial modules
  * isWellDefined(SimplicialModuleMap)
    -- whether a map of simplicial
    modules is well-defined
  * isSimplicialMorphism(SimplicialModuleMap)
    -- whether a simplicial
    module map is a morphism of simplicial modules

Ways to use this method:
========================

  * id _ SimplicialModule -- the identity map
    of a simplicial module
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1580:0.

image(SimplicialModuleMap) -- make the image of a map of simplicial modules
***************************************************************************

  * Function: image (Macaulay2Doc),
  * Usage:
        E = image f
  * Inputs:
      * f, a Map of Simplicial Modules,
  * Outputs:
      * E, a Simplicial Module,

Description
===========

If $f : C \to D$ is a map of simplicial modules of degree $d$, then the image
is the simplicial module $E$ whose $i$-th term is $image(f_{i-d})$, and whose
face/degeneracy map is induced from the face/degeneracy map on the target.

In the following example, we first construct a random simplicial morphism $f :
C \to D$.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a,b,c,d];                                                    |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution ideal(b^2-a*c, b*c-a*d, c^2-b*d),3, |
|                                                                             |
|      1      4      9      16                                                |
|o2 = S  <-- S  <-- S  <-- S  <-- ...                                         |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i3 : D = simplicialModule(freeResolution ideal(a,b,c), Degeneracy => true)   |
|                                                                             |
|      1      4      10      20                                               |
|o3 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : f = randomSimplicialMap(D, C, Cycle => true, InternalDegree => 0)       |
|                                                                             |
|          1               1                                                  |
|o4 = 0 : S  <----------- S  : 0                                              |
|               | -22 |                                                       |
|                                                                             |
|          4                                                        4         |
|     1 : S  <---------------------------------------------------- S  : 1     |
|               {0} | -22 0            0           0           |              |
|               {1} | 0   36b+3c       30b-19c+22d -29b-10c    |              |
|               {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d |              |
|               {1} | 0   19a-29b      19a-8b      10a-29b-22c |              |
|                                                                             |
|          10                                                                 |
|     2 : S   <---------------------------------------------------------------|
|                {0} | -22 0            0           0           0            0|
|                {1} | 0   36b+3c       30b-19c+22d -29b-10c    0            0|
|                {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d 0            0|
|                {1} | 0   19a-29b      19a-8b      10a-29b-22c 0            0|
|                {1} | 0   0            0           0           36b+3c       3|
|                {1} | 0   0            0           0           -36a-22b+29c -|
|                {1} | 0   0            0           0           19a-29b      1|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                                                                             |
|          20                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {0} | -22 0            0           0           0            0|
|                {1} | 0   36b+3c       30b-19c+22d -29b-10c    0            0|
|                {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d 0            0|
|                {1} | 0   19a-29b      19a-8b      10a-29b-22c 0            0|
|                {1} | 0   0            0           0           36b+3c       3|
|                {1} | 0   0            0           0           -36a-22b+29c -|
|                {1} | 0   0            0           0           19a-29b      1|
|                {1} | 0   0            0           0           0            0|
|                {1} | 0   0            0           0           0            0|
|                {1} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {3} | 0   0            0           0           0            0|
|                                                                             |
|o4 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                                              9              |
|------------------------------------------------------------ S  : 2          |
|           0           0                0                |                   |
|           0           0                0                |                   |
|           0           0                0                |                   |
|           0           0                0                |                   |
|0b-19c+22d -29b-10c    0                0                |                   |
|30a-14c    29a+29c+22d 0                0                |                   |
|9a-8b      10a-29b-22c 0                0                |                   |
|           0           -29a-30b+31c-22d 29b+6c-36d       |                   |
|           0           -10a+24b+3c      34b-19c+19d      |                   |
|           0           24a-8b+29c       -24a-29b-14c-29d |                   |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|0b-19c+22d -29b-10c    0            0           0           0                |
|30a-14c    29a+29c+22d 0            0           0           0                |
|9a-8b      10a-29b-22c 0            0           0           0                |
|           0           36b+3c       30b-19c+22d -29b-10c    0                |
|           0           -36a-22b+29c -30a-14c    29a+29c+22d 0                |
|           0           19a-29b      19a-8b      10a-29b-22c 0                |
|           0           0            0           0           -29a-30b+31c-22d |
|           0           0            0           0           -10a+24b+3c      |
|           0           0            0           0           24a-8b+29c       |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|29b+6c-36d       0                0                0                0        |
|34b-19c+19d      0                0                0                0        |
|-24a-29b-14c-29d 0                0                0                0        |
|0                -29a-30b+31c-22d 29b+6c-36d       0                0        |
|0                -10a+24b+3c      34b-19c+19d      0                0        |
|0                24a-8b+29c       -24a-29b-14c-29d 0                0        |
|0                0                0                -29a-30b+31c-22d 29b+6c-36|
|0                0                0                -10a+24b+3c      34b-19c+1|
|0                0                0                24a-8b+29c       -24a-29b-|
|0                0                0                0                0        |
|-----------------------------------------------------------------------------|
|             16                                                              |
|----------- S   : 3                                                          |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|        |                                                                    |
|d       |                                                                    |
|9d      |                                                                    |
|14c-29d |                                                                    |
|        |                                                                    |
+-----------------------------------------------------------------------------+
|i5 : prune image f                                                           |
|                                                                             |
|      1      4      9      16                                                |
|o5 = S  <-- S  <-- S  <-- S  <-- ...                                         |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o5 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i6 : prune normalize oo                                                      |
|                                                                             |
|      1      3      2                                                        |
|o6 = S  <-- S  <-- S                                                         |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o6 : Complex                                                                 |
+-----------------------------------------------------------------------------+
|i7 : i = inducedMap(forgetComplex target f, image f)                         |
|                                                                             |
|          1                                                                  |
|o7 = 0 : S  <----------- image | -22 | : 0                                   |
|               | -22 |                                                       |
|                                                                             |
|          4                                                                  |
|     1 : S  <---------------------------------------------------- image {0} ||
|               {0} | -22 0            0           0           |         {1} ||
|               {1} | 0   36b+3c       30b-19c+22d -29b-10c    |         {1} ||
|               {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d |         {1} ||
|               {1} | 0   19a-29b      19a-8b      10a-29b-22c |              |
|                                                                             |
|          10                                                                 |
|     2 : S   <---------------------------------------------------------------|
|                {0} | -22 0            0           0           0            0|
|                {1} | 0   36b+3c       30b-19c+22d -29b-10c    0            0|
|                {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d 0            0|
|                {1} | 0   19a-29b      19a-8b      10a-29b-22c 0            0|
|                {1} | 0   0            0           0           36b+3c       3|
|                {1} | 0   0            0           0           -36a-22b+29c -|
|                {1} | 0   0            0           0           19a-29b      1|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                                                                             |
|          20                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {0} | -22 0            0           0           0            0|
|                {1} | 0   36b+3c       30b-19c+22d -29b-10c    0            0|
|                {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d 0            0|
|                {1} | 0   19a-29b      19a-8b      10a-29b-22c 0            0|
|                {1} | 0   0            0           0           36b+3c       3|
|                {1} | 0   0            0           0           -36a-22b+29c -|
|                {1} | 0   0            0           0           19a-29b      1|
|                {1} | 0   0            0           0           0            0|
|                {1} | 0   0            0           0           0            0|
|                {1} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {2} | 0   0            0           0           0            0|
|                {3} | 0   0            0           0           0            0|
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
| -22 0            0           0           | : 1                              |
| 0   36b+3c       30b-19c+22d -29b-10c    |                                  |
| 0   -36a-22b+29c -30a-14c    29a+29c+22d |                                  |
| 0   19a-29b      19a-8b      10a-29b-22c |                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|------------------------------------------------------------ image {0} | -22 |
|           0           0                0                |         {1} | 0   |
|           0           0                0                |         {1} | 0   |
|           0           0                0                |         {1} | 0   |
|           0           0                0                |         {1} | 0   |
|0b-19c+22d -29b-10c    0                0                |         {1} | 0   |
|30a-14c    29a+29c+22d 0                0                |         {1} | 0   |
|9a-8b      10a-29b-22c 0                0                |         {2} | 0   |
|           0           -29a-30b+31c-22d 29b+6c-36d       |         {2} | 0   |
|           0           -10a+24b+3c      34b-19c+19d      |         {2} | 0   |
|           0           24a-8b+29c       -24a-29b-14c-29d |                   |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|0b-19c+22d -29b-10c    0            0           0           0                |
|30a-14c    29a+29c+22d 0            0           0           0                |
|9a-8b      10a-29b-22c 0            0           0           0                |
|           0           36b+3c       30b-19c+22d -29b-10c    0                |
|           0           -36a-22b+29c -30a-14c    29a+29c+22d 0                |
|           0           19a-29b      19a-8b      10a-29b-22c 0                |
|           0           0            0           0           -29a-30b+31c-22d |
|           0           0            0           0           -10a+24b+3c      |
|           0           0            0           0           24a-8b+29c       |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|           0           0            0           0           0                |
|-----------------------------------------------------------------------------|
|0            0           0           0            0           0           0  |
|36b+3c       30b-19c+22d -29b-10c    0            0           0           0  |
|-36a-22b+29c -30a-14c    29a+29c+22d 0            0           0           0  |
|19a-29b      19a-8b      10a-29b-22c 0            0           0           0  |
|0            0           0           36b+3c       30b-19c+22d -29b-10c    0  |
|0            0           0           -36a-22b+29c -30a-14c    29a+29c+22d 0  |
|0            0           0           19a-29b      19a-8b      10a-29b-22c 0  |
|0            0           0           0            0           0           -29|
|0            0           0           0            0           0           -10|
|0            0           0           0            0           0           24a|
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|0                0                0                0                0        |
|29b+6c-36d       0                0                0                0        |
|34b-19c+19d      0                0                0                0        |
|-24a-29b-14c-29d 0                0                0                0        |
|0                -29a-30b+31c-22d 29b+6c-36d       0                0        |
|0                -10a+24b+3c      34b-19c+19d      0                0        |
|0                24a-8b+29c       -24a-29b-14c-29d 0                0        |
|0                0                0                -29a-30b+31c-22d 29b+6c-36|
|0                0                0                -10a+24b+3c      34b-19c+1|
|0                0                0                24a-8b+29c       -24a-29b-|
|0                0                0                0                0        |
|-----------------------------------------------------------------------------|
|              0                | : 2                                         |
|              0                |                                             |
|              0                |                                             |
|              0                |                                             |
|              0                |                                             |
|              0                |                                             |
|              0                |                                             |
|a-30b+31c-22d 29b+6c-36d       |                                             |
|a+24b+3c      34b-19c+19d      |                                             |
|-8b+29c       -24a-29b-14c-29d |                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|----------- image {0} | -22 0            0           0           0           |
|        |         {1} | 0   36b+3c       30b-19c+22d -29b-10c    0           |
|        |         {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d 0           |
|        |         {1} | 0   19a-29b      19a-8b      10a-29b-22c 0           |
|        |         {1} | 0   0            0           0           36b+3c      |
|        |         {1} | 0   0            0           0           -36a-22b+29c|
|        |         {1} | 0   0            0           0           19a-29b     |
|        |         {1} | 0   0            0           0           0           |
|        |         {1} | 0   0            0           0           0           |
|        |         {1} | 0   0            0           0           0           |
|        |         {2} | 0   0            0           0           0           |
|        |         {2} | 0   0            0           0           0           |
|        |         {2} | 0   0            0           0           0           |
|        |         {2} | 0   0            0           0           0           |
|        |         {2} | 0   0            0           0           0           |
|        |         {2} | 0   0            0           0           0           |
|        |         {2} | 0   0            0           0           0           |
|d       |         {2} | 0   0            0           0           0           |
|9d      |         {2} | 0   0            0           0           0           |
|14c-29d |         {3} | 0   0            0           0           0           |
|        |                                                                    |
|-----------------------------------------------------------------------------|
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|30b-19c+22d -29b-10c    0            0           0           0               |
|-30a-14c    29a+29c+22d 0            0           0           0               |
|19a-8b      10a-29b-22c 0            0           0           0               |
|0           0           36b+3c       30b-19c+22d -29b-10c    0               |
|0           0           -36a-22b+29c -30a-14c    29a+29c+22d 0               |
|0           0           19a-29b      19a-8b      10a-29b-22c 0               |
|0           0           0            0           0           -29a-30b+31c-22d|
|0           0           0            0           0           -10a+24b+3c     |
|0           0           0            0           0           24a-8b+29c      |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|-----------------------------------------------------------------------------|
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|29b+6c-36d       0                0                0                         |
|34b-19c+19d      0                0                0                         |
|-24a-29b-14c-29d 0                0                0                         |
|0                -29a-30b+31c-22d 29b+6c-36d       0                         |
|0                -10a+24b+3c      34b-19c+19d      0                         |
|0                24a-8b+29c       -24a-29b-14c-29d 0                         |
|0                0                0                -29a-30b+31c-22d          |
|0                0                0                -10a+24b+3c               |
|0                0                0                24a-8b+29c                |
|0                0                0                0                         |
|-----------------------------------------------------------------------------|
|0                | : 3                                                       |
|0                |                                                           |
|0                |                                                           |
|0                |                                                           |
|0                |                                                           |
|0                |                                                           |
|0                |                                                           |
|0                |                                                           |
|0                |                                                           |
|0                |                                                           |
|0                |                                                           |
|0                |                                                           |
|0                |                                                           |
|0                |                                                           |
|0                |                                                           |
|0                |                                                           |
|29b+6c-36d       |                                                           |
|34b-19c+19d      |                                                           |
|-24a-29b-14c-29d |                                                           |
|0                |                                                           |
+-----------------------------------------------------------------------------+
|i8 : isSimplicialMorphism i                                                  |
|                                                                             |
|o8 = true                                                                    |
+-----------------------------------------------------------------------------+
|i9 : normalize i                                                             |
|                                                                             |
|          1                                                                  |
|o9 = 0 : S  <-----------                                                     |
|               | -22 |                                                       |
|                                                                             |
|     1 : image {0} | 0 0                                                     |
|               {1} | 1 0                                                     |
|               {1} | 0 1                                                     |
|               {1} | 0 0                                                     |
|                                                                             |
|     2 : image {0} | 0 0                                                     |
|               {1} | 0 0                                                     |
|               {1} | 0 0                                                     |
|               {1} | 0 0                                                     |
|               {1} | 0 0                                                     |
|               {1} | 0 0                                                     |
|               {1} | 0 0                                                     |
|               {2} | 1 0                                                     |
|               {2} | 0 1                                                     |
|               {2} | 0 0                                                     |
|                                                                             |
|o9 : ComplexMap                                                              |
|-----------------------------------------------------------------------------|
|image | -22 | : 0                                                            |
|                                                                             |
|                                                                             |
|0 | <------------------------------------------------                        |
|0 |    {1} | -b-43c-36d    -8c+45d    50c+43d     |                          |
|0 |    {1} | a+43b+36c+39d 8b-46c-40d 50b-43c-35d |                          |
|1 |    {1} | -39c          b+40c      a+35c       |                          |
|                                                                             |
|0 | <-------------------------------------- image {0}                        |
|0 |    {2} | 13a-10b-34c+26d 3b+25c-49d |         {1}                        |
|0 |    {2} | a-26b+42c-12d   7b+5c-5d   |         {1}                        |
|0 |    {2} | 34b+49c+13d     a-3b+9c-3d |         {1}                        |
|0 |                                               {1}                        |
|0 |                                               {1}                        |
|0 |                                               {1}                        |
|0 |                                               {2}                        |
|0 |                                               {2}                        |
|1 |                                               {2}                        |
|-----------------------------------------------------------------------------|
|image {0} | 0             0          0           | : 1                       |
|      {1} | -b-43c-36d    -8c+45d    50c+43d     |                           |
|      {1} | a+43b+36c+39d 8b-46c-40d 50b-43c-35d |                           |
|      {1} | -39c          b+40c      a+35c       |                           |
|                                                                             |
|| 0               0          | : 2                                           |
|| 0               0          |                                               |
|| 0               0          |                                               |
|| 0               0          |                                               |
|| 0               0          |                                               |
|| 0               0          |                                               |
|| 0               0          |                                               |
|| 13a-10b-34c+26d 3b+25c-49d |                                               |
|| a-26b+42c-12d   7b+5c-5d   |                                               |
|| 34b+49c+13d     a-3b+9c-3d |                                               |
+-----------------------------------------------------------------------------+

There is a canonical map of simplicial modules from the image to the target.

+-----------------------------------------------------------------------------+
|i10 : g1 = inducedMap(target f, image f)                                     |
|                                                                             |
|           1                                                                 |
|o10 = 0 : S  <----------- image | -22 | : 0                                  |
|                | -22 |                                                      |
|                                                                             |
|           4                                                                 |
|      1 : S  <---------------------------------------------------- image {0} |
|                {0} | -22 0            0           0           |         {1} |
|                {1} | 0   36b+3c       30b-19c+22d -29b-10c    |         {1} |
|                {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d |         {1} |
|                {1} | 0   19a-29b      19a-8b      10a-29b-22c |             |
|                                                                             |
|           10                                                                |
|      2 : S   <--------------------------------------------------------------|
|                 {0} | -22 0            0           0           0            |
|                 {1} | 0   36b+3c       30b-19c+22d -29b-10c    0            |
|                 {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d 0            |
|                 {1} | 0   19a-29b      19a-8b      10a-29b-22c 0            |
|                 {1} | 0   0            0           0           36b+3c       |
|                 {1} | 0   0            0           0           -36a-22b+29c |
|                 {1} | 0   0            0           0           19a-29b      |
|                 {2} | 0   0            0           0           0            |
|                 {2} | 0   0            0           0           0            |
|                 {2} | 0   0            0           0           0            |
|                                                                             |
|           20                                                                |
|      3 : S   <--------------------------------------------------------------|
|                 {0} | -22 0            0           0           0            |
|                 {1} | 0   36b+3c       30b-19c+22d -29b-10c    0            |
|                 {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d 0            |
|                 {1} | 0   19a-29b      19a-8b      10a-29b-22c 0            |
|                 {1} | 0   0            0           0           36b+3c       |
|                 {1} | 0   0            0           0           -36a-22b+29c |
|                 {1} | 0   0            0           0           19a-29b      |
|                 {1} | 0   0            0           0           0            |
|                 {1} | 0   0            0           0           0            |
|                 {1} | 0   0            0           0           0            |
|                 {2} | 0   0            0           0           0            |
|                 {2} | 0   0            0           0           0            |
|                 {2} | 0   0            0           0           0            |
|                 {2} | 0   0            0           0           0            |
|                 {2} | 0   0            0           0           0            |
|                 {2} | 0   0            0           0           0            |
|                 {2} | 0   0            0           0           0            |
|                 {2} | 0   0            0           0           0            |
|                 {2} | 0   0            0           0           0            |
|                 {3} | 0   0            0           0           0            |
|                                                                             |
|o10 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|| -22 0            0           0           | : 1                             |
|| 0   36b+3c       30b-19c+22d -29b-10c    |                                 |
|| 0   -36a-22b+29c -30a-14c    29a+29c+22d |                                 |
|| 0   19a-29b      19a-8b      10a-29b-22c |                                 |
|                                                                             |
|                                                                             |
|                                                                             |
|------------------------------------------------------------- image {0} | -22|
|0           0           0                0                |         {1} | 0  |
|0           0           0                0                |         {1} | 0  |
|0           0           0                0                |         {1} | 0  |
|0           0           0                0                |         {1} | 0  |
|30b-19c+22d -29b-10c    0                0                |         {1} | 0  |
|-30a-14c    29a+29c+22d 0                0                |         {1} | 0  |
|19a-8b      10a-29b-22c 0                0                |         {2} | 0  |
|0           0           -29a-30b+31c-22d 29b+6c-36d       |         {2} | 0  |
|0           0           -10a+24b+3c      34b-19c+19d      |         {2} | 0  |
|0           0           24a-8b+29c       -24a-29b-14c-29d |                  |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|30b-19c+22d -29b-10c    0            0           0           0               |
|-30a-14c    29a+29c+22d 0            0           0           0               |
|19a-8b      10a-29b-22c 0            0           0           0               |
|0           0           36b+3c       30b-19c+22d -29b-10c    0               |
|0           0           -36a-22b+29c -30a-14c    29a+29c+22d 0               |
|0           0           19a-29b      19a-8b      10a-29b-22c 0               |
|0           0           0            0           0           -29a-30b+31c-22d|
|0           0           0            0           0           -10a+24b+3c     |
|0           0           0            0           0           24a-8b+29c      |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|0           0           0            0           0           0               |
|-----------------------------------------------------------------------------|
| 0            0           0           0            0           0           0 |
| 36b+3c       30b-19c+22d -29b-10c    0            0           0           0 |
| -36a-22b+29c -30a-14c    29a+29c+22d 0            0           0           0 |
| 19a-29b      19a-8b      10a-29b-22c 0            0           0           0 |
| 0            0           0           36b+3c       30b-19c+22d -29b-10c    0 |
| 0            0           0           -36a-22b+29c -30a-14c    29a+29c+22d 0 |
| 0            0           0           19a-29b      19a-8b      10a-29b-22c 0 |
| 0            0           0           0            0           0           -2|
| 0            0           0           0            0           0           -1|
| 0            0           0           0            0           0           24|
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
| 0                0                0                0                0       |
| 0                0                0                0                0       |
| 0                0                0                0                0       |
| 0                0                0                0                0       |
| 0                0                0                0                0       |
| 0                0                0                0                0       |
| 0                0                0                0                0       |
| 0                0                0                0                0       |
| 0                0                0                0                0       |
| 0                0                0                0                0       |
| 29b+6c-36d       0                0                0                0       |
| 34b-19c+19d      0                0                0                0       |
| -24a-29b-14c-29d 0                0                0                0       |
| 0                -29a-30b+31c-22d 29b+6c-36d       0                0       |
| 0                -10a+24b+3c      34b-19c+19d      0                0       |
| 0                24a-8b+29c       -24a-29b-14c-29d 0                0       |
| 0                0                0                -29a-30b+31c-22d 29b+6c-3|
| 0                0                0                -10a+24b+3c      34b-19c+|
| 0                0                0                24a-8b+29c       -24a-29b|
| 0                0                0                0                0       |
|-----------------------------------------------------------------------------|
|               0                | : 2                                        |
|               0                |                                            |
|               0                |                                            |
|               0                |                                            |
|               0                |                                            |
|               0                |                                            |
|               0                |                                            |
|9a-30b+31c-22d 29b+6c-36d       |                                            |
|0a+24b+3c      34b-19c+19d      |                                            |
|a-8b+29c       -24a-29b-14c-29d |                                            |
|                                                                             |
|                                                                             |
|                                                                             |
|------------ image {0} | -22 0            0           0                      |
|         |         {1} | 0   36b+3c       30b-19c+22d -29b-10c               |
|         |         {1} | 0   -36a-22b+29c -30a-14c    29a+29c+22d            |
|         |         {1} | 0   19a-29b      19a-8b      10a-29b-22c            |
|         |         {1} | 0   0            0           0                      |
|         |         {1} | 0   0            0           0                      |
|         |         {1} | 0   0            0           0                      |
|         |         {1} | 0   0            0           0                      |
|         |         {1} | 0   0            0           0                      |
|         |         {1} | 0   0            0           0                      |
|         |         {2} | 0   0            0           0                      |
|         |         {2} | 0   0            0           0                      |
|         |         {2} | 0   0            0           0                      |
|         |         {2} | 0   0            0           0                      |
|         |         {2} | 0   0            0           0                      |
|         |         {2} | 0   0            0           0                      |
|         |         {2} | 0   0            0           0                      |
|6d       |         {2} | 0   0            0           0                      |
|19d      |         {2} | 0   0            0           0                      |
|-14c-29d |         {3} | 0   0            0           0                      |
|         |                                                                   |
|-----------------------------------------------------------------------------|
|0            0           0           0            0           0              |
|0            0           0           0            0           0              |
|0            0           0           0            0           0              |
|0            0           0           0            0           0              |
|36b+3c       30b-19c+22d -29b-10c    0            0           0              |
|-36a-22b+29c -30a-14c    29a+29c+22d 0            0           0              |
|19a-29b      19a-8b      10a-29b-22c 0            0           0              |
|0            0           0           36b+3c       30b-19c+22d -29b-10c       |
|0            0           0           -36a-22b+29c -30a-14c    29a+29c+22d    |
|0            0           0           19a-29b      19a-8b      10a-29b-22c    |
|0            0           0           0            0           0              |
|0            0           0           0            0           0              |
|0            0           0           0            0           0              |
|0            0           0           0            0           0              |
|0            0           0           0            0           0              |
|0            0           0           0            0           0              |
|0            0           0           0            0           0              |
|0            0           0           0            0           0              |
|0            0           0           0            0           0              |
|0            0           0           0            0           0              |
|-----------------------------------------------------------------------------|
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|-29a-30b+31c-22d 29b+6c-36d       0                0                         |
|-10a+24b+3c      34b-19c+19d      0                0                         |
|24a-8b+29c       -24a-29b-14c-29d 0                0                         |
|0                0                -29a-30b+31c-22d 29b+6c-36d                |
|0                0                -10a+24b+3c      34b-19c+19d               |
|0                0                24a-8b+29c       -24a-29b-14c-29d          |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|0                0                0                0                         |
|-----------------------------------------------------------------------------|
|0                0                | : 3                                      |
|0                0                |                                          |
|0                0                |                                          |
|0                0                |                                          |
|0                0                |                                          |
|0                0                |                                          |
|0                0                |                                          |
|0                0                |                                          |
|0                0                |                                          |
|0                0                |                                          |
|0                0                |                                          |
|0                0                |                                          |
|0                0                |                                          |
|0                0                |                                          |
|0                0                |                                          |
|0                0                |                                          |
|-29a-30b+31c-22d 29b+6c-36d       |                                          |
|-10a+24b+3c      34b-19c+19d      |                                          |
|24a-8b+29c       -24a-29b-14c-29d |                                          |
|0                0                |                                          |
+-----------------------------------------------------------------------------+
|i11 : ker g1 == 0                                                            |
|                                                                             |
|o11 = true                                                                   |
+-----------------------------------------------------------------------------+
|i12 : image g1 == image f                                                    |
|                                                                             |
|o12 = true                                                                   |
+-----------------------------------------------------------------------------+

See also
========

  * Making simplicial modules -- information
    about the basic constructors
  * Making maps between simplicial modules
    -- information about the basic constructors
  * image (Macaulay2Doc) -- image of a map
  * coimage(SimplicialModuleMap), --
    make the coimage of a map of simplicial modules
  * kernel(SimplicialModuleMap) -- make
    the kernel of a map of simplicial modules
  * cokernel(SimplicialModuleMap), --
    make the cokernel of a map of simplicial modules

Ways to use this method:
========================

  * image(SimplicialModuleMap) -- make
    the image of a map of simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2775:0.

inducedMap(...,Degree=>...) -- specify the degree of an induced map of simplicial modules
*****************************************************************************************

Description
===========

When constructing an induced map of simplicial modules, this option specifies
the degree of the resulting map. By default, the degree is 0.

See also
========

  * inducedMap(SimplicialModule,SimplicialModule)
    -- make the map of
    simplicial modules induced at each term by the identity map

Functions with optional argument named Degree:
==============================================

  * "basis(...,Degree=>...)" -- see basis (Macaulay2Doc), -- basis
    or generating set of all or part of a ring, ideal or module
  * "cohomology(...,Degree=>...)" -- see cohomology (Macaulay2Doc),
    -- general cohomology functor
  * inducedMap(...,Degree=>...) (Macaulay2Doc)
    -- specify the
    degree of a map
  * map(...,Degree=>...) (Macaulay2Doc)
    -- specify the degree
    of a map
  * "matrix(...,Degree=>...)" -- see matrix(List) (Macaulay2Doc),
    -- create a matrix from a doubly-nested
    list of ring elements or matrices
  * "randomComplexMap(...,Degree=>...)" -- see
    randomComplexMap(Complex,Complex) (Complexes),
    -- a random map of
    chain complexes
  * "randomSimplicialMap(...,Degree=>...)" -- see
    randomSimplicialMap(SimplicialModule,SimplicialModule),
    -- a random
    map of simplicial modules

Further information
===================

  * Default value: null
  * Function: inducedMap (Macaulay2Doc), -- compute an induced
    map
  * Option key: Degree (Macaulay2Doc), -- an optional argument
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:4176:0.

inducedMap(SimplicialModule,SimplicialModule) -- make the map of simplicial modules induced at each term by the identity map
****************************************************************************************************************************

  * Function: inducedMap (Macaulay2Doc),
  * Usage:
        f = inducedMap(D, C)
  * Inputs:
      * C, a Simplicial Module,
      * D, a Simplicial Module,
  * Optional inputs (Macaulay2Doc),:
      * Degree, => ...,
        default value null, specify the degree of an induced map of simplicial
        modules
      * Verify (Macaulay2Doc),
        => ..., default value true,
  * Outputs:
      * f, a Map of Simplicial Modules,

Description
===========

Let $d$ be the value of the optional argument Degree, or zero, if not given.
For each $i$, the terms $D_{i+d}$ and $C_i$ must be subquotients of the same
ambient free module.  This method returns the simplicial module map induced by
the identity on each of these free modules.

If Verify => true is given, then this method also checks that these identity
maps induce well-defined maps.  This can be a relatively expensive computation.

We illustrate this method by truncating a free resolution at two distinct
internal degrees.  We check that the various induced maps compose to give
another induced map.

+-----------------------------------------------------------------------------+
|i1 : needsPackage "Truncations"                                              |
|                                                                             |
|o1 = Truncations                                                             |
|                                                                             |
|o1 : Package                                                                 |
+-----------------------------------------------------------------------------+
|i2 : kk = ZZ/32003                                                           |
|                                                                             |
|o2 = kk                                                                      |
|                                                                             |
|o2 : QuotientRing                                                            |
+-----------------------------------------------------------------------------+
|i3 : R = kk[a,b,c]                                                           |
|                                                                             |
|o3 = R                                                                       |
|                                                                             |
|o3 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i4 : F = simplicialModule(freeResolution (ideal gens R)^2, 2, Degeneracy =>  |
|                                                                             |
|      1      7      21                                                       |
|o4 = R  <-- R  <-- R  <-- ...                                                |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o4 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|true)                                                                        |
+-----------------------------------------------------------------------------+
|i5 : C1 = truncate(3, F);                                                    |
+-----------------------------------------------------------------------------+
|i6 : prune normalize C1                                                      |
|                                                                             |
|                                                                             |
|o6 = cokernel {3} | b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  | <--       |
|              {3} | -c 0  b  a  0  0  0  0  0  0  0  0  0  0  0  |           |
|              {3} | 0  -c 0  0  b  a  0  0  0  0  0  0  0  0  0  |           |
|              {3} | 0  0  -c 0  0  0  b  a  0  0  0  0  0  0  0  |           |
|              {3} | 0  0  0  -c -c 0  0  0  b  0  a  0  0  0  0  |           |
|              {3} | 0  0  0  0  0  -c 0  0  0  0  0  b  0  a  0  |           |
|              {3} | 0  0  0  0  0  0  -c 0  0  a  0  0  0  0  0  |           |
|              {3} | 0  0  0  0  0  0  0  -c -c -b 0  0  a  0  0  |           |
|              {3} | 0  0  0  0  0  0  0  0  0  0  -c -c -b 0  a  |           |
|              {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  -c -b |           |
|                                                                             |
|     0                                                                       |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o6 : Complex                                                                 |
|-----------------------------------------------------------------------------|
|                                                                            8|
|cokernel {3} | b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  | <-- R |
|         {3} | -c 0  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |       |
|         {3} | 0  -c -b 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |     2 |
|         {3} | 0  0  0  b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  |       |
|         {3} | 0  0  0  -c 0  a  0  0  0  0  0  0  0  0  0  0  0  0  |       |
|         {3} | 0  0  0  0  -c -b 0  0  0  0  0  0  0  0  0  0  0  0  |       |
|         {3} | 0  0  0  0  0  0  b  a  0  0  0  0  0  0  0  0  0  0  |       |
|         {3} | 0  0  0  0  0  0  -c 0  a  0  0  0  0  0  0  0  0  0  |       |
|         {3} | 0  0  0  0  0  0  0  -c -b 0  0  0  0  0  0  0  0  0  |       |
|         {3} | 0  0  0  0  0  0  0  0  0  b  a  0  0  0  0  0  0  0  |       |
|         {3} | 0  0  0  0  0  0  0  0  0  -c 0  a  0  0  0  0  0  0  |       |
|         {3} | 0  0  0  0  0  0  0  0  0  0  -c -b 0  0  0  0  0  0  |       |
|         {3} | 0  0  0  0  0  0  0  0  0  0  0  0  b  a  0  0  0  0  |       |
|         {3} | 0  0  0  0  0  0  0  0  0  0  0  0  -c 0  a  0  0  0  |       |
|         {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  -c -b 0  0  0  |       |
|         {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  b  a  0  |       |
|         {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c 0  a  |       |
|         {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c -b |       |
|                                                                             |
|1                                                                            |
|-----------------------------------------------------------------------------|
|     3                                                                       |
|<-- R                                                                        |
|                                                                             |
|    3                                                                        |
+-----------------------------------------------------------------------------+
|i7 : C2 = truncate(4, F);                                                    |
+-----------------------------------------------------------------------------+
|i8 : prune normalize C2                                                      |
|                                                                             |
|                                                                             |
|o8 = cokernel {4} | b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
|              {4} | -c 0  b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
|              {4} | 0  -c 0  0  b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  |
|              {4} | 0  0  -c 0  0  0  b  a  0  0  0  0  0  0  0  0  0  0  0  |
|              {4} | 0  0  0  -c -c 0  0  0  b  a  0  0  0  0  0  0  0  0  0  |
|              {4} | 0  0  0  0  0  -c 0  0  0  0  b  a  0  0  0  0  0  0  0  |
|              {4} | 0  0  0  0  0  0  -c 0  0  0  0  0  b  a  0  0  0  0  0  |
|              {4} | 0  0  0  0  0  0  0  -c -c 0  0  0  0  0  b  0  a  0  0  |
|              {4} | 0  0  0  0  0  0  0  0  0  -c -c 0  0  0  0  0  0  b  0  |
|              {4} | 0  0  0  0  0  0  0  0  0  0  0  -c 0  0  0  0  0  0  0  |
|              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  -c 0  0  a  0  0  0  |
|              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  -c -c -b 0  0  a  |
|              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c -c -b |
|              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
|              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
|                                                                             |
|     0                                                                       |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o8 : Complex                                                                 |
|-----------------------------------------------------------------------------|
|                                                                             |
|0  0  0  0  0  | <-- cokernel {4} | b  a  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  |              {4} | -c 0  b  a  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  |              {4} | 0  -c 0  0  b  0  a  0  0  0  0  0  0  0 |
|0  0  0  0  0  |              {4} | 0  0  -c 0  0  a  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  |              {4} | 0  0  0  -c -c -b 0  a  0  0  0  0  0  0 |
|0  0  0  0  0  |              {4} | 0  0  0  0  0  0  -c -b 0  0  0  0  0  0 |
|0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  b  a  0  0  0  0 |
|0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  -c 0  b  a  0  0 |
|a  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  -c 0  0  b  0 |
|0  b  0  a  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  -c 0  0  a |
|0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0  -c -c -b|
|0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  a  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|-c -c -b 0  a  |              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  -c -b |              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                              {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                                                                             |
|                     1                                                       |
|-----------------------------------------------------------------------------|
|                                                                             |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|-c -b 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  -c 0  b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  -c 0  0  b  0  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  -c 0  0  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  -c -c -b 0  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  -c -b 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  -c 0  b  a  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  -c 0  0  b  0  a  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  -c 0  0  a  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  -c -c -b 0  a  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c -b 0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  b  a  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c 0  b  a  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c 0  0  b  0  a  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c 0  0  a  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c -c -b 0  a |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c -b|
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|-----------------------------------------------------------------------------|
|                                                                             |
|0  0  0  0  0  0  0  0  | <-- cokernel {4} | b  a  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | -c 0  a  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  -c -b 0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  b  a  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  -c 0  a  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  -c -b 0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  b  a  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  -c 0  a  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  -c -b 0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  b  a |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  -c 0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  -c|
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |              {4} | 0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  |                                                    |
|0  0  0  0  0  0  0  0  |     2                                              |
|0  0  0  0  0  0  0  0  |                                                    |
|0  0  0  0  0  0  0  0  |                                                    |
|0  0  0  0  0  0  0  0  |                                                    |
|0  0  0  0  0  0  0  0  |                                                    |
|b  a  0  0  0  0  0  0  |                                                    |
|-c 0  b  a  0  0  0  0  |                                                    |
|0  -c 0  0  b  0  a  0  |                                                    |
|0  0  -c 0  0  a  0  0  |                                                    |
|0  0  0  -c -c -b 0  a  |                                                    |
|0  0  0  0  0  0  -c -b |                                                    |
|-----------------------------------------------------------------------------|
|                                              3                              |
|0  0  0  0  0  0  0  0  0  0  0  0  0  | <-- R                               |
|0  0  0  0  0  0  0  0  0  0  0  0  0  |                                     |
|0  0  0  0  0  0  0  0  0  0  0  0  0  |     3                               |
|0  0  0  0  0  0  0  0  0  0  0  0  0  |                                     |
|0  0  0  0  0  0  0  0  0  0  0  0  0  |                                     |
|0  0  0  0  0  0  0  0  0  0  0  0  0  |                                     |
|0  0  0  0  0  0  0  0  0  0  0  0  0  |                                     |
|0  0  0  0  0  0  0  0  0  0  0  0  0  |                                     |
|0  0  0  0  0  0  0  0  0  0  0  0  0  |                                     |
|0  0  0  0  0  0  0  0  0  0  0  0  0  |                                     |
|a  0  0  0  0  0  0  0  0  0  0  0  0  |                                     |
|-b 0  0  0  0  0  0  0  0  0  0  0  0  |                                     |
|0  b  a  0  0  0  0  0  0  0  0  0  0  |                                     |
|0  -c 0  a  0  0  0  0  0  0  0  0  0  |                                     |
|0  0  -c -b 0  0  0  0  0  0  0  0  0  |                                     |
|0  0  0  0  b  a  0  0  0  0  0  0  0  |                                     |
|0  0  0  0  -c 0  a  0  0  0  0  0  0  |                                     |
|0  0  0  0  0  -c -b 0  0  0  0  0  0  |                                     |
|0  0  0  0  0  0  0  b  a  0  0  0  0  |                                     |
|0  0  0  0  0  0  0  -c 0  a  0  0  0  |                                     |
|0  0  0  0  0  0  0  0  -c -b 0  0  0  |                                     |
|0  0  0  0  0  0  0  0  0  0  b  a  0  |                                     |
|0  0  0  0  0  0  0  0  0  0  -c 0  a  |                                     |
|0  0  0  0  0  0  0  0  0  0  0  -c -b |                                     |
+-----------------------------------------------------------------------------+
|i9 : assert isWellDefined C1                                                 |
+-----------------------------------------------------------------------------+
|i10 : assert isWellDefined C2                                                |
+-----------------------------------------------------------------------------+
|i11 : f = inducedMap(C1, C2);                                                |
+-----------------------------------------------------------------------------+
|i12 : prune normalize f                                                      |
|                                                                             |
|o12 = 0 : cokernel {3} | b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  | <----|
|                   {3} | -c 0  b  a  0  0  0  0  0  0  0  0  0  0  0  |    {3|
|                   {3} | 0  -c 0  0  b  a  0  0  0  0  0  0  0  0  0  |    {3|
|                   {3} | 0  0  -c 0  0  0  b  a  0  0  0  0  0  0  0  |    {3|
|                   {3} | 0  0  0  -c -c 0  0  0  b  0  a  0  0  0  0  |    {3|
|                   {3} | 0  0  0  0  0  -c 0  0  0  0  0  b  0  a  0  |    {3|
|                   {3} | 0  0  0  0  0  0  -c 0  0  a  0  0  0  0  0  |    {3|
|                   {3} | 0  0  0  0  0  0  0  -c -c -b 0  0  a  0  0  |    {3|
|                   {3} | 0  0  0  0  0  0  0  0  0  0  -c -c -b 0  a  |    {3|
|                   {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  -c -b |    {3|
|                                                                           {3|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      1 : cokernel {3} | b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                   {3} | -c 0  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                   {3} | 0  -c -b 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                   {3} | 0  0  0  b  a  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                   {3} | 0  0  0  -c 0  a  0  0  0  0  0  0  0  0  0  0  0  0|
|                   {3} | 0  0  0  0  -c -b 0  0  0  0  0  0  0  0  0  0  0  0|
|                   {3} | 0  0  0  0  0  0  b  a  0  0  0  0  0  0  0  0  0  0|
|                   {3} | 0  0  0  0  0  0  -c 0  a  0  0  0  0  0  0  0  0  0|
|                   {3} | 0  0  0  0  0  0  0  -c -b 0  0  0  0  0  0  0  0  0|
|                   {3} | 0  0  0  0  0  0  0  0  0  b  a  0  0  0  0  0  0  0|
|                   {3} | 0  0  0  0  0  0  0  0  0  -c 0  a  0  0  0  0  0  0|
|                   {3} | 0  0  0  0  0  0  0  0  0  0  -c -b 0  0  0  0  0  0|
|                   {3} | 0  0  0  0  0  0  0  0  0  0  0  0  b  a  0  0  0  0|
|                   {3} | 0  0  0  0  0  0  0  0  0  0  0  0  -c 0  a  0  0  0|
|                   {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  -c -b 0  0  0|
|                   {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  b  a  0|
|                   {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c 0  a|
|                   {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c -|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
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|                                                                             |
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|                                                                             |
|                                                                             |
|                                                                             |
|           8                                                                 |
|      2 : R  <----------------------------------------------------------- cok|
|                {3} | c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|                {3} | 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|                {3} | 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|                {3} | 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 |      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 |      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 |      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a |      |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
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|                                                                             |
|                                                                             |
|o12 : ComplexMap                                                             |
|-----------------------------------------------------------------------------|
|------------------------------------- cokernel {4} | b  a  0  0  0  0  0  0  |
|} | c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            {4} | -c 0  b  a  0  0  0  0  |
|} | 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 |            {4} | 0  -c 0  0  b  a  0  0  |
|} | 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 |            {4} | 0  0  -c 0  0  0  b  a  |
|} | 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 |            {4} | 0  0  0  -c -c 0  0  0  |
|} | 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 |            {4} | 0  0  0  0  0  -c 0  0  |
|} | 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 |            {4} | 0  0  0  0  0  0  -c 0  |
|} | 0 0 0 0 0 0 c 0 0 0 b 0 0 0 0 |            {4} | 0  0  0  0  0  0  0  -c |
|} | 0 0 0 0 0 0 0 c 0 0 0 b 0 0 0 |            {4} | 0  0  0  0  0  0  0  0  |
|} | 0 0 0 0 0 0 0 0 c 0 0 0 b 0 0 |            {4} | 0  0  0  0  0  0  0  0  |
|} | 0 0 0 0 0 0 0 0 0 c 0 0 0 b a |            {4} | 0  0  0  0  0  0  0  0  |
|                                               {4} | 0  0  0  0  0  0  0  0  |
|                                               {4} | 0  0  0  0  0  0  0  0  |
|                                               {4} | 0  0  0  0  0  0  0  0  |
|                                               {4} | 0  0  0  0  0  0  0  0  |
|                                                                             |
|  | <------------------------------------------------------------------------|
|  |    {3} | c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|  |    {3} | 0 c 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|  |    {3} | 0 0 c 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|  |    {3} | 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|  |    {3} | 0 0 0 0 0 0 0 c 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|  |    {3} | 0 0 0 0 0 0 0 0 c 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|  |    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|  |    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|  |    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|  |    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|  |    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 b 0 0 0 0 0 0 0 0 0 0 |
|  |    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 b a 0 0 0 0 0 0 0 0 |
|  |    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 |
|  |    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 b 0 0 0 0 |
|  |    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 b a 0 0 |
|  |    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 |
|b |    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c |
|       {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
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|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|ernel {4} | b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|      {4} | -c 0  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|      {4} | 0  -c -b 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|      {4} | 0  0  0  b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|      {4} | 0  0  0  -c 0  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|      {4} | 0  0  0  0  -c -b 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|      {4} | 0  0  0  0  0  0  b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|      {4} | 0  0  0  0  0  0  -c 0  a  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|      {4} | 0  0  0  0  0  0  0  -c -b 0  0  0  0  0  0  0  0  0  0  0  0  0 |
|      {4} | 0  0  0  0  0  0  0  0  0  b  a  0  0  0  0  0  0  0  0  0  0  0 |
|      {4} | 0  0  0  0  0  0  0  0  0  -c 0  a  0  0  0  0  0  0  0  0  0  0 |
|      {4} | 0  0  0  0  0  0  0  0  0  0  -c -b 0  0  0  0  0  0  0  0  0  0 |
|      {4} | 0  0  0  0  0  0  0  0  0  0  0  0  b  a  0  0  0  0  0  0  0  0 |
|      {4} | 0  0  0  0  0  0  0  0  0  0  0  0  -c 0  a  0  0  0  0  0  0  0 |
|      {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  -c -b 0  0  0  0  0  0  0 |
|      {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  b  a  0  0  0  0  0 |
|      {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c 0  a  0  0  0  0 |
|      {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c -b 0  0  0  0 |
|      {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  b  a  0  0 |
|      {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c 0  a  0 |
|      {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c -b 0 |
|      {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  b |
|      {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c|
|      {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|-----------------------------------------------------------------------------|
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  | : 0                        |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |                            |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |                            |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |                            |
|b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |                            |
|0  0  b  a  0  0  0  0  0  0  0  0  0  0  0  0  |                            |
|0  0  0  0  b  a  0  0  0  0  0  0  0  0  0  0  |                            |
|-c 0  0  0  0  0  b  0  a  0  0  0  0  0  0  0  |                            |
|0  -c -c 0  0  0  0  0  0  b  0  a  0  0  0  0  |                            |
|0  0  0  -c 0  0  0  0  0  0  0  0  b  0  a  0  |                            |
|0  0  0  0  -c 0  0  a  0  0  0  0  0  0  0  0  |                            |
|0  0  0  0  0  -c -c -b 0  0  a  0  0  0  0  0  |                            |
|0  0  0  0  0  0  0  0  -c -c -b 0  0  a  0  0  |                            |
|0  0  0  0  0  0  0  0  0  0  0  -c -c -b 0  a  |                            |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c -b |                            |
|                                                                             |
|----------- cokernel {4} | b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0 0 0 0 |            {4} | -c 0  b  a  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0 0 0 0 |            {4} | 0  -c 0  0  b  0  a  0  0  0  0  0  0  0  0  0  0 |
|0 0 0 0 |            {4} | 0  0  -c 0  0  a  0  0  0  0  0  0  0  0  0  0  0 |
|0 0 0 0 |            {4} | 0  0  0  -c -c -b 0  a  0  0  0  0  0  0  0  0  0 |
|0 0 0 0 |            {4} | 0  0  0  0  0  0  -c -b 0  0  0  0  0  0  0  0  0 |
|0 0 0 0 |            {4} | 0  0  0  0  0  0  0  0  b  a  0  0  0  0  0  0  0 |
|0 0 0 0 |            {4} | 0  0  0  0  0  0  0  0  -c 0  b  a  0  0  0  0  0 |
|0 0 0 0 |            {4} | 0  0  0  0  0  0  0  0  0  -c 0  0  b  0  a  0  0 |
|0 0 0 0 |            {4} | 0  0  0  0  0  0  0  0  0  0  -c 0  0  a  0  0  0 |
|0 0 0 0 |            {4} | 0  0  0  0  0  0  0  0  0  0  0  -c -c -b 0  a  0 |
|0 0 0 0 |            {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c -b 0 |
|0 0 0 0 |            {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  b |
|0 0 0 0 |            {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c|
|0 0 0 0 |            {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0 0 0 0 |            {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0 0 0 0 |            {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0 b 0 0 |            {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|c 0 b a |            {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                     {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                                                                             |
|                                                                             |
| 0  0  | : 2                                                                 |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| 0  0  |                                                                     |
| a  0  |                                                                     |
| 0  a  |                                                                     |
| -c -b |                                                                     |
|-----------------------------------------------------------------------------|
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|-c 0  0  b  0  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  -c 0  0  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  -c -c -b 0  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  -c -b 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  -c 0  b  a  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  -c 0  0  b  0  a  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  -c 0  0  a  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  -c -c -b 0  a  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  -c -b 0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  b  a  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c 0  b  a  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c 0  0  b  0  a  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c 0  0  a  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c -c -b 0  a  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c -b 0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  b  a  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c 0  b |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -c|
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|-----------------------------------------------------------------------------|
|0  0  0  0  0  | : 1                                                         |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|0  0  0  0  0  |                                                             |
|a  0  0  0  0  |                                                             |
|0  b  0  a  0  |                                                             |
|0  0  a  0  0  |                                                             |
|-c -c -b 0  a  |                                                             |
|0  0  0  -c -b |                                                             |
+-----------------------------------------------------------------------------+
|i13 : assert isWellDefined f                                                 |
+-----------------------------------------------------------------------------+
|i14 : f1 = inducedMap(F, C1)                                                 |
|                                                                             |
|           1                                                                 |
|o14 = 0 : R  <-------------------------------------------- image | c3 bc2 ac2|
|                | c3 bc2 ac2 b2c abc a2c b3 ab2 a2b a3 |                     |
|                                                                             |
|           7                                                                 |
|      1 : R  <---------------------------------------------------------------|
|                {0} | c3 bc2 ac2 b2c abc a2c b3 ab2 a2b a3 0 0 0 0 0 0 0 0 0 |
|                {2} | 0  0   0   0   0   0   0  0   0   0  c b a 0 0 0 0 0 0 |
|                {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 c b a 0 0 0 |
|                {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 c b a |
|                {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 |
|                {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 |
|                {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 |
|                                                                             |
|           21                                                                |
|      2 : R   <--------------------------------------------------------------|
|                 {0} | c3 bc2 ac2 b2c abc a2c b3 ab2 a2b a3 0 0 0 0 0 0 0 0 0|
|                 {2} | 0  0   0   0   0   0   0  0   0   0  c b a 0 0 0 0 0 0|
|                 {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 c b a 0 0 0|
|                 {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 c b a|
|                 {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                 {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0|
|                                                                             |
|o14 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
| b2c abc a2c b3 ab2 a2b a3 | : 0                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|--------------------- image {0} | c3 bc2 ac2 b2c abc a2c b3 ab2 a2b a3 0 0   |
|0 0 0 0 0 0 0 0 0 |         {2} | 0  0   0   0   0   0   0  0   0   0  c b   |
|0 0 0 0 0 0 0 0 0 |         {2} | 0  0   0   0   0   0   0  0   0   0  0 0   |
|0 0 0 0 0 0 0 0 0 |         {2} | 0  0   0   0   0   0   0  0   0   0  0 0   |
|0 0 0 0 0 0 0 0 0 |         {2} | 0  0   0   0   0   0   0  0   0   0  0 0   |
|c b a 0 0 0 0 0 0 |         {2} | 0  0   0   0   0   0   0  0   0   0  0 0   |
|0 0 0 c b a 0 0 0 |         {2} | 0  0   0   0   0   0   0  0   0   0  0 0   |
|0 0 0 0 0 0 c b a |                                                          |
|                                                                             |
|                                                                             |
|--------------------------------------------------------------------------   |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     |
| c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     |
| 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     |
| 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |     |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |     |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | : 1                                        |
|a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 c b a |                                            |
|                                                                             |
|                                                                             |
|                                                                             |
|image {0} | c3 bc2 ac2 b2c abc a2c b3 ab2 a2b a3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2} | 0  0   0   0   0   0   0  0   0   0  c b a 0 0 0 0 0 0 0 0 0 0 0 |
|      {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 c b a 0 0 0 0 0 0 0 0 |
|      {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 c b a 0 0 0 0 0 |
|      {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 c b a 0 0 |
|      {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 c b |
|      {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | : 2            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 |                |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 |                |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                |
+-----------------------------------------------------------------------------+
|i15 : f2 = inducedMap(F, C2)                                                 |
|                                                                             |
|           1                                                                 |
|o15 = 0 : R  <---------------------------------------------------------------|
|                | c4 bc3 ac3 b2c2 abc2 a2c2 b3c ab2c a2bc a3c b4 ab3 a2b2 a3b|
|                                                                             |
|           7                                                                 |
|      1 : R  <---------------------------------------------------------------|
|                {0} | c4 bc3 ac3 b2c2 abc2 a2c2 b3c ab2c a2bc a3c b4 ab3 a2b2|
|                {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0   |
|                {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0   |
|                {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0   |
|                {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0   |
|                {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0   |
|                {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0   |
|                                                                             |
|           21                                                                |
|      2 : R   <--------------------------------------------------------------|
|                 {0} | c4 bc3 ac3 b2c2 abc2 a2c2 b3c ab2c a2bc a3c b4 ab3 a2b|
|                 {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {2} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {3} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {3} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {3} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {3} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {3} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {3} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {3} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                 {3} | 0  0   0   0    0    0    0   0    0    0   0  0   0  |
|                                                                             |
|o15 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|------- image | c4 bc3 ac3 b2c2 abc2 a2c2 b3c ab2c a2bc a3c b4 ab3 a2b2 a3b a|
| a4 |                                                                        |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
| a3b a4 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
| 0   0  c2 bc ac b2 ab a2 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
| 0   0  0  0  0  0  0  0  c2 bc ac b2 ab a2 0  0  0  0  0  0  0  0  0  0  0  |
| 0   0  0  0  0  0  0  0  0  0  0  0  0  0  c2 bc ac b2 ab a2 0  0  0  0  0  |
| 0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c2 bc ac b2 ab |
| 0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
| 0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|2 a3b a4 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|  0   0  c2 bc ac b2 ab a2 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
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|-----------------------------------------------------------------------------|
|4 | : 0                                                                      |
|                                                                             |
|                                                                             |
|                                                                             |
|------------------------------------------ image {0} | c4 bc3 ac3 b2c2 abc2 a|
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|0  0  0  0  0  0  0  c2 bc ac b2 ab a2 |                                     |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
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|-----------------------------------------------------------------------------|
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|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
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|-----------------------------------------------------------------------------|
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|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c2 bc ac b2 ab a2 |    |
|                                                                             |
|                                                                             |
|                                                                             |
|--------------------------------------------- image {0} | c4 bc3 ac3 b2c2    |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {2} | 0  0   0   0       |
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|0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 |         {3} | 0  0   0   0       |
|0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 |         {3} | 0  0   0   0       |
|0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 |         {3} | 0  0   0   0       |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 |         {3} | 0  0   0   0       |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a |                                  |
|-----------------------------------------------------------------------------|
|: 1                                                                          |
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|                                                                             |
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|                                                                             |
|                                                                             |
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|-----------------------------------------------------------------------------|
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|-----------------------------------------------------------------------------|
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|0  0  0  0  0  0  0  0  0  0  0  0  c2 bc ac b2 ab a2 0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c2 bc ac b2 ab a2 0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  c2 bc|
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|-----------------------------------------------------------------------------|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|ac b2 ab a2 0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  c2 bc ac b2 ab a2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0|
|0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a|
|-----------------------------------------------------------------------------|
|| : 2                                                                        |
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||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
+-----------------------------------------------------------------------------+
|i16 : assert isWellDefined f1                                                |
+-----------------------------------------------------------------------------+
|i17 : assert isWellDefined f2                                                |
+-----------------------------------------------------------------------------+
|i18 : assert(f2 == f1 * f)                                                   |
+-----------------------------------------------------------------------------+

See also
========

  * inducedMap(Module,Module) (Macaulay2Doc)
    -- compute the map induced
    by the identity
  * truncate(List,SimplicialModule)
    -- truncation of a simplicial module
    at a specified degree or set of degrees

Ways to use this method:
========================

  * inducedMap(SimplicialModule,SimplicialModule)
    -- make the map of
    simplicial modules induced at each term by the identity map
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2507:0.

isCommutative(SimplicialModuleMap) -- whether a simplicial module map commutes with the face/degeneracy maps
************************************************************************************************************

  * Function: isCommutative (Macaulay2Doc),
  * Usage:
        isCommutative f
  * Inputs:
      * f, a Map of Simplicial Modules,
  * Outputs:
      * a Boolean value (Macaulay2Doc), that is true when $f$
        commutes with the face/degeneracy maps

Description
===========

For a simplicial module map $f : C \to D$, this method checks whether, for all
$i$, we have $dd^D_{i} * f_i =  f_{i-1} * dd^C_i$ and, if the source at target
are equipped with degeneracy maps, it also checks the equality $ss^D_{i} * f_i
=  f_{i+1} * ss^C_i$.

We first construct a random simplicial module map which commutes with the
face/degeneracy map.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a,b,c];                                                      |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution coker vars S, Degeneracy => true)   |
|                                                                             |
|      1      4      10      20                                               |
|o2 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o2 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i3 : D = C ** C                                                              |
|                                                                             |
|      1      16      100      400                                            |
|o3 = S  <-- S   <-- S    <-- S   <-- ...                                     |
|                                                                             |
|     0      1       2        3                                               |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : isWellDefined D                                                         |
|                                                                             |
|o4 = true                                                                    |
+-----------------------------------------------------------------------------+
|i5 : f1 = prune randomSimplicialMap(D, C, Cycle => true, InternalDegree =>   |
|-----------------------------------------------------------------------------|
|1);                                                                          |
+-----------------------------------------------------------------------------+
|i6 : prune normalize f1                                                      |
|                                                                             |
|          1                       1                                          |
|o6 = 0 : S  <------------------- S  : 0                                      |
|               | -14a-6b+26c |                                               |
|                                                                             |
|          15                                                      3          |
|     1 : S   <-------------------------------------------------- S  : 1      |
|                {1} | 4a-b-32c     4a+25b+10c   -44a+41b+26c |               |
|                {1} | -20a+28b-27c -23a-14b+13c -22a+28b+34c |               |
|                {1} | -34a+25b-43c -2a-9b+27c   14a-12b-28c  |               |
|                {1} | -40a+43b+17c -31a+48b-12c -35a+16b+29c |               |
|                {1} | -27a-13b+8c  30a+17b-8c   -49a+29b+37c |               |
|                {1} | -39a-5b-23c  -38a-27b-12c -11a+12b+7c  |               |
|                {2} | 22           27           -22          |               |
|                {2} | 15           34           20           |               |
|                {2} | 7            -11          -50          |               |
|                {2} | -16          -27          -6           |               |
|                {2} | -15          -9           44           |               |
|                {2} | 7            10           -2           |               |
|                {2} | 6            -48          -22          |               |
|                {2} | -8           47           26           |               |
|                {2} | -35          -15          47           |               |
|                                                                             |
|          69                                                      3          |
|     2 : S   <-------------------------------------------------- S  : 2      |
|                {2} | -17a-15b-41c 21a-9b+6c    11a-35b-17c  |               |
|                {2} | -28a-19b+5c  11a+4b-22c   -2a+4b+31c   |               |
|                {2} | 28a+46b+40c  16a-9b-35c   -38a-47b+10c |               |
|                {2} | -31a-31b+25c -35a-47b-17c 9b-48c       |               |
|                {2} | 35a-28b-38c  50a+13b+40c  -29a-5c      |               |
|                {2} | -45a+41b+2c  20a+47b+29c  -10a-27b+2c  |               |
|                {2} | 27a+40b-48c  -22a-14b+13c -28b+5c      |               |
|                {2} | -46a+12b-24c -41a+6b-47c  -22a-30b-25c |               |
|                {2} | 19a-9b+3c    42a+15b+2c   8a-31b-22c   |               |
|                {2} | 13a+27b-33c  -36a-16b-32c -15a+37b-18c |               |
|                {2} | 2a+13b+19c   32a+31c      31a+29b+45c  |               |
|                {2} | 48a-31b-19c  18a+39b+13c  17a+23b+21c  |               |
|                {2} | -9a-47b+12c  -32a-43b+25c 28a+32b-5c   |               |
|                {2} | 27a+15b-31c  -20a+26c     12a+44b-25c  |               |
|                {2} | -28a+42b+27c 28a+36b-12c  -2a+31b-14c  |               |
|                {2} | -30a-38c     -46a-12b+25c -12a-25b+41c |               |
|                {2} | -31a+9b-22c  -28a+44b+26c -2a-36b-25c  |               |
|                {2} | 24a-37b      -25a-42b+23c 7a+46b+12c   |               |
|                {2} | -30a-13b-3c  -48a-38b-36c -40a-2b+24c  |               |
|                {2} | 5a-10b-22c   42a-47b-42c  -14a+27b-38c |               |
|                {2} | -15a+30b     -33a-46b+35c -44a+42b+15c |               |
|                {2} | 12a+20b+49c  28a-47b+50c  45a+48b+48c  |               |
|                {2} | -17a-47b+11c -14a-45b+5c  -23a-15b-5c  |               |
|                {2} | 6a-45b+22c   26a-46b-14c  -15a+14b-32c |               |
|                {3} | -17          -34          39           |               |
|                {3} | 0            -36          0            |               |
|                {3} | -37          23           49           |               |
|                {3} | 34           -33          26           |               |
|                {3} | 4            -48          -44          |               |
|                {3} | 41           -17          -12          |               |
|                {3} | -43          -7           5            |               |
|                {3} | 36           1            15           |               |
|                {3} | 4            24           -35          |               |
|                {3} | -21          -40          22           |               |
|                {3} | -30          9            -8           |               |
|                {3} | 26           36           -8           |               |
|                {3} | 38           -43          38           |               |
|                {3} | 14           20           -9           |               |
|                {3} | -49          -36          -33          |               |
|                {3} | -9           -49          -3           |               |
|                {3} | -3           -30          -24          |               |
|                {3} | -22          -50          9            |               |
|                {3} | 5            -41          -30          |               |
|                {3} | -5           -41          37           |               |
|                {3} | 10           14           18           |               |
|                {3} | -18          26           -28          |               |
|                {3} | 27           6            -50          |               |
|                {3} | 24           -50          11           |               |
|                {3} | 9            49           35           |               |
|                {3} | 8            10           37           |               |
|                {3} | -9           41           -5           |               |
|                {3} | -18          26           40           |               |
|                {3} | -50          33           17           |               |
|                {3} | 3            9            -33          |               |
|                {3} | -25          -40          9            |               |
|                {3} | 41           2            34           |               |
|                {3} | -41          35           45           |               |
|                {3} | -2           -11          -23          |               |
|                {3} | 18           47           -1           |               |
|                {3} | 27           -5           -4           |               |
|                {4} | 0            0            0            |               |
|                {4} | 0            0            0            |               |
|                {4} | 0            0            0            |               |
|                {4} | 0            0            0            |               |
|                {4} | 0            0            0            |               |
|                {4} | 0            0            0            |               |
|                {4} | 0            0            0            |               |
|                {4} | 0            0            0            |               |
|                {4} | 0            0            0            |               |
|                                                                             |
|          147                            1                                   |
|     3 : S    <------------------------ S  : 3                               |
|                 {3} | 24a-36b-30c  |                                        |
|                 {3} | 39a+8b+15c   |                                        |
|                 {3} | 23b-47c      |                                        |
|                 {3} | 37a-39b+5c   |                                        |
|                 {3} | -49a+33b-23c |                                        |
|                 {3} | -19a-33b-29c |                                        |
|                 {3} | -16a+17b+4c  |                                        |
|                 {3} | -39a+44b-20c |                                        |
|                 {3} | -39a+9b+36c  |                                        |
|                 {3} | 47a+13b+4c   |                                        |
|                 {3} | -29a+19b+19c |                                        |
|                 {3} | -10a-29b-8c  |                                        |
|                 {3} | -22a-29b-24c |                                        |
|                 {3} | 22a-26b-18c  |                                        |
|                 {3} | -8a-11b-49c  |                                        |
|                 {3} | -35a-31b-48c |                                        |
|                 {3} | -38a-16b+39c |                                        |
|                 {3} | 21a+34b+19c  |                                        |
|                 {3} | -47a-39b-18c |                                        |
|                 {3} | 3a+43b-22c   |                                        |
|                 {3} | 20a+33b-29c  |                                        |
|                 {3} | 10a+36b-8c   |                                        |
|                 {3} | -13a-43b-15c |                                        |
|                 {3} | -28a-47b+38c |                                        |
|                 {3} | 2a+16b+22c   |                                        |
|                 {3} | -33a-45b-26c |                                        |
|                 {3} | -5a-11b-3c   |                                        |
|                 {3} | -35a+9b-22c  |                                        |
|                 {3} | 45a-34b-48c  |                                        |
|                 {3} | -47a+47b+19c |                                        |
|                 {3} | -16a+7b+15c  |                                        |
|                 {3} | -30a-28b-49c |                                        |
|                 {3} | 5a+41b-40c   |                                        |
|                 {3} | -13a+15b-48c |                                        |
|                 {3} | -28a+16b+46c |                                        |
|                 {3} | 9a-6b+10c    |                                        |
|                 {3} | -26a+16b+30c |                                        |
|                 {3} | -23a+39b+43c |                                        |
|                 {3} | -17a-11b+48c |                                        |
|                 {3} | 36a+35b+11c  |                                        |
|                 {3} | 35a-49b+8c   |                                        |
|                 {3} | 21a-4b-44c   |                                        |
|                 {3} | -4a-32b-9c   |                                        |
|                 {3} | 40a+6b-35c   |                                        |
|                 {3} | -42a+18b+28c |                                        |
|                 {3} | -16a+8b+3c   |                                        |
|                 {3} | -38a+33b+40c |                                        |
|                 {3} | 11a+46b-28c  |                                        |
|                 {3} | a-3b+22c     |                                        |
|                 {3} | 25a-31b-45c  |                                        |
|                 {3} | -2a-2b+24c   |                                        |
|                 {3} | 7a-18b-41c   |                                        |
|                 {3} | 4a-13b-49c   |                                        |
|                 {3} | -28a-47b+30c |                                        |
|                 {3} | 49a-7b+27c   |                                        |
|                 {3} | -47a-23b-7c  |                                        |
|                 {4} | -38          |                                        |
|                 {4} | -27          |                                        |
|                 {4} | -42          |                                        |
|                 {4} | 12           |                                        |
|                 {4} | 20           |                                        |
|                 {4} | -23          |                                        |
|                 {4} | 27           |                                        |
|                 {4} | -48          |                                        |
|                 {4} | 17           |                                        |
|                 {4} | -26          |                                        |
|                 {4} | -25          |                                        |
|                 {4} | -16          |                                        |
|                 {4} | 44           |                                        |
|                 {4} | -47          |                                        |
|                 {4} | 42           |                                        |
|                 {4} | -37          |                                        |
|                 {4} | -33          |                                        |
|                 {4} | -37          |                                        |
|                 {4} | 27           |                                        |
|                 {4} | -33          |                                        |
|                 {4} | 40           |                                        |
|                 {4} | 35           |                                        |
|                 {4} | -29          |                                        |
|                 {4} | 5            |                                        |
|                 {4} | 15           |                                        |
|                 {4} | -34          |                                        |
|                 {4} | 35           |                                        |
|                 {4} | 30           |                                        |
|                 {4} | -19          |                                        |
|                 {4} | 27           |                                        |
|                 {4} | 29           |                                        |
|                 {4} | -9           |                                        |
|                 {4} | 5            |                                        |
|                 {4} | -7           |                                        |
|                 {4} | -43          |                                        |
|                 {4} | 41           |                                        |
|                 {4} | 32           |                                        |
|                 {4} | 31           |                                        |
|                 {4} | -29          |                                        |
|                 {4} | 42           |                                        |
|                 {4} | 37           |                                        |
|                 {4} | 34           |                                        |
|                 {4} | 0            |                                        |
|                 {4} | 0            |                                        |
|                 {4} | 0            |                                        |
|                 {4} | 16           |                                        |
|                 {4} | -1           |                                        |
|                 {4} | 49           |                                        |
|                 {4} | -42          |                                        |
|                 {4} | -17          |                                        |
|                 {4} | -23          |                                        |
|                 {4} | 0            |                                        |
|                 {4} | 0            |                                        |
|                 {4} | 0            |                                        |
|                 {4} | 30           |                                        |
|                 {4} | -20          |                                        |
|                 {4} | -17          |                                        |
|                 {4} | -24          |                                        |
|                 {4} | -47          |                                        |
|                 {4} | 12           |                                        |
|                 {4} | 0            |                                        |
|                 {4} | 0            |                                        |
|                 {4} | 0            |                                        |
|                 {4} | -31          |                                        |
|                 {4} | -33          |                                        |
|                 {4} | 41           |                                        |
|                 {4} | -16          |                                        |
|                 {4} | 6            |                                        |
|                 {4} | -43          |                                        |
|                 {4} | -27          |                                        |
|                 {4} | -50          |                                        |
|                 {4} | -6           |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {5} | 0            |                                        |
|                 {6} | 0            |                                        |
|                                                                             |
|o6 : ComplexMap                                                              |
+-----------------------------------------------------------------------------+
|i7 : isCommutative oo                                                        |
|                                                                             |
|o7 = true                                                                    |
+-----------------------------------------------------------------------------+
|i8 : isCommutative f1                                                        |
|                                                                             |
|o8 = true                                                                    |
+-----------------------------------------------------------------------------+
|i9 : assert(degree f1 == 0)                                                  |
+-----------------------------------------------------------------------------+

We next generate a simplicial module map that is commutative and (likely)
induces a nontrivial map on homology.

+----------------------------------------------------+
|i10 : f2 = randomSimplicialMap(D, C, Cycle => true);|
+----------------------------------------------------+
|i11 : isCommutative f2                              |
|                                                    |
|o11 = true                                          |
+----------------------------------------------------+
|i12 : assert(degree f2 == 0)                        |
+----------------------------------------------------+
|i13 : assert isSimplicialMorphism f2                |
+----------------------------------------------------+

If the debugLevel (Macaulay2Doc), is greater than zero, then
the location of the first failure of commutativity is displayed.

See also
========

  * isSimplicialMorphism,
    -- whether a simplicial module map is a morphism of simplicial modules
  * randomSimplicialMap
    -- a random
    map of simplicial modules

Ways to use this method:
========================

  * isCommutative(SimplicialModuleMap)
    -- whether a simplicial module map
    commutes with the face/degeneracy maps
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2134:0.

isHomogeneous(SimplicialModuleMap) -- whether a map of simplicial modules is homogeneous
****************************************************************************************

  * Function: isHomogeneous (Macaulay2Doc),
  * Usage:
        isHomogeneous f
  * Inputs:
      * f, a Map of Simplicial Modules,
  * Outputs:
      * a Boolean value (Macaulay2Doc), that is true when $f_i$
        is homogeneous, for all $i$

Description
===========

A map of simplicial modules $f \colon C \to D$ is homogeneous (graded) if its
underlying ring is graded, and all the component maps $f_i \colon C_i \to
D_{d+i}$ are graded of degree zero, where $f$ has degree $d$.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a,b,c,d];                                                    |
+-----------------------------------------------------------------------------+
|i2 : I = minors(2, matrix{{a,b,c},{b,c,d}})                                  |
|                                                                             |
|               2                        2                                    |
|o2 = ideal (- b  + a*c, - b*c + a*d, - c  + b*d)                             |
|                                                                             |
|o2 : Ideal of S                                                              |
+-----------------------------------------------------------------------------+
|i3 : C = simplicialModule(freeResolution (S^1/I), Degeneracy => true)        |
|                                                                             |
|      1      4      9                                                        |
|o3 = S  <-- S  <-- S <-- ...                                                 |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : assert isHomogeneous dd^C                                               |
+-----------------------------------------------------------------------------+
|i5 : f = randomSimplicialMap(C, C, Degree => -1)                             |
|                                                                             |
|          1                                                                  |
|o5 = 0 : S  <----------------------------------------------------------------|
|               | 24a2-36ab+19b2-30ac+19bc-29c2-29ad-10bd-8cd-22d2 -29a2-24ab+|
|                                                                             |
|          4                                                                  |
|     1 : S  <----------------------------------------------------------------|
|               {0} | 24a2-36ab+19b2-30ac+19bc-29c2-29ad-10bd-8cd-22d2 -29a2-2|
|               {2} | 0                                                0      |
|               {2} | 0                                                0      |
|               {2} | 0                                                0      |
|                                                                             |
|          9                                                                  |
|     2 : S  <----------------------------------------------------------------|
|               {0} | 24a2-36ab+19b2-30ac+19bc-29c2-29ad-10bd-8cd-22d2 -29a2-2|
|               {2} | 0                                                0      |
|               {2} | 0                                                0      |
|               {2} | 0                                                0      |
|               {2} | 0                                                0      |
|               {2} | 0                                                0      |
|               {2} | 0                                                0      |
|               {3} | 0                                                0      |
|               {3} | 0                                                0      |
|                                                                             |
|o5 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|39b2-38ac+21bc+19c2-16ad+34bd-47cd-39d2 -18a2-13ab-28b2-43ac-47bc+2c2-15ad+38|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|4ab+39b2-38ac+21bc+19c2-16ad+34bd-47cd-39d2 -18a2-13ab-28b2-43ac-47bc+2c2-15a|
|                                            0                                |
|                                            0                                |
|                                            0                                |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|4ab+39b2-38ac+21bc+19c2-16ad+34bd-47cd-39d2 -18a2-13ab-28b2-43ac-47bc+2c2-15a|
|                                            0                                |
|                                            0                                |
|                                            0                                |
|                                            0                                |
|                                            0                                |
|                                            0                                |
|                                            0                                |
|                                            0                                |
|-----------------------------------------------------------------------------|
|                  3                                                          |
|---------------- S  : 0                                                      |
|bd+16cd+22d2 |                                                               |
|                                                                             |
|                                                       5                     |
|----------------------------------------------------- S  : 1                 |
|d+38bd+16cd+22d2 0               0                |                          |
|                 45a-34b-48c-47d -17a-11b+48c+36d |                          |
|                 47a+19b-16c+7d  35a+11b-38c+33d  |                          |
|                 15a-23b+39c+43d 40a+11b+46c-28d  |                          |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|d+38bd+16cd+22d2 0               0                0               0          |
|                 45a-34b-48c-47d -17a-11b+48c+36d 0               0          |
|                 47a+19b-16c+7d  35a+11b-38c+33d  0               0          |
|                 15a-23b+39c+43d 40a+11b+46c-28d  0               0          |
|                 0               0                45a-34b-48c-47d -17a-11b+48|
|                 0               0                47a+19b-16c+7d  35a+11b-38c|
|                 0               0                15a-23b+39c+43d 40a+11b+46c|
|                 0               0                0               0          |
|                 0               0                0               0          |
|-----------------------------------------------------------------------------|
|           7                                                                 |
|--------- S  : 2                                                             |
|      |                                                                      |
|      |                                                                      |
|      |                                                                      |
|      |                                                                      |
|c+36d |                                                                      |
|+33d  |                                                                      |
|-28d  |                                                                      |
|      |                                                                      |
|      |                                                                      |
+-----------------------------------------------------------------------------+
|i6 : assert isHomogeneous f                                                  |
+-----------------------------------------------------------------------------+
|i7 : f = randomSimplicialMap(C, C, InternalDegree => 2)                      |
|                                                                             |
|          1                                                        1         |
|o7 = 0 : S  <---------------------------------------------------- S  : 0     |
|               | a2-3ab-23b2+22ac-7bc+29c2-47ad+2bd-47cd+15d2 |              |
|                                                                             |
|          4                                                                  |
|     1 : S  <----------------------------------------------------------------|
|               {0} | a2-3ab-23b2+22ac-7bc+29c2-47ad+2bd-47cd+15d2 0          |
|               {2} | 0                                            -37a2-13ab-|
|               {2} | 0                                            -32a2-20ab-|
|               {2} | 0                                            -33a2-19ab+|
|                                                                             |
|          9                                                                  |
|     2 : S  <----------------------------------------------------------------|
|               {0} | a2-3ab-23b2+22ac-7bc+29c2-47ad+2bd-47cd+15d2 0          |
|               {2} | 0                                            -37a2-13ab-|
|               {2} | 0                                            -32a2-20ab-|
|               {2} | 0                                            -33a2-19ab+|
|               {2} | 0                                            0          |
|               {2} | 0                                            0          |
|               {2} | 0                                            0          |
|               {3} | 0                                            0          |
|               {3} | 0                                            0          |
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|                                       0                                     |
|18b2-10ac+39bc-22c2+30ad+27bd+32cd-9d2 13a2-26ab-11b2+22ac-8bc-8c2-49ad+43bd+|
|48b2+24ac-15bc-30ad+39bd+33cd-49d2     -22a2-30ab-28b2+41ac-6bc-9c2+16ad+35bd|
|44b2+17ac-39bc+9c2-20ad+36bd-39cd+4d2  40a2+3ab-2b2-31ac-41bc-13c2+25ad-49bd+|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                       0                                     |
|18b2-10ac+39bc-22c2+30ad+27bd+32cd-9d2 13a2-26ab-11b2+22ac-8bc-8c2-49ad+43bd+|
|48b2+24ac-15bc-30ad+39bd+33cd-49d2     -22a2-30ab-28b2+41ac-6bc-9c2+16ad+35bd|
|44b2+17ac-39bc+9c2-20ad+36bd-39cd+4d2  40a2+3ab-2b2-31ac-41bc-13c2+25ad-49bd+|
|                                       0                                     |
|                                       0                                     |
|                                       0                                     |
|                                       0                                     |
|                                       0                                     |
|-----------------------------------------------------------------------------|
|                                                                  4          |
|---------------------------------------------------------------- S  : 1      |
|          0                                                  |               |
|36cd-3d2  -47a2+27ab-35b2-40ac-31bc-31c2+37ad-39bd-48cd-29d2 |               |
|-35cd+6d2 -48a2+30ab-49b2-37ac+28bc+46c2+47ad-18bd+cd+40d2   |               |
|4cd+30d2  -22a2+10ab+13b2+7ac-17bc+3c2+30ad-13bd-41cd+8d2    |               |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|          0                                                  0               |
|36cd-3d2  -47a2+27ab-35b2-40ac-31bc-31c2+37ad-39bd-48cd-29d2 0               |
|-35cd+6d2 -48a2+30ab-49b2-37ac+28bc+46c2+47ad-18bd+cd+40d2   0               |
|4cd+30d2  -22a2+10ab+13b2+7ac-17bc+3c2+30ad-13bd-41cd+8d2    0               |
|          0                                                  -37a2-13ab-18b2-|
|          0                                                  -32a2-20ab-48b2+|
|          0                                                  -33a2-19ab+44b2+|
|          0                                                  0               |
|          0                                                  0               |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|                                  0                                          |
|                                  0                                          |
|                                  0                                          |
|                                  0                                          |
|10ac+39bc-22c2+30ad+27bd+32cd-9d2 13a2-26ab-11b2+22ac-8bc-8c2-49ad+43bd+36cd-|
|24ac-15bc-30ad+39bd+33cd-49d2     -22a2-30ab-28b2+41ac-6bc-9c2+16ad+35bd-35cd|
|17ac-39bc+9c2-20ad+36bd-39cd+4d2  40a2+3ab-2b2-31ac-41bc-13c2+25ad-49bd+4cd+3|
|                                  0                                          |
|                                  0                                          |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|     0                                                  0                    |
|     0                                                  0                    |
|     0                                                  0                    |
|     0                                                  0                    |
|3d2  -47a2+27ab-35b2-40ac-31bc-31c2+37ad-39bd-48cd-29d2 0                    |
|+6d2 -48a2+30ab-49b2-37ac+28bc+46c2+47ad-18bd+cd+40d2   0                    |
|0d2  -22a2+10ab+13b2+7ac-17bc+3c2+30ad-13bd-41cd+8d2    0                    |
|     0                                                  8a2-29ab+49b2+30ac-18|
|     0                                                  18a2-16ab-18b2-46ac+2|
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|                             0                                               |
|                             0                                               |
|                             0                                               |
|                             0                                               |
|                             0                                               |
|                             0                                               |
|                             0                                               |
|bc+23c2-46ad+42bd-28cd+15d2  44a2-39ab+20ac-47bc+47c2+19ad-28bd-28cd+6d2     |
|7bc+23c2+12ad-21bd-37cd-23d2 -9a2-33ab+26b2+28ac+5bc-33c2-29ad-37bd-28cd+42d2|
|-----------------------------------------------------------------------------|
|      9                                                                      |
|---- S  : 2                                                                  |
| |                                                                           |
| |                                                                           |
| |                                                                           |
| |                                                                           |
| |                                                                           |
| |                                                                           |
| |                                                                           |
| |                                                                           |
| |                                                                           |
+-----------------------------------------------------------------------------+

A map of simplicial modules may be homogeneous even if the source or the target
is not homogeneous.

+------------------------------------------------------------------------+
|i8 : phi = map(S, S, {1,b,c,d})                                         |
|                                                                        |
|o8 = map (S, S, {1, b, c, d})                                           |
|                                                                        |
|o8 : RingMap S <-- S                                                    |
+------------------------------------------------------------------------+
|i9 : D = phi C                                                          |
|                                                                        |
|      1      4      9                                                   |
|o9 = S  <-- S  <-- S <-- ...                                            |
|                                                                        |
|     0      1      2                                                    |
|                                                                        |
|o9 : SimplicialModule                                                   |
+------------------------------------------------------------------------+
|i10 : dd^D                                                              |
|                                                                        |
|                1                             4                         |
|o10 = (0, 0) : S  <------------------------- S  : (1, 0)                |
|                     | 1 b2-c bc-d c2-bd |                              |
|                                                                        |
|                1                   4                                   |
|      (0, 1) : S  <--------------- S  : (1, 1)                          |
|                     | 1 0 0 0 |                                        |
|                                                                        |
|                4                                             9         |
|      (1, 0) : S  <----------------------------------------- S  : (2, 0)|
|                     {0} | 1 b2-c bc-d c2-bd 0 0 0 0  0  |              |
|                     {2} | 0 0    0    0     1 0 0 -c d  |              |
|                     {2} | 0 0    0    0     0 1 0 b  -c |              |
|                     {2} | 0 0    0    0     0 0 1 -1 b  |              |
|                                                                        |
|                4                                 9                     |
|      (1, 1) : S  <----------------------------- S  : (2, 1)            |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                          |
|                     {2} | 0 1 0 0 1 0 0 0 0 |                          |
|                     {2} | 0 0 1 0 0 1 0 0 0 |                          |
|                     {2} | 0 0 0 1 0 0 1 0 0 |                          |
|                                                                        |
|                4                                 9                     |
|      (1, 2) : S  <----------------------------- S  : (2, 2)            |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                          |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                          |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                          |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                          |
|                                                                        |
|o10 : SimplicialModuleMap                                               |
+------------------------------------------------------------------------+
|i11 : assert not isHomogeneous dd^D                                     |
+------------------------------------------------------------------------+

See also
========

  * Basic invariants and properties, --
    information about accessing basic features
  * isHomogeneous (Macaulay2Doc) -- whether something is
    homogeneous (graded)
  * randomSimplicialMap(SimplicialModule,SimplicialModule)
    -- a random
    map of simplicial modules

Ways to use this method:
========================

  * isHomogeneous(SimplicialModuleMap)
    -- whether a map of simplicial
    modules is homogeneous
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1899:0.

isShortExactSequence(SimplicialModuleMap,SimplicialModuleMap) -- whether a pair of simplicial module maps forms a short exact sequence
**************************************************************************************************************************************

  * Function: isShortExactSequence (Complexes),
  * Usage:
        isShortExactSequence(g, f)
  * Inputs:
      * f, a Map of Simplicial Modules,
      * g, a Map of Simplicial Modules,
  * Outputs:
      * a Boolean value (Macaulay2Doc), that is true (Macaulay2Doc),
        if these form a short exact sequence

Description
===========

A short exact sequence of simplicial modules \[ 0 \to B \xrightarrow{f} C
\xrightarrow{g} D \to 0\] consists of two morphisms of simplicial modules $f
\colon B \to C$ and $g \colon C \to D$ such that $g f = 0$,
$\operatorname{image} f = \operatorname{ker} g$, $\operatorname{ker} f = 0$,
and $\operatorname{coker} g = 0$.

From a simplicial morphism $h \colon B \to C$, one obtains a short exact
sequence \[ 0 \to \operatorname{image} h \to C \to \operatorname{coker} h \to
0. \]

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[a,b,c];                                                      |
+-----------------------------------------------------------------------------+
|i2 : B = simplicialModule(freeResolution coker matrix{{a^2*b, a*b*c, c^3}},  |
|                                                                             |
|      1      4      9                                                        |
|o2 = R  <-- R  <-- R <-- ...                                                 |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i3 : C = simplicialModule(freeResolution coker vars R, 2, Degeneracy => true)|
|                                                                             |
|      1      4      10                                                       |
|o3 = R  <-- R  <-- R  <-- ...                                                |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : h = randomSimplicialMap(C, B, Cycle => true)                            |
|                                                                             |
|          1               1                                                  |
|o4 = 0 : R  <----------- R  : 0                                              |
|               | -47 |                                                       |
|                                                                             |
|          4                                                                  |
|     1 : R  <----------------------------------------------------------------|
|               {0} | -47 0                        0                        0 |
|               {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 39|
|               {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -3|
|               {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2a|
|                                                                             |
|          10                                                                 |
|     2 : R   <---------------------------------------------------------------|
|                {0} | -47 0                        0                        0|
|                {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 3|
|                {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -|
|                {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2|
|                {1} | 0   0                        0                        0|
|                {1} | 0   0                        0                        0|
|                {1} | 0   0                        0                        0|
|                {2} | 0   0                        0                        0|
|                {2} | 0   0                        0                        0|
|                {2} | 0   0                        0                        0|
|                                                                             |
|o4 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                4                                            |
|------------------------------ R  : 1                                        |
|                           |                                                 |
|ab+18b2-2ac-3bc-22c2       |                                                 |
|9a2-18ab-13ac+47bc-38c2    |                                                 |
|2+16ab-47b2+22ac+38bc-47c2 |                                                 |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                            0                        0                       |
|9ab+18b2-2ac-3bc-22c2       0                        0                       |
|39a2-18ab-13ac+47bc-38c2    0                        0                       |
|a2+16ab-47b2+22ac+38bc-47c2 0                        0                       |
|                            35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2|
|                            19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2|
|                            -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc|
|                            0                        0                       |
|                            0                        0                       |
|                            0                        0                       |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
| 0                            0                            0                 |
| 0                            0                            0                 |
| 0                            0                            0                 |
| 0                            0                            0                 |
| 39ab+18b2-2ac-3bc-22c2       0                            0                 |
| -39a2-18ab-13ac+47bc-38c2    0                            0                 |
| 2a2+16ab-47b2+22ac+38bc-47c2 0                            0                 |
| 0                            -34a2-19ab-44ac+15bc+2c2     -39a2b-18ab2+24a2c|
| 0                            43a2+40ab-5b2+36ac+49bc+29c2 -22a2b-49ab2-18b3-|
| 0                            -16a2+39ab+21ac+24bc+38c2    24a3-36a2b-29ab2-3|
|-----------------------------------------------------------------------------|
|                                  9                                          |
|-------------------------------- R  : 2                                      |
|                             |                                               |
|                             |                                               |
|                             |                                               |
|                             |                                               |
|                             |                                               |
|                             |                                               |
|                             |                                               |
|-49abc+18b2c+4ac2-47c3       |                                               |
|49abc+19b2c-43ac2-15bc2-28c3 |                                               |
|0a2c+19abc+45ac2-34bc2-48c3  |                                               |
+-----------------------------------------------------------------------------+
|i5 : f = inducedMap(C, image h)                                              |
|                                                                             |
|          1                                                                  |
|o5 = 0 : R  <----------- image | -47 | : 0                                   |
|               | -47 |                                                       |
|                                                                             |
|          4                                                                  |
|     1 : R  <----------------------------------------------------------------|
|               {0} | -47 0                        0                        0 |
|               {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 39|
|               {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -3|
|               {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2a|
|                                                                             |
|          10                                                                 |
|     2 : R   <---------------------------------------------------------------|
|                {0} | -47 0                        0                        0|
|                {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 3|
|                {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -|
|                {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2|
|                {1} | 0   0                        0                        0|
|                {1} | 0   0                        0                        0|
|                {1} | 0   0                        0                        0|
|                {2} | 0   0                        0                        0|
|                {2} | 0   0                        0                        0|
|                {2} | 0   0                        0                        0|
|                                                                             |
|o5 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|------------------------------ image {0} | -47 0                        0    |
|                           |         {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+|
|ab+18b2-2ac-3bc-22c2       |         {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2|
|9a2-18ab-13ac+47bc-38c2    |         {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+|
|2+16ab-47b2+22ac+38bc-47c2 |                                                 |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                            0                        0                       |
|9ab+18b2-2ac-3bc-22c2       0                        0                       |
|39a2-18ab-13ac+47bc-38c2    0                        0                       |
|a2+16ab-47b2+22ac+38bc-47c2 0                        0                       |
|                            35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2|
|                            19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2|
|                            -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc|
|                            0                        0                       |
|                            0                        0                       |
|                            0                        0                       |
|-----------------------------------------------------------------------------|
|                    0                            | : 1                       |
|19b2-43ac+39bc-28c2 39ab+18b2-2ac-3bc-22c2       |                           |
|-19ab-9ac-34bc-48c2 -39a2-18ab-13ac+47bc-38c2    |                           |
|24ab+34b2+28ac+48bc 2a2+16ab-47b2+22ac+38bc-47c2 |                           |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
| 0                            0                            0                 |
| 0                            0                            0                 |
| 0                            0                            0                 |
| 0                            0                            0                 |
| 39ab+18b2-2ac-3bc-22c2       0                            0                 |
| -39a2-18ab-13ac+47bc-38c2    0                            0                 |
| 2a2+16ab-47b2+22ac+38bc-47c2 0                            0                 |
| 0                            -34a2-19ab-44ac+15bc+2c2     -39a2b-18ab2+24a2c|
| 0                            43a2+40ab-5b2+36ac+49bc+29c2 -22a2b-49ab2-18b3-|
| 0                            -16a2+39ab+21ac+24bc+38c2    24a3-36a2b-29ab2-3|
|-----------------------------------------------------------------------------|
|-------------------------------- image {0} | -47 0                           |
|                             |         {1} | 0   35ab+10b2+8ac-50bc+29c2     |
|                             |         {1} | 0   19a2-10ab-29ac+24bc+38c2    |
|                             |         {1} | 0   -8a2-22ab-24b2-29ac-38bc    |
|                             |         {1} | 0   0                           |
|                             |         {1} | 0   0                           |
|                             |         {1} | 0   0                           |
|                             |         {2} | 0   0                           |
|-49abc+18b2c+4ac2-47c3       |         {2} | 0   0                           |
|49abc+19b2c-43ac2-15bc2-28c3 |         {2} | 0   0                           |
|0a2c+19abc+45ac2-34bc2-48c3  |                                               |
|-----------------------------------------------------------------------------|
|0                        0                                                   |
|34ab+19b2-43ac+39bc-28c2 39ab+18b2-2ac-3bc-22c2                              |
|-34a2-19ab-9ac-34bc-48c2 -39a2-18ab-13ac+47bc-38c2                           |
|43a2+24ab+34b2+28ac+48bc 2a2+16ab-47b2+22ac+38bc-47c2                        |
|0                        0                                                   |
|0                        0                                                   |
|0                        0                                                   |
|0                        0                                                   |
|0                        0                                                   |
|0                        0                                                   |
|-----------------------------------------------------------------------------|
|0                        0                                                   |
|0                        0                                                   |
|0                        0                                                   |
|0                        0                                                   |
|35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2                            |
|19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2                            |
|-8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc                            |
|0                        0                                                   |
|0                        0                                                   |
|0                        0                                                   |
|-----------------------------------------------------------------------------|
|0                            0                                               |
|0                            0                                               |
|0                            0                                               |
|0                            0                                               |
|39ab+18b2-2ac-3bc-22c2       0                                               |
|-39a2-18ab-13ac+47bc-38c2    0                                               |
|2a2+16ab-47b2+22ac+38bc-47c2 0                                               |
|0                            -34a2-19ab-44ac+15bc+2c2                        |
|0                            43a2+40ab-5b2+36ac+49bc+29c2                    |
|0                            -16a2+39ab+21ac+24bc+38c2                       |
|-----------------------------------------------------------------------------|
|0                                              | : 2                         |
|0                                              |                             |
|0                                              |                             |
|0                                              |                             |
|0                                              |                             |
|0                                              |                             |
|0                                              |                             |
|-39a2b-18ab2+24a2c-49abc+18b2c+4ac2-47c3       |                             |
|-22a2b-49ab2-18b3-49abc+19b2c-43ac2-15bc2-28c3 |                             |
|24a3-36a2b-29ab2-30a2c+19abc+45ac2-34bc2-48c3  |                             |
+-----------------------------------------------------------------------------+
|i6 : g = inducedMap(coker h, C)                                              |
|                                                                             |
|                                  1                                          |
|o6 = 0 : cokernel | -47 | <----- R  : 0                                      |
|                             0                                               |
|                                                                             |
|                                                                             |
|     1 : cokernel {0} | -47 0                        0                       |
|                  {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2|
|                  {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2|
|                  {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc|
|                                                                             |
|                                                                             |
|                                                                             |
|     2 : cokernel {0} | -47 0                        0                       |
|                  {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2|
|                  {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2|
|                  {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc|
|                  {1} | 0   0                        0                       |
|                  {1} | 0   0                        0                       |
|                  {1} | 0   0                        0                       |
|                  {2} | 0   0                        0                       |
|                  {2} | 0   0                        0                       |
|                  {2} | 0   0                        0                       |
|                                                                             |
|                                                                             |
|o6 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                                                             |
|0                                                                            |
|39ab+18b2-2ac-3bc-22c2                                                       |
|-39a2-18ab-13ac+47bc-38c2                                                    |
|2a2+16ab-47b2+22ac+38bc-47c2                                                 |
|                                                                             |
|                                                                             |
|                                                                             |
|0                                                                            |
|39ab+18b2-2ac-3bc-22c2                                                       |
|-39a2-18ab-13ac+47bc-38c2                                                    |
|2a2+16ab-47b2+22ac+38bc-47c2                                                 |
|0                                                                            |
|0                                                                            |
|0                                                                            |
|0                                                                            |
|0                                                                            |
|0                                                                            |
|                                                                             |
|-----------------------------------------------------------------------------|
|                        4                                                    |
|| <------------------- R  : 1                                                |
||    {0} | 0 0 0 0 |                                                         |
||    {1} | 0 1 0 0 |                                                         |
||    {1} | 0 0 1 0 |                                                         |
|     {1} | 0 0 0 1 |                                                         |
|                                                                             |
|                                                                             |
|0                        0                                                   |
|0                        0                                                   |
|0                        0                                                   |
|0                        0                                                   |
|35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2                            |
|19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2                            |
|-8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc                            |
|0                        0                                                   |
|0                        0                                                   |
|0                        0                                                   |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|0                            0                                               |
|0                            0                                               |
|0                            0                                               |
|0                            0                                               |
|39ab+18b2-2ac-3bc-22c2       0                                               |
|-39a2-18ab-13ac+47bc-38c2    0                                               |
|2a2+16ab-47b2+22ac+38bc-47c2 0                                               |
|0                            -34a2-19ab-44ac+15bc+2c2                        |
|0                            43a2+40ab-5b2+36ac+49bc+29c2                    |
|0                            -16a2+39ab+21ac+24bc+38c2                       |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|0                                              |                             |
|0                                              |                             |
|0                                              |                             |
|0                                              |                             |
|0                                              |                             |
|0                                              |                             |
|0                                              |                             |
|-39a2b-18ab2+24a2c-49abc+18b2c+4ac2-47c3       |                             |
|-22a2b-49ab2-18b3-49abc+19b2c-43ac2-15bc2-28c3 |                             |
|24a3-36a2b-29ab2-30a2c+19abc+45ac2-34bc2-48c3  |                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                  10                                         |
|<------------------------------- R   : 2                                     |
|   {0} | 0 0 0 0 0 0 0 0 0 0 |                                               |
|   {1} | 0 1 0 0 0 0 0 0 0 0 |                                               |
|   {1} | 0 0 1 0 0 0 0 0 0 0 |                                               |
|   {1} | 0 0 0 1 0 0 0 0 0 0 |                                               |
|   {1} | 0 0 0 0 1 0 0 0 0 0 |                                               |
|   {1} | 0 0 0 0 0 1 0 0 0 0 |                                               |
|   {1} | 0 0 0 0 0 0 1 0 0 0 |                                               |
|   {2} | 0 0 0 0 0 0 0 1 0 0 |                                               |
|   {2} | 0 0 0 0 0 0 0 0 1 0 |                                               |
|   {2} | 0 0 0 0 0 0 0 0 0 1 |                                               |
+-----------------------------------------------------------------------------+
|i7 : assert isShortExactSequence(g,f)                                        |
+-----------------------------------------------------------------------------+

A short exact sequence of modules gives rise to a short exact sequence of
simplicial modules.  These simplicial modules arise as free resolutions of the
modules.

+-----------------------------------------------------------------------------+
|i8 : I = ideal(a^3, b^3, c^3)                                                |
|                                                                             |
|             3   3   3                                                       |
|o8 = ideal (a , b , c )                                                      |
|                                                                             |
|o8 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i9 : J = I + ideal(a*b*c)                                                    |
|                                                                             |
|             3   3   3                                                       |
|o9 = ideal (a , b , c , a*b*c)                                               |
|                                                                             |
|o9 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i10 : K = I : ideal(a*b*c)                                                   |
|                                                                             |
|              2   2   2                                                      |
|o10 = ideal (c , b , a )                                                     |
|                                                                             |
|o10 : Ideal of R                                                             |
+-----------------------------------------------------------------------------+
|i11 : SES = complex{                                                         |
|          map(comodule J, comodule I, 1),                                    |
|          map(comodule I, (comodule K) ** R^{-3}, {{a*b*c}})                 |
|          }                                                                  |
|                                                                             |
|o11 = cokernel | a3 b3 c3 abc | <-- cokernel | a3 b3 c3 | <-- cokernel {3} | |
|                                                                             |
|      0                             1                         2              |
|                                                                             |
|o11 : Complex                                                                |
|-----------------------------------------------------------------------------|
|c2 b2 a2 |                                                                   |
+-----------------------------------------------------------------------------+
|i12 : assert isWellDefined SES                                               |
+-----------------------------------------------------------------------------+
|i13 : assert isShortExactSequence(dd^SES_1, dd^SES_2)                        |
+-----------------------------------------------------------------------------+
|i14 : (g,f) = (horseshoeResolution SES)/simplicialModule;                    |
+-----------------------------------------------------------------------------+
|i15 : assert isShortExactSequence(g,f)                                       |
+-----------------------------------------------------------------------------+

See also
========

  * Basic invariants and properties, --
    information about accessing basic features

Ways to use this method:
========================

  * isShortExactSequence(SimplicialModuleMap,SimplicialModuleMap),
    --
    whether a pair of simplicial module maps forms a short exact sequence
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3095:0.

isSimplicialModule(SimplicialModule) -- check if the simplicial identities hold for a simplicial object with degeneracy map keys
********************************************************************************************************************************

  * Function: isSimplicialModule,
  * Usage:
        isSimplicialModule(S)
  * Inputs:
      * S, a Simplicial Module, A simplicial module
        with face maps `dd` and degeneracy maps `ss`.
  * Outputs:
      * a Boolean value (Macaulay2Doc), true if the simplicial
        identities hold, false otherwise

Description
===========

This function checks if the simplicial identities hold for a given simplicial
object `S` with face maps `dd` and degeneracy maps `ss`. The simplicial
identities are a set of equations that need to be satisfied by the face and
degeneracy maps of the simplicial object.

Specifically, the function verifies the following identities: 1. For face maps:
\[ d_j d_i = d_i d_{j-1} \text{ for } 0 \leq i < j \leq n \] 2. For face and
degeneracy maps: \[ d_i s_j = s_{j-1} d_i \text{ for } i < j \] \[ d_j s_j =
\text{id} \] \[ d_{j+1} s_j = \text{id} \] \[ d_k s_j = s_j d_{k-1} \text{ for
} k > j+1 \] 3. For degeneracy maps: \[ s_j s_i = s_i s_{j+1} \text{ for } i
\leq j \]

If any of these identities fail, the function returns `false`. If all
identities are satisfied, the function returns `true`.

+-----------------------------------------------------------------------------+
|i1 : R = QQ[a..d];                                                           |
+-----------------------------------------------------------------------------+
|i2 : f0 = matrix {{-b^2+a*c, b*c-a*d, -c^2+b*d}}                             |
|                                                                             |
|o2 = | -b2+ac bc-ad -c2+bd |                                                 |
|                                                                             |
|             1      3                                                        |
|o2 : Matrix R  <-- R                                                         |
+-----------------------------------------------------------------------------+
|i3 : f1 = map(source f0,, {{d, c}, {c, b}, {b, a}})                          |
|                                                                             |
|o3 = {2} | d c |                                                             |
|     {2} | c b |                                                             |
|     {2} | b a |                                                             |
|                                                                             |
|             3      2                                                        |
|o3 : Matrix R  <-- R                                                         |
+-----------------------------------------------------------------------------+
|i4 : C = simplicialModule(complex {f0, f1}, 3, Degeneracy => true)           |
|                                                                             |
|      1      4      9      16                                                |
|o4 = R  <-- R  <-- R  <-- R  <-- ...                                         |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o4 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i5 : isSimplicialModule C                                                    |
|                                                                             |
|o5 = true                                                                    |
+-----------------------------------------------------------------------------+
|i6 : dd^C                                                                    |
|                                                                             |
|               1                                 4                           |
|o6 = (0, 0) : R  <----------------------------- R  : (1, 0)                  |
|                    | 1 -b2+ac bc-ad -c2+bd |                                |
|                                                                             |
|               1                   4                                         |
|     (0, 1) : R  <--------------- R  : (1, 1)                                |
|                    | 1 0 0 0 |                                              |
|                                                                             |
|               4                                               9             |
|     (1, 0) : R  <------------------------------------------- R  : (2, 0)    |
|                    {0} | 1 -b2+ac bc-ad -c2+bd 0 0 0 0 0 |                  |
|                    {2} | 0 0      0     0      1 0 0 d c |                  |
|                    {2} | 0 0      0     0      0 1 0 c b |                  |
|                    {2} | 0 0      0     0      0 0 1 b a |                  |
|                                                                             |
|               4                                 9                           |
|     (1, 1) : R  <----------------------------- R  : (2, 1)                  |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 1 0 0 1 0 0 0 0 |                                |
|                    {2} | 0 0 1 0 0 1 0 0 0 |                                |
|                    {2} | 0 0 0 1 0 0 1 0 0 |                                |
|                                                                             |
|               4                                 9                           |
|     (1, 2) : R  <----------------------------- R  : (2, 2)                  |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 1 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 1 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 1 0 0 0 0 0 |                                |
|                                                                             |
|               9                                                             |
|     (2, 0) : R  <---------------------------------------------------------  |
|                    {0} | 1 -b2+ac bc-ad -c2+bd 0 0 0 0 0 0 0 0 0 0 0 0 |    |
|                    {2} | 0 0      0     0      1 0 0 0 0 0 d c 0 0 0 0 |    |
|                    {2} | 0 0      0     0      0 1 0 0 0 0 c b 0 0 0 0 |    |
|                    {2} | 0 0      0     0      0 0 1 0 0 0 b a 0 0 0 0 |    |
|                    {2} | 0 0      0     0      0 0 0 1 0 0 0 0 d c 0 0 |    |
|                    {2} | 0 0      0     0      0 0 0 0 1 0 0 0 c b 0 0 |    |
|                    {2} | 0 0      0     0      0 0 0 0 0 1 0 0 b a 0 0 |    |
|                    {3} | 0 0      0     0      0 0 0 0 0 0 0 0 0 0 1 0 |    |
|                    {3} | 0 0      0     0      0 0 0 0 0 0 0 0 0 0 0 1 |    |
|                                                                             |
|               9                                               16            |
|     (2, 1) : R  <------------------------------------------- R   : (3, 1)   |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |                  |
|                                                                             |
|               9                                               16            |
|     (2, 2) : R  <------------------------------------------- R   : (3, 2)   |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |                  |
|                                                                             |
|               9                                               16            |
|     (2, 3) : R  <------------------------------------------- R   : (3, 3)   |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                  |
|                                                                             |
|o6 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
| 16                                                                          |
|R   : (3, 0)                                                                 |
+-----------------------------------------------------------------------------+
|i7 : ss^C                                                                    |
|                                                                             |
|               4                 1                                           |
|o7 = (1, 0) : R  <------------- R  : (0, 0)                                  |
|                    {0} | 1 |                                                |
|                    {2} | 0 |                                                |
|                    {2} | 0 |                                                |
|                    {2} | 0 |                                                |
|                                                                             |
|               9                       4                                     |
|     (2, 0) : R  <------------------- R  : (1, 0)                            |
|                    {0} | 1 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 1 0 0 |                                          |
|                    {2} | 0 0 1 0 |                                          |
|                    {2} | 0 0 0 1 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                                                                             |
|               9                       4                                     |
|     (2, 1) : R  <------------------- R  : (1, 1)                            |
|                    {0} | 1 0 0 0 |                                          |
|                    {2} | 0 1 0 0 |                                          |
|                    {2} | 0 0 1 0 |                                          |
|                    {2} | 0 0 0 1 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                                                                             |
|               16                                 9                          |
|     (3, 0) : R   <----------------------------- R  : (2, 0)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 1 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 1 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 1 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 1 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 1 |                               |
|                                                                             |
|               16                                 9                          |
|     (3, 1) : R   <----------------------------- R  : (2, 1)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 1 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 1 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 1 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 1 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 1 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                                                                             |
|               16                                 9                          |
|     (3, 2) : R   <----------------------------- R  : (2, 2)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 1 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 1 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 1 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 1 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 1 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+
|i8 : dd^C*ss^C  --if C is simplicial, this should be all identity maps       |
|                                                                             |
|               1             1                                               |
|o8 = (0, 0) : R  <--------- R  : (0, 0)                                      |
|                    | 1 |                                                    |
|                                                                             |
|               4                       4                                     |
|     (1, 0) : R  <------------------- R  : (1, 0)                            |
|                    {0} | 1 0 0 0 |                                          |
|                    {2} | 0 1 0 0 |                                          |
|                    {2} | 0 0 1 0 |                                          |
|                    {2} | 0 0 0 1 |                                          |
|                                                                             |
|               4                       4                                     |
|     (1, 1) : R  <------------------- R  : (1, 1)                            |
|                    {0} | 1 0 0 0 |                                          |
|                    {2} | 0 1 0 0 |                                          |
|                    {2} | 0 0 1 0 |                                          |
|                    {2} | 0 0 0 1 |                                          |
|                                                                             |
|               9                                 9                           |
|     (2, 0) : R  <----------------------------- R  : (2, 0)                  |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 1 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 1 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 1 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 0 1 0 0 0 0 |                                |
|                    {2} | 0 0 0 0 0 1 0 0 0 |                                |
|                    {2} | 0 0 0 0 0 0 1 0 0 |                                |
|                    {3} | 0 0 0 0 0 0 0 1 0 |                                |
|                    {3} | 0 0 0 0 0 0 0 0 1 |                                |
|                                                                             |
|               9                                 9                           |
|     (2, 1) : R  <----------------------------- R  : (2, 1)                  |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 1 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 1 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 1 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 0 1 0 0 0 0 |                                |
|                    {2} | 0 0 0 0 0 1 0 0 0 |                                |
|                    {2} | 0 0 0 0 0 0 1 0 0 |                                |
|                    {3} | 0 0 0 0 0 0 0 1 0 |                                |
|                    {3} | 0 0 0 0 0 0 0 0 1 |                                |
|                                                                             |
|               9                                 9                           |
|     (2, 2) : R  <----------------------------- R  : (2, 2)                  |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 1 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 1 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 1 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 0 1 0 0 0 0 |                                |
|                    {2} | 0 0 0 0 0 1 0 0 0 |                                |
|                    {2} | 0 0 0 0 0 0 1 0 0 |                                |
|                    {3} | 0 0 0 0 0 0 0 1 0 |                                |
|                    {3} | 0 0 0 0 0 0 0 0 1 |                                |
|                                                                             |
|o8 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+

The zero simplicial module is well-defined.

+-----------------------------------------------------+
|i9 : C = simplicialModule(R^0, 6, Degeneracy => true)|
|                                                     |
|o9 = 0 <-- 0 <-- 0 <-- 0 <-- 0 <-- 0 <-- 0<-- ...    |
|                                                     |
|     0     1     2     3     4     5     6           |
|                                                     |
|o9 : SimplicialModule                                |
+-----------------------------------------------------+
|i10 : isSimplicialModule C                           |
|                                                     |
|o10 = true                                           |
+-----------------------------------------------------+

See also
========

  * isWellDefined (Macaulay2Doc) -- whether a map is well
    defined

Ways to use this method:
========================

  * isSimplicialModule(SimplicialModule)
    -- check if the simplicial
    identities hold for a simplicial object with degeneracy map keys
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3265:0.

isSimplicialMorphism(SimplicialModuleMap) -- whether a simplicial module map is a morphism of simplicial modules
****************************************************************************************************************

  * Function: isSimplicialMorphism,
  * Usage:
        isSimplicialMorphism f
  * Inputs:
      * f, a Map of Simplicial Modules,
  * Outputs:
      * a Boolean value (Macaulay2Doc), that is true when $f$
        commutes with the face/degeneracy maps and has degree $0$

Description
===========

For a simplicial module map $f : C \to D$ of degree $d$, this method checks
whether $d = 0$ and, for all $i$, we have $dd^D_{i+d} * f_i =  f_{i-1} *
dd^C_i$ and if the simplicial module also has degeneracy maps, it checks that
$ss^D_{i} * f_i =  f_{i+1} * ss^C_i$.

We first construct a random simplicial module morphism.

+--------------------------------------------------------------------------+
|i1 : S = ZZ/101[a,b,c];                                                   |
+--------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution coker vars S, Degeneracy => true)|
|                                                                          |
|      1      4      10      20                                            |
|o2 = S  <-- S  <-- S   <-- S  <-- ...                                     |
|                                                                          |
|     0      1      2       3                                              |
|                                                                          |
|o2 : SimplicialModule                                                     |
+--------------------------------------------------------------------------+
|i3 : D = C ** C                                                           |
|                                                                          |
|      1      16      100      400                                         |
|o3 = S  <-- S   <-- S    <-- S   <-- ...                                  |
|                                                                          |
|     0      1       2        3                                            |
|                                                                          |
|o3 : SimplicialModule                                                     |
+--------------------------------------------------------------------------+
|i4 : f1 = randomSimplicialMap(D, C, Cycle => true, InternalDegree => 1);  |
+--------------------------------------------------------------------------+
|i5 : isSimplicialMorphism f1                                              |
|                                                                          |
|o5 = true                                                                 |
+--------------------------------------------------------------------------+
|i6 : assert(degree f1 == 0)                                               |
+--------------------------------------------------------------------------+

We next generate a complex morphism that (likely) induces a nontrivial map on
homology.

+-----------------------------------------------------------------------------+
|i7 : f2 = randomSimplicialMap(D, C, Cycle => true);                          |
+-----------------------------------------------------------------------------+
|i8 : isSimplicialMorphism f2                                                 |
|                                                                             |
|o8 = true                                                                    |
+-----------------------------------------------------------------------------+
|i9 : assert(degree f2 == 0)                                                  |
+-----------------------------------------------------------------------------+
|i10 : assert isSimplicialMorphism f2                                         |
+-----------------------------------------------------------------------------+
|i11 : prune HH f2                                                            |
|                                                                             |
|o11 = 0 : cokernel | c b a | <----------- cokernel | c b a | : 0             |
|                                | -50 |                                      |
|                                                                             |
|      1 : cokernel {0} | c b a 0 0 0 0 0 0 0 0 0 | <--------------- cokernel |
|                   {1} | 0 0 0 c b a 0 0 0 0 0 0 |    {0} | -50 |            |
|                   {1} | 0 0 0 0 0 0 c b a 0 0 0 |    {1} | 0   |            |
|                   {1} | 0 0 0 0 0 0 0 0 0 c b a |    {1} | 0   |            |
|                                                      {1} | 0   |            |
|                                                                             |
|      2 : cokernel {0} | c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {1} | 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {1} | 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {1} | 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 |
|                   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 |
|                   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 |
|                   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 |
|                   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b |
|                   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|                                                                             |
|      3 : cokernel {0} | c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {1} | 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {1} | 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {1} | 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 |
|                   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 |
|                   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 |
|                   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 |
|                   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b |
|                   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                   {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|                                                                             |
|o11 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|| c b a | : 1                                                                |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|0 0 0 0 | <--------------- cokernel | c b a | : 2                            |
|0 0 0 0 |    {0} | -50 |                                                     |
|0 0 0 0 |    {1} | 0   |                                                     |
|0 0 0 0 |    {1} | 0   |                                                     |
|0 0 0 0 |    {1} | 0   |                                                     |
|0 0 0 0 |    {1} | 0   |                                                     |
|0 0 0 0 |    {1} | 0   |                                                     |
|0 0 0 0 |    {1} | 0   |                                                     |
|a 0 0 0 |    {2} | 0   |                                                     |
|0 c b a |    {2} | 0   |                                                     |
|             {2} | 0   |                                                     |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -b c  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b  -c 0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a  0  -c |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  -a b  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -b 0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -b 0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  c  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a  0  -c |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  -c 0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  -c 0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  b  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  b  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  -b |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  -a 0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  c  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  -a 0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  a  -b |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  a  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  a  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  c  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  -c |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  -b |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  b  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  |     |
|                                                                             |
|-----------------------------------------------------------------------------|
|<--------------- cokernel | c b a | : 3                                      |
|   {0} | -50 |                                                               |
|   {1} | 0   |                                                               |
|   {1} | 0   |                                                               |
|   {1} | 0   |                                                               |
|   {1} | 0   |                                                               |
|   {1} | 0   |                                                               |
|   {1} | 0   |                                                               |
|   {1} | 0   |                                                               |
|   {1} | 0   |                                                               |
|   {1} | 0   |                                                               |
|   {2} | 0   |                                                               |
|   {2} | 0   |                                                               |
|   {2} | 0   |                                                               |
|   {2} | 0   |                                                               |
|   {2} | 0   |                                                               |
|   {2} | 0   |                                                               |
|   {2} | 0   |                                                               |
|   {2} | 0   |                                                               |
|   {2} | 0   |                                                               |
|   {3} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {4} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {5} | 0   |                                                               |
|   {6} | 0   |                                                               |
+-----------------------------------------------------------------------------+

If the debugLevel (Macaulay2Doc), is greater than zero, then
information about the failure is displayed.

See also
========

  * isCommutative(SimplicialModuleMap)
    -- whether a simplicial module map
    commutes with the face/degeneracy maps
  * randomSimplicialMap
    -- a random
    map of simplicial modules

Ways to use this method:
========================

  * isSimplicialMorphism(SimplicialModuleMap)
    -- whether a simplicial
    module map is a morphism of simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2179:0.

isWellDefined(SimplicialModule) -- whether a simplicial module is well-defined
******************************************************************************

  * Function: isWellDefined (Macaulay2Doc),
  * Usage:
        isWellDefined C
  * Inputs:
      * C, a Simplicial Module,
  * Outputs:
      * a Boolean value (Macaulay2Doc), that is true when C
        determines a well defined simplicial module

Description
===========

This routine checks that the face/degeneracy maps of C satisfy the simplicial
identities (see isSimplicialModule,
for a list of these identities). If
there are no degeneracy maps stored, it instead checks that the naive
normalization of C is a well-defined complex. Additionally, it checks that the
underlying data in C is a properly formed SimplicialModule,
object in Macaulay2. If the variable debugLevel is set to a
value greater than zero, then information about the nature of any failure is
displayed.

As a first example, we construct the Dold-Kan image of a free resolution and
verify that it is well-defined.

+-----------------------------------------------------------------------------+
|i1 : R = QQ[a..d];                                                           |
+-----------------------------------------------------------------------------+
|i2 : f0 = matrix {{-b^2+a*c, b*c-a*d, -c^2+b*d}}                             |
|                                                                             |
|o2 = | -b2+ac bc-ad -c2+bd |                                                 |
|                                                                             |
|             1      3                                                        |
|o2 : Matrix R  <-- R                                                         |
+-----------------------------------------------------------------------------+
|i3 : f1 = map(source f0,, {{d, c}, {c, b}, {b, a}})                          |
|                                                                             |
|o3 = {2} | d c |                                                             |
|     {2} | c b |                                                             |
|     {2} | b a |                                                             |
|                                                                             |
|             3      2                                                        |
|o3 : Matrix R  <-- R                                                         |
+-----------------------------------------------------------------------------+
|i4 : C = simplicialModule(complex {f0, f1}, 3, Degeneracy => true)           |
|                                                                             |
|      1      4      9      16                                                |
|o4 = R  <-- R  <-- R  <-- R  <-- ...                                         |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o4 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i5 : isWellDefined C                                                         |
|                                                                             |
|o5 = true                                                                    |
+-----------------------------------------------------------------------------+
|i6 : dd^C                                                                    |
|                                                                             |
|               1                                 4                           |
|o6 = (0, 0) : R  <----------------------------- R  : (1, 0)                  |
|                    | 1 -b2+ac bc-ad -c2+bd |                                |
|                                                                             |
|               1                   4                                         |
|     (0, 1) : R  <--------------- R  : (1, 1)                                |
|                    | 1 0 0 0 |                                              |
|                                                                             |
|               4                                               9             |
|     (1, 0) : R  <------------------------------------------- R  : (2, 0)    |
|                    {0} | 1 -b2+ac bc-ad -c2+bd 0 0 0 0 0 |                  |
|                    {2} | 0 0      0     0      1 0 0 d c |                  |
|                    {2} | 0 0      0     0      0 1 0 c b |                  |
|                    {2} | 0 0      0     0      0 0 1 b a |                  |
|                                                                             |
|               4                                 9                           |
|     (1, 1) : R  <----------------------------- R  : (2, 1)                  |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 1 0 0 1 0 0 0 0 |                                |
|                    {2} | 0 0 1 0 0 1 0 0 0 |                                |
|                    {2} | 0 0 0 1 0 0 1 0 0 |                                |
|                                                                             |
|               4                                 9                           |
|     (1, 2) : R  <----------------------------- R  : (2, 2)                  |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 1 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 1 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 1 0 0 0 0 0 |                                |
|                                                                             |
|               9                                                             |
|     (2, 0) : R  <---------------------------------------------------------  |
|                    {0} | 1 -b2+ac bc-ad -c2+bd 0 0 0 0 0 0 0 0 0 0 0 0 |    |
|                    {2} | 0 0      0     0      1 0 0 0 0 0 d c 0 0 0 0 |    |
|                    {2} | 0 0      0     0      0 1 0 0 0 0 c b 0 0 0 0 |    |
|                    {2} | 0 0      0     0      0 0 1 0 0 0 b a 0 0 0 0 |    |
|                    {2} | 0 0      0     0      0 0 0 1 0 0 0 0 d c 0 0 |    |
|                    {2} | 0 0      0     0      0 0 0 0 1 0 0 0 c b 0 0 |    |
|                    {2} | 0 0      0     0      0 0 0 0 0 1 0 0 b a 0 0 |    |
|                    {3} | 0 0      0     0      0 0 0 0 0 0 0 0 0 0 1 0 |    |
|                    {3} | 0 0      0     0      0 0 0 0 0 0 0 0 0 0 0 1 |    |
|                                                                             |
|               9                                               16            |
|     (2, 1) : R  <------------------------------------------- R   : (3, 1)   |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |                  |
|                                                                             |
|               9                                               16            |
|     (2, 2) : R  <------------------------------------------- R   : (3, 2)   |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |                  |
|                                                                             |
|               9                                               16            |
|     (2, 3) : R  <------------------------------------------- R   : (3, 3)   |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                  |
|                    {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                  |
|                    {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                  |
|                                                                             |
|o6 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
| 16                                                                          |
|R   : (3, 0)                                                                 |
+-----------------------------------------------------------------------------+
|i7 : ss^C                                                                    |
|                                                                             |
|               4                 1                                           |
|o7 = (1, 0) : R  <------------- R  : (0, 0)                                  |
|                    {0} | 1 |                                                |
|                    {2} | 0 |                                                |
|                    {2} | 0 |                                                |
|                    {2} | 0 |                                                |
|                                                                             |
|               9                       4                                     |
|     (2, 0) : R  <------------------- R  : (1, 0)                            |
|                    {0} | 1 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 1 0 0 |                                          |
|                    {2} | 0 0 1 0 |                                          |
|                    {2} | 0 0 0 1 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                                                                             |
|               9                       4                                     |
|     (2, 1) : R  <------------------- R  : (1, 1)                            |
|                    {0} | 1 0 0 0 |                                          |
|                    {2} | 0 1 0 0 |                                          |
|                    {2} | 0 0 1 0 |                                          |
|                    {2} | 0 0 0 1 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {2} | 0 0 0 0 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                    {3} | 0 0 0 0 |                                          |
|                                                                             |
|               16                                 9                          |
|     (3, 0) : R   <----------------------------- R  : (2, 0)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 1 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 1 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 1 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 1 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 1 |                               |
|                                                                             |
|               16                                 9                          |
|     (3, 1) : R   <----------------------------- R  : (2, 1)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 1 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 1 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 1 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 1 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 1 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                                                                             |
|               16                                 9                          |
|     (3, 2) : R   <----------------------------- R  : (2, 2)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 1 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 1 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 1 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 1 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 1 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                     {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+
|i8 : dd^C*ss^C  --if C is well-defined, this should be all identity maps     |
|                                                                             |
|               1             1                                               |
|o8 = (0, 0) : R  <--------- R  : (0, 0)                                      |
|                    | 1 |                                                    |
|                                                                             |
|               4                       4                                     |
|     (1, 0) : R  <------------------- R  : (1, 0)                            |
|                    {0} | 1 0 0 0 |                                          |
|                    {2} | 0 1 0 0 |                                          |
|                    {2} | 0 0 1 0 |                                          |
|                    {2} | 0 0 0 1 |                                          |
|                                                                             |
|               4                       4                                     |
|     (1, 1) : R  <------------------- R  : (1, 1)                            |
|                    {0} | 1 0 0 0 |                                          |
|                    {2} | 0 1 0 0 |                                          |
|                    {2} | 0 0 1 0 |                                          |
|                    {2} | 0 0 0 1 |                                          |
|                                                                             |
|               9                                 9                           |
|     (2, 0) : R  <----------------------------- R  : (2, 0)                  |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 1 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 1 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 1 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 0 1 0 0 0 0 |                                |
|                    {2} | 0 0 0 0 0 1 0 0 0 |                                |
|                    {2} | 0 0 0 0 0 0 1 0 0 |                                |
|                    {3} | 0 0 0 0 0 0 0 1 0 |                                |
|                    {3} | 0 0 0 0 0 0 0 0 1 |                                |
|                                                                             |
|               9                                 9                           |
|     (2, 1) : R  <----------------------------- R  : (2, 1)                  |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 1 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 1 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 1 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 0 1 0 0 0 0 |                                |
|                    {2} | 0 0 0 0 0 1 0 0 0 |                                |
|                    {2} | 0 0 0 0 0 0 1 0 0 |                                |
|                    {3} | 0 0 0 0 0 0 0 1 0 |                                |
|                    {3} | 0 0 0 0 0 0 0 0 1 |                                |
|                                                                             |
|               9                                 9                           |
|     (2, 2) : R  <----------------------------- R  : (2, 2)                  |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 1 0 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 1 0 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 1 0 0 0 0 0 |                                |
|                    {2} | 0 0 0 0 1 0 0 0 0 |                                |
|                    {2} | 0 0 0 0 0 1 0 0 0 |                                |
|                    {2} | 0 0 0 0 0 0 1 0 0 |                                |
|                    {3} | 0 0 0 0 0 0 0 1 0 |                                |
|                    {3} | 0 0 0 0 0 0 0 0 1 |                                |
|                                                                             |
|o8 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+

The zero simplicial module is well-defined.

+-----------------------------------------------------+
|i9 : C = simplicialModule(R^0, 6, Degeneracy => true)|
|                                                     |
|o9 = 0 <-- 0 <-- 0 <-- 0 <-- 0 <-- 0 <-- 0<-- ...    |
|                                                     |
|     0     1     2     3     4     5     6           |
|                                                     |
|o9 : SimplicialModule                                |
+-----------------------------------------------------+
|i10 : isWellDefined C                                |
|                                                     |
|o10 = true                                           |
+-----------------------------------------------------+

See also
========

  * isWellDefined(SimplicialModuleMap)
    -- whether a map of simplicial
    modules is well-defined
  * map (Macaulay2Doc) -- make a map

Ways to use this method:
========================

  * isWellDefined(SimplicialModule),
    -- whether a simplicial module is well-defined
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:626:0.

isWellDefined(SimplicialModuleMap) -- whether a map of simplicial modules is well-defined
*****************************************************************************************

  * Function: isWellDefined (Macaulay2Doc),
  * Usage:
        isWellDefined f
  * Inputs:
      * f, a Map of Simplicial Modules,
  * Outputs:
      * a Boolean value (Macaulay2Doc), that is true when f
        determines a well defined simplicial module map

Description
===========

A map of simplicial modules $f : C \to D$ of degree $d$ is a sequence of maps
$f_i : C_i \to D_{d+i}$.  No relationship is required between these maps and
the face/degeneracy maps in the source and target.

This routine checks that $C$ and $D$ are well-defined simplicial modules, and
that, for each $f_i$, the source and target equal $C_i$ and $D_{d+i}$,
respectively.  If the variable debugLevel is set to a value greater than zero,
then information about the nature of any failure is displayed.

Unlike the corresponding function for SimplicialModules,
the basic constructors for simplicial
module maps are all but assured to be well defined. The only case that could
cause a problem is if one constructs the source or target complex, and those
are not well defined.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[a,b,c];                                                      |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(F = freeResolution coker                           |
|                                                                             |
|      1      4      10      20                                               |
|o2 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|matrix{{a^2-b^2,b^3-c^3,c^4}}, Degeneracy => true)                           |
+-----------------------------------------------------------------------------+
|i3 : D = simplicialModule(G = freeResolution coker vars R, Degeneracy =>     |
|                                                                             |
|      1      4      10      20                                               |
|o3 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o3 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|true)                                                                        |
+-----------------------------------------------------------------------------+
|i4 : H = hashTable { 0 => map(D_0, C_0, 1),                                  |
|         1 => map(D_1, C_1, {{1, 0, 0, 0}, {0, a, 0, 0}, {0, -b, b^2, 0},    |
|         2 => map(D_2, C_2, {{1, 0, 0, 0, 0, 0, 0, 0, 0, 0},                 |
|                              {0, a, 0, 0, 0, 0,0, 0, 0, 0},                 |
|                              {0, -b, b^2, 0, 0, 0, 0, 0, 0, 0},             |
|                              {0, 0,-c^2, c^3, 0, 0, 0, 0, 0, 0},            |
|                              {0, 0, 0, 0, a, 0, 0, 0, 0,0},                 |
|                              {0, 0, 0, 0, -b, b^2, 0, 0, 0, 0},             |
|                              {0, 0, 0, 0, 0,-c^2, c^3, 0, 0, 0},            |
|                              {0, 0, 0, 0, 0, 0, 0, a*b^2, 0, 0},            |
|                              {0, 0, 0, 0, 0, 0, 0, -a*c^2, a*c^3, 0},       |
|                              {0, 0, 0, 0, 0, 0, 0, b*c^2, -b*c^3, b^2*c^    |
|         3 => map(D_3, C_3, {{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0    |
|                             {0, a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0    |
|                             {0, -b, b^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0,    |
|                             {0, 0, -c^2, c^3, 0, 0, 0, 0, 0, 0,0, 0, 0,     |
|                             {0, 0, 0, 0, a, 0, 0, 0, 0,0, 0, 0, 0, 0, 0,    |
|                             {0, 0, 0, 0, -b, b^2, 0,0, 0, 0, 0, 0, 0, 0,    |
|                             {0, 0, 0, 0, 0,-c^2, c^3, 0, 0, 0, 0, 0, 0,     |
|                             {0, 0,0, 0, 0, 0, 0, a, 0, 0, 0, 0, 0, 0, 0,    |
|                             {0,0, 0, 0, 0, 0, 0, -b, b^2, 0, 0, 0, 0, 0,    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, -c^2, c^3, 0, 0, 0,    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a*b^2, 0, 0,0    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-a*c^2, a*c^3,    |
|                             {0, 0, 0, 0, 0, 0,0, 0, 0, 0, b*c^2, -b*c^3,    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a*b^    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -a*c    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, b*c^2    |
|                             {0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0,    |
|                             {0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0,    |
|                     }                                                       |
|                                                                             |
|o4 = HashTable{0 => | 1 |                                                    |
|               1 => {0} | 1 0  0   0  |                                      |
|                    {1} | 0 a  0   0  |                                      |
|                    {1} | 0 -b b2  0  |                                      |
|                    {1} | 0 0  -c2 c3 |                                      |
|               2 => {0} | 1 0  0   0  0  0   0  0    0    0    |             |
|                    {1} | 0 a  0   0  0  0   0  0    0    0    |             |
|                    {1} | 0 -b b2  0  0  0   0  0    0    0    |             |
|                    {1} | 0 0  -c2 c3 0  0   0  0    0    0    |             |
|                    {1} | 0 0  0   0  a  0   0  0    0    0    |             |
|                    {1} | 0 0  0   0  -b b2  0  0    0    0    |             |
|                    {1} | 0 0  0   0  0  -c2 c3 0    0    0    |             |
|                    {2} | 0 0  0   0  0  0   0  ab2  0    0    |             |
|                    {2} | 0 0  0   0  0  0   0  -ac2 ac3  0    |             |
|                    {2} | 0 0  0   0  0  0   0  bc2  -bc3 b2c3 |             |
|               3 => {0} | 1 0  0   0  0  0   0  0  0   0  0    0    0        |
|                    {1} | 0 a  0   0  0  0   0  0  0   0  0    0    0        |
|                    {1} | 0 -b b2  0  0  0   0  0  0   0  0    0    0        |
|                    {1} | 0 0  -c2 c3 0  0   0  0  0   0  0    0    0        |
|                    {1} | 0 0  0   0  a  0   0  0  0   0  0    0    0        |
|                    {1} | 0 0  0   0  -b b2  0  0  0   0  0    0    0        |
|                    {1} | 0 0  0   0  0  -c2 c3 0  0   0  0    0    0        |
|                    {1} | 0 0  0   0  0  0   0  a  0   0  0    0    0        |
|                    {1} | 0 0  0   0  0  0   0  -b b2  0  0    0    0        |
|                    {1} | 0 0  0   0  0  0   0  0  -c2 c3 0    0    0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  ab2  0    0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  -ac2 ac3  0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  bc2  -bc3 b2c3     |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0        |
|                    {3} | 0 0  0   0  0  0   0  0  0   0  0    0    0        |
|                                                                             |
|o4 : HashTable                                                               |
|-----------------------------------------------------------------------------|
| {0, 0, -c^2, c^3}}),                                                        |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|3}}),                                                                        |
|, 0,0, 0, 0, 0},                                                             |
|,0, 0, 0, 0, 0},                                                             |
| 0, 0, 0, 0, 0, 0},                                                          |
|0, 0, 0, 0, 0, 0, 0},                                                        |
| 0, 0, 0, 0, 0},                                                             |
| 0, 0, 0, 0, 0, 0},                                                          |
|0, 0, 0, 0, 0, 0, 0},                                                        |
| 0, 0, 0, 0, 0},                                                             |
| 0, 0, 0, 0, 0,0},                                                           |
| 0, 0, 0,0, 0, 0, 0},                                                        |
|, 0, 0, 0, 0, 0, 0},                                                         |
| 0, 0, 0, 0, 0, 0, 0,0},                                                     |
| b^2*c^3, 0, 0, 0, 0, 0, 0, 0},                                              |
|2, 0, 0, 0, 0,0, 0},                                                         |
|^2,a*c^3, 0, 0, 0, 0, 0},                                                    |
|, -b*c^3, b^2*c^3, 0, 0, 0, 0},                                              |
| 0, a*b^2, 0, 0, 0},                                                         |
| 0, -a*c^2, a*c^3,0, 0},                                                     |
|, 0, b*c^2, -b*c^3, b^2*c^3, 0},                                             |
| 0, 0, 0, 0, a*b^2*c^3}})                                                    |
|                                                                             |
|                                                                             |
|                                     }                                       |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|ab2  0    0    0    0    0    0     |                                        |
|-ac2 ac3  0    0    0    0    0     |                                        |
|bc2  -bc3 b2c3 0    0    0    0     |                                        |
|0    0    0    ab2  0    0    0     |                                        |
|0    0    0    -ac2 ac3  0    0     |                                        |
|0    0    0    bc2  -bc3 b2c3 0     |                                        |
|0    0    0    0    0    0    ab2c3 |                                        |
+-----------------------------------------------------------------------------+
|i5 : f = map(D, C, H)                                                        |
|                                                                             |
|          1             1                                                    |
|o5 = 0 : R  <--------- R  : 0                                                |
|               | 1 |                                                         |
|                                                                             |
|          4                           4                                      |
|     1 : R  <----------------------- R  : 1                                  |
|               {0} | 1 0  0   0  |                                           |
|               {1} | 0 a  0   0  |                                           |
|               {1} | 0 -b b2  0  |                                           |
|               {1} | 0 0  -c2 c3 |                                           |
|                                                                             |
|          10                                                    10           |
|     2 : R   <------------------------------------------------ R   : 2       |
|                {0} | 1 0  0   0  0  0   0  0    0    0    |                 |
|                {1} | 0 a  0   0  0  0   0  0    0    0    |                 |
|                {1} | 0 -b b2  0  0  0   0  0    0    0    |                 |
|                {1} | 0 0  -c2 c3 0  0   0  0    0    0    |                 |
|                {1} | 0 0  0   0  a  0   0  0    0    0    |                 |
|                {1} | 0 0  0   0  -b b2  0  0    0    0    |                 |
|                {1} | 0 0  0   0  0  -c2 c3 0    0    0    |                 |
|                {2} | 0 0  0   0  0  0   0  ab2  0    0    |                 |
|                {2} | 0 0  0   0  0  0   0  -ac2 ac3  0    |                 |
|                {2} | 0 0  0   0  0  0   0  bc2  -bc3 b2c3 |                 |
|                                                                             |
|          20                                                                 |
|     3 : R   <---------------------------------------------------------------|
|                {0} | 1 0  0   0  0  0   0  0  0   0  0    0    0    0    0  |
|                {1} | 0 a  0   0  0  0   0  0  0   0  0    0    0    0    0  |
|                {1} | 0 -b b2  0  0  0   0  0  0   0  0    0    0    0    0  |
|                {1} | 0 0  -c2 c3 0  0   0  0  0   0  0    0    0    0    0  |
|                {1} | 0 0  0   0  a  0   0  0  0   0  0    0    0    0    0  |
|                {1} | 0 0  0   0  -b b2  0  0  0   0  0    0    0    0    0  |
|                {1} | 0 0  0   0  0  -c2 c3 0  0   0  0    0    0    0    0  |
|                {1} | 0 0  0   0  0  0   0  a  0   0  0    0    0    0    0  |
|                {1} | 0 0  0   0  0  0   0  -b b2  0  0    0    0    0    0  |
|                {1} | 0 0  0   0  0  0   0  0  -c2 c3 0    0    0    0    0  |
|                {2} | 0 0  0   0  0  0   0  0  0   0  ab2  0    0    0    0  |
|                {2} | 0 0  0   0  0  0   0  0  0   0  -ac2 ac3  0    0    0  |
|                {2} | 0 0  0   0  0  0   0  0  0   0  bc2  -bc3 b2c3 0    0  |
|                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    ab2  0  |
|                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    -ac2 ac3|
|                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    bc2  -bc|
|                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0  |
|                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0  |
|                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0  |
|                {3} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0  |
|                                                                             |
|o5 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                 20                                          |
|------------------------------- R   : 3                                      |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|3 b2c3 0    0    0    0     |                                                |
|  0    ab2  0    0    0     |                                                |
|  0    -ac2 ac3  0    0     |                                                |
|  0    bc2  -bc3 b2c3 0     |                                                |
|  0    0    0    0    ab2c3 |                                                |
+-----------------------------------------------------------------------------+
|i6 : assert isWellDefined f                                                  |
+-----------------------------------------------------------------------------+
|i7 : assert isHomogeneous f                                                  |
+-----------------------------------------------------------------------------+
|i8 : assert(degree f == 0)                                                   |
+-----------------------------------------------------------------------------+
|i9 : assert isSimplicialMorphism f                                           |
+-----------------------------------------------------------------------------+

We construct two random maps of simplicial modules, and check to see that, as
should be the case, both are well defined.

+-----------------------------------------------------------------------------+
|i10 : g = randomSimplicialMap(D,C); --outputs are large                      |
+-----------------------------------------------------------------------------+
|i11 : normalize g                                                            |
|                                                                             |
|           1              1                                                  |
|o11 = 0 : R  <---------- R  : 0                                              |
|                | 24 |                                                       |
|                                                                             |
|           3                                                                 |
|      1 : R  <---------------------------------------------------------------|
|                {1} | -36a-30b-29c -29a2-24ab-16b2-38ac+39bc+21c2 16a3+22a2b-|
|                {1} | 19a+19b-10c  34a2+19ab-39b2-47ac-18bc-13c2  15a3-23a2b+|
|                {1} | -29a-8b-22c  -43a2-15ab-47b2-28ac+38bc+2c2  -38a3+33a2b|
|                                                                             |
|           3                                                                 |
|      2 : R  <---------------------------------------------------------------|
|                {2} | -23a3-7a2b+29ab2-37b3+2a2c-47abc-13b2c+15ac2-10bc2+30c3|
|                {2} | -18a3+39a2b-22ab2-32b3+27a2c+32abc-20b2c-9ac2+24bc2-30c|
|                {2} | -48a3-15a2b-33b3+39a2c+33abc-19b2c-49ac2+17bc2-20c3    |
|                                                                             |
|           1                                                                 |
|      3 : R  <---------------------------------------------------------------|
|                {3} | 4a6+22a5b-20a4b2+15a3b3+9a2b4+16ab5-30b6+5a5c-13a4bc-4a|
|                                                                             |
|o11 : ComplexMap                                                             |
|-----------------------------------------------------------------------------|
|                                                   3                         |
|------------------------------------------------- R  : 1                     |
|34ab2+47b3+45a2c-48abc+19b2c-47ac2-16bc2+7c3  |                              |
|43ab2+48b3+39a2c-17abc+36b2c-11ac2+35bc2+11c3 |                              |
|+11ab2+b3+40a2c+46abc-3b2c-28ac2+22bc2-47c3   |                              |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|  44a4-39a3b+9a2b2+13ab3-11b4+36a3c-39a2bc-26ab2c-8b3c+4a2c2+22abc2+43b2c2-49|
|3 -3a4-22a3b+41a2b2-6ab3+6b4-30a3c+16a2bc+35ab2c+40b3c-28a2c2-9abc2+3b2c2-35a|
|  -2a4-41a3b-13a2b2-47ab3-35b4-49a3c+4a2bc+27ab2c-31b3c+30a2c2-40abc2-39b2c2+|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|3b2c-2a2b3c+10ab4c-4b5c-29a4c2+12a3bc2+20a2b2c2+31ab3c2-14b4c2+3a3c3-26a2bc3+|
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|ac3-8bc3+36c4    -29a5-48a4b-37a3b2+28a2b3+40ab4+13b5+30a4c+47a3bc-18a2b2c-22|
|c3-31bc3+25c4    8a5-29a4b-46a3b2+42a2b3+18ab4+27b5+30a4c+49a3bc+23a2b2c-16ab|
|37ac3-31bc3-48c4 -39a5+20a4b+47a2b3-33ab4-37b5+19a4c-47a3bc-28a2b2c+28ab3c-33|
|                                                                             |
|                                                        1                    |
|------------------------------------------------------ R  : 3                |
|28ab2c3-33b3c3+33a2c4-6abc4-42b2c4+21ac5-44bc5-5c6 |                         |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|ab3c-17b4c-49a3c2+46a2bc2+10ab2c2-13b3c2+a2c3+7abc3+3b2c3+30ac4-41bc4+8c5    |
|3c-21b4c-18a3c2-28a2bc2-46ab2c2+23b3c2+15a2c3+12abc3-37b2c3-18ac4-23bc4+44c5 |
|b4c-28a3c2+6a2bc2-29ab2c2-28b3c2-9a2c3+26abc3+42b2c3+5ac4+44bc4+30c5         |
|-----------------------------------------------------------------------------|
|     3                                                                       |
|--- R  : 2                                                                   |
||                                                                            |
||                                                                            |
||                                                                            |
+-----------------------------------------------------------------------------+
|i12 : assert isWellDefined g                                                 |
+-----------------------------------------------------------------------------+
|i13 : assert not isCommutative g                                             |
+-----------------------------------------------------------------------------+
+-----------------------------------------------------------------------------+
|i14 : h = randomSimplicialMap(D,C, Cycle => true);                           |
+-----------------------------------------------------------------------------+
|i15 : normalize h                                                            |
|                                                                             |
|           1               1                                                 |
|o15 = 0 : R  <----------- R  : 0                                             |
|                | -17 |                                                      |
|                                                                             |
|           3                                                                 |
|      1 : R  <---------------------------------------------------------------|
|                {1} | -17a-29b-44c -49ab-b2+48ac-50bc-40c2        33a2b-9ab2+|
|                {1} | 29a+17b-42c  49a2+ab-17b2+11ac+19bc-45c2    -33a3+9a2b-|
|                {1} | 44a+42b      -48a2+39ab-19b2+40ac+45bc+17c2 43a3+6a2b+2|
|                                                                             |
|           3                                                                 |
|      2 : R  <---------------------------------------------------------------|
|                {2} | 49a3+a2b+35ab2-30b3-a2c+37abc+48b2c+34ac2+16bc2-2c3    |
|                {2} | -48a3-50a2b+11ab2-41b3+40a2c-34abc+45b2c+17ac2+31bc2+44|
|                {2} | -12a3+18a2b-42ab2-23b3-22a2c+16abc-45b2c-31ac2-17bc2+42|
|                                                                             |
|           1                                                                 |
|      3 : R  <---------------------------------------------------------------|
|                {3} | -16a6-35a5b+12a4b2+7a3b3+46a2b4+8ab5+5b6-39a5c-24a4bc+1|
|                                                                             |
|o15 : ComplexMap                                                             |
|-----------------------------------------------------------------------------|
|                                                3                            |
|---------------------------------------------- R  : 1                        |
|46b3-43a2c+35abc+27b2c+50ac2+45bc2+49c3    |                                 |
|46ab2-41a2c-29abc-36b2c-47ac2+40bc2-45c3   |                                 |
|ab2+36b3-50a2c+2abc-40b2c-49ac2+45bc2-17c3 |                                 |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|   -33a4+9a3b-13a2b2-9ab3+46b4+21a3c-40a2bc-ab2c-7b3c-8a2c2-42abc2-44b2c2-49a|
|c3 43a4+45a3b-30a2b2+36ab3+34b4-50a3c-37a2bc-9ab2c-12b3c-49a2c2+49abc2+23b2c2|
|c3 -39a4-11a3b-6a2b2-36ab3-36b4+39a3c+19a2bc+10ab2c+40b3c-4a2c2+26abc2-45b2c2|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|6a3b2c+13a2b3c-36ab4c-38b5c-32a4c2-18a3bc2-13a2b2c2-34ab3c2+19b4c2-46a3c3-24a|
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|c3+26bc3+44c4     -33a2b3+9ab4-46b5-16a4c-35a3bc-4a2b2c-7ab3c+2b4c-39a3c2-24a|
|-17ac3+28bc3-44c4 16a4b+35a3b2+47a2b3-28ab4-29b5+39a3bc+24a2b2c-27ab3c-26b4c+|
|-28ac3+17bc3-42c4 -16a5-35a4b-4a3b2+34a2b3+31ab4+36b5-39a4c-24a3bc-23a2b2c+28|
|                                                                             |
|                                                             1               |
|----------------------------------------------------------- R  : 3           |
|2bc3+49ab2c3+5b3c3-48a2c4-5abc4+43b2c4-15ac5+42bc5+42c6 |                    |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|2bc2-23ab2c2-19b3c2+a2c3-27abc3+b2c3-15ac4+15bc4-39c5                        |
|32a2bc2+18ab2c2-4b3c2-43a2c3+abc3+11b2c3-3ac4+34bc4+9c5                      |
|ab3c-40b4c-32a3c2-18a2bc2-45ab2c2+45b3c2-7a2c3-13abc3+48b2c3+26ac4-42bc4+11c5|
|-----------------------------------------------------------------------------|
|      3                                                                      |
|---- R  : 2                                                                  |
| |                                                                           |
| |                                                                           |
| |                                                                           |
+-----------------------------------------------------------------------------+
|i16 : assert isWellDefined h                                                 |
+-----------------------------------------------------------------------------+
|i17 : assert isSimplicialMorphism h                                          |
+-----------------------------------------------------------------------------+

This method also checks the following aspects of the data structure:

  * The underlying hash table has exactly the expected keys, namely, source,
    target, degree, map, cache
  * The ring of the source and target are the same
  * The source and target are well defined simplicial modules
  * The degree is an integer
  * All keys in the map field are integers or sequences, in the range of the
    top degree of the source
  * The source and target of each $f_i$ is as expected
  * If the isCommutative key is present in the cache table, then commutativity
    of the map with the face/degeneracy maps is checked

See also
========

  * isWellDefined(SimplicialModule),
    -- whether a simplicial module is well-defined
  * isCommutative(SimplicialModuleMap)
    -- whether a simplicial module map
    commutes with the face/degeneracy maps
  * isSimplicialMorphism(SimplicialModuleMap)
    -- whether a simplicial
    module map is a morphism of simplicial modules
  * map(SimplicialModule,SimplicialModule,HashTable)
    -- make a map of
    simplicial modules

Ways to use this method:
========================

  * isWellDefined(SimplicialModuleMap)
    -- whether a map of simplicial
    modules is well-defined
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1691:0.

kernel(SimplicialModuleMap) -- make the kernel of a map of simplicial modules
*****************************************************************************

  * Function: kernel (Macaulay2Doc),
  * Usage:
        kernel f
        ker f
  * Inputs:
      * f, a Map of Simplicial Modules,
  * Optional inputs (Macaulay2Doc),:
      * DegreeLimit (missing documentation) => ..., default value {},
      * Strategy (missing documentation) => ..., default value {},
      * SubringLimit (Macaulay2Doc),
        => ..., default value infinity,
  * Outputs:
      * a Simplicial Module,

Description
===========

The kernel of a simplicial module map $f : C \to D$ is the simplicial module
$E$ whose $i$-th term is $kernel(f_i)$, and whose face/degeneracy map is
induced from the face/degeneracy map on the source.

In the following example, we first construct a random simplicial morphism $f :
C \to D$.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a,b,c,d];                                                    |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution ideal(b^2-a*c, b*c-a*d, c^2-b*d), 3,|
|                                                                             |
|      1      4      9      16                                                |
|o2 = S  <-- S  <-- S  <-- S  <-- ...                                         |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i3 : D = simplicialModule(freeResolution ideal(a,b,c), Degeneracy => true)   |
|                                                                             |
|      1      4      10      20                                               |
|o3 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : f = randomSimplicialMap(D, C, Boundary => true, InternalDegree => 0)    |
|                                                                             |
|          1         1                                                        |
|o4 = 0 : S  <----- S  : 0                                                    |
|               0                                                             |
|                                                                             |
|          4                                           4                      |
|     1 : S  <--------------------------------------- S  : 1                  |
|               {0} | 0 0        0        0       |                           |
|               {1} | 0 -24b+36c 29b-19c  10b+29c |                           |
|               {1} | 0 24a+30c  -29a-19c -10a+8c |                           |
|               {1} | 0 -36a-30b 19a+19b  -29a-8b |                           |
|                                                                             |
|          10                                                                 |
|     2 : S   <---------------------------------------------------------------|
|                {0} | 0 0        0        0       0        0        0       0|
|                {1} | 0 -24b+36c 29b-19c  10b+29c 0        0        0       0|
|                {1} | 0 24a+30c  -29a-19c -10a+8c 0        0        0       0|
|                {1} | 0 -36a-30b 19a+19b  -29a-8b 0        0        0       0|
|                {1} | 0 0        0        0       -24b+36c 29b-19c  10b+29c 0|
|                {1} | 0 0        0        0       24a+30c  -29a-19c -10a+8c 0|
|                {1} | 0 0        0        0       -36a-30b 19a+19b  -29a-8b 0|
|                {2} | 0 0        0        0       0        0        0       1|
|                {2} | 0 0        0        0       0        0        0       2|
|                {2} | 0 0        0        0       0        0        0       -|
|                                                                             |
|          20                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {0} | 0 0        0        0       0        0        0       0|
|                {1} | 0 -24b+36c 29b-19c  10b+29c 0        0        0       0|
|                {1} | 0 24a+30c  -29a-19c -10a+8c 0        0        0       0|
|                {1} | 0 -36a-30b 19a+19b  -29a-8b 0        0        0       0|
|                {1} | 0 0        0        0       -24b+36c 29b-19c  10b+29c 0|
|                {1} | 0 0        0        0       24a+30c  -29a-19c -10a+8c 0|
|                {1} | 0 0        0        0       -36a-30b 19a+19b  -29a-8b 0|
|                {1} | 0 0        0        0       0        0        0       -|
|                {1} | 0 0        0        0       0        0        0       2|
|                {1} | 0 0        0        0       0        0        0       -|
|                {2} | 0 0        0        0       0        0        0       0|
|                {2} | 0 0        0        0       0        0        0       0|
|                {2} | 0 0        0        0       0        0        0       0|
|                {2} | 0 0        0        0       0        0        0       0|
|                {2} | 0 0        0        0       0        0        0       0|
|                {2} | 0 0        0        0       0        0        0       0|
|                {2} | 0 0        0        0       0        0        0       0|
|                {2} | 0 0        0        0       0        0        0       0|
|                {2} | 0 0        0        0       0        0        0       0|
|                {3} | 0 0        0        0       0        0        0       0|
|                                                                             |
|o4 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                 9                                           |
|------------------------------- S  : 2                                       |
|            0               |                                                |
|            0               |                                                |
|            0               |                                                |
|            0               |                                                |
|            0               |                                                |
|            0               |                                                |
|            0               |                                                |
|0a-29b-46c  -10b+24d        |                                                |
|9a+41b+36c  -19c-36d        |                                                |
|14a+19b+30c -29a-8b-19c-30d |                                                |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|        0        0       0            0               0            0         |
|        0        0       0            0               0            0         |
|        0        0       0            0               0            0         |
|        0        0       0            0               0            0         |
|        0        0       0            0               0            0         |
|        0        0       0            0               0            0         |
|        0        0       0            0               0            0         |
|24b+36c 29b-19c  10b+29c 0            0               0            0         |
|4a+30c  -29a-19c -10a+8c 0            0               0            0         |
|36a-30b 19a+19b  -29a-8b 0            0               0            0         |
|        0        0       10a-29b-46c  -10b+24d        0            0         |
|        0        0       29a+41b+36c  -19c-36d        0            0         |
|        0        0       -14a+19b+30c -29a-8b-19c-30d 0            0         |
|        0        0       0            0               10a-29b-46c  -10b+24d  |
|        0        0       0            0               29a+41b+36c  -19c-36d  |
|        0        0       0            0               -14a+19b+30c -29a-8b-19|
|        0        0       0            0               0            0         |
|        0        0       0            0               0            0         |
|        0        0       0            0               0            0         |
|        0        0       0            0               0            0         |
|-----------------------------------------------------------------------------|
|                                        16                                   |
|-------------------------------------- S   : 3                               |
|      0            0               |                                         |
|      0            0               |                                         |
|      0            0               |                                         |
|      0            0               |                                         |
|      0            0               |                                         |
|      0            0               |                                         |
|      0            0               |                                         |
|      0            0               |                                         |
|      0            0               |                                         |
|      0            0               |                                         |
|      0            0               |                                         |
|      0            0               |                                         |
|      0            0               |                                         |
|      0            0               |                                         |
|      0            0               |                                         |
|c-30d 0            0               |                                         |
|      10a-29b-46c  -10b+24d        |                                         |
|      29a+41b+36c  -19c-36d        |                                         |
|      -14a+19b+30c -29a-8b-19c-30d |                                         |
|      0            0               |                                         |
+-----------------------------------------------------------------------------+
|i5 : prune ker f                                                             |
|                                                                             |
|      1      2      3      4                                                 |
|o5 = S  <-- S  <-- S  <-- S <-- ...                                          |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o5 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i6 : h1 = inducedMap(source f, ker f)                                        |
|                                                                             |
|          1             1                                                    |
|o6 = 0 : S  <--------- S  : 0                                                |
|               | 1 |                                                         |
|                                                                             |
|          4                                                                  |
|     1 : S  <------------------------- image {0} | 1 0           | : 1       |
|               {0} | 1 0           |         {2} | 0 18a+b+39c   |           |
|               {2} | 0 18a+b+39c   |         {2} | 0 25a-40b-35c |           |
|               {2} | 0 25a-40b-35c |         {2} | 0 a-23b-40c   |           |
|               {2} | 0 a-23b-40c   |                                         |
|                                                                             |
|          9                                                                  |
|     2 : S  <------------------------------------- image {0} | 1 0           |
|               {0} | 1 0           0           |         {2} | 0 18a+b+39c   |
|               {2} | 0 18a+b+39c   0           |         {2} | 0 25a-40b-35c |
|               {2} | 0 25a-40b-35c 0           |         {2} | 0 a-23b-40c   |
|               {2} | 0 a-23b-40c   0           |         {2} | 0 0           |
|               {2} | 0 0           18a+b+39c   |         {2} | 0 0           |
|               {2} | 0 0           25a-40b-35c |         {2} | 0 0           |
|               {2} | 0 0           a-23b-40c   |         {3} | 0 0           |
|               {3} | 0 0           0           |         {3} | 0 0           |
|               {3} | 0 0           0           |                             |
|                                                                             |
|          16                                                                 |
|     3 : S   <------------------------------------------------- image {0} | 1|
|                {0} | 1 0           0           0           |         {2} | 0|
|                {2} | 0 18a+b+39c   0           0           |         {2} | 0|
|                {2} | 0 25a-40b-35c 0           0           |         {2} | 0|
|                {2} | 0 a-23b-40c   0           0           |         {2} | 0|
|                {2} | 0 0           18a+b+39c   0           |         {2} | 0|
|                {2} | 0 0           25a-40b-35c 0           |         {2} | 0|
|                {2} | 0 0           a-23b-40c   0           |         {2} | 0|
|                {2} | 0 0           0           18a+b+39c   |         {2} | 0|
|                {2} | 0 0           0           25a-40b-35c |         {2} | 0|
|                {2} | 0 0           0           a-23b-40c   |         {3} | 0|
|                {3} | 0 0           0           0           |         {3} | 0|
|                {3} | 0 0           0           0           |         {3} | 0|
|                {3} | 0 0           0           0           |         {3} | 0|
|                {3} | 0 0           0           0           |         {3} | 0|
|                {3} | 0 0           0           0           |         {3} | 0|
|                {3} | 0 0           0           0           |                |
|                                                                             |
|o6 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|0           | : 2                                                            |
|0           |                                                                |
|0           |                                                                |
|0           |                                                                |
|18a+b+39c   |                                                                |
|25a-40b-35c |                                                                |
|a-23b-40c   |                                                                |
|0           |                                                                |
|0           |                                                                |
|                                                                             |
|                                                                             |
|                                                                             |
| 0           0           0           | : 3                                   |
| 18a+b+39c   0           0           |                                       |
| 25a-40b-35c 0           0           |                                       |
| a-23b-40c   0           0           |                                       |
| 0           18a+b+39c   0           |                                       |
| 0           25a-40b-35c 0           |                                       |
| 0           a-23b-40c   0           |                                       |
| 0           0           18a+b+39c   |                                       |
| 0           0           25a-40b-35c |                                       |
| 0           0           a-23b-40c   |                                       |
| 0           0           0           |                                       |
| 0           0           0           |                                       |
| 0           0           0           |                                       |
| 0           0           0           |                                       |
| 0           0           0           |                                       |
| 0           0           0           |                                       |
+-----------------------------------------------------------------------------+
|i7 : ker f == image h1                                                       |
|                                                                             |
|o7 = true                                                                    |
+-----------------------------------------------------------------------------+
|i8 : ker h1 == 0                                                             |
|                                                                             |
|o8 = true                                                                    |
+-----------------------------------------------------------------------------+

See also
========

  * Making simplicial modules -- information
    about the basic constructors
  * Making maps between simplicial modules
    -- information about the basic constructors
  * kernel (Macaulay2Doc) -- kernel of a map
  * image(SimplicialModuleMap) -- make
    the image of a map of simplicial modules
  * coimage(SimplicialModuleMap), --
    make the coimage of a map of simplicial modules
  * cokernel(SimplicialModuleMap), --
    make the cokernel of a map of simplicial modules

Ways to use this method:
========================

  * kernel(SimplicialModuleMap) -- make
    the kernel of a map of simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2854:0.

Making maps between simplicial modules -- information about the basic constructors
**********************************************************************************

Basic constructors
==================

  * map(SimplicialModule,SimplicialModule,HashTable)
    -- make a map of
    simplicial modules
  * map(SimplicialModule,SimplicialModule,ZZ)
    -- make the zero map or
    identity between simplicial modules
  * map(SimplicialModule,SimplicialModule,SimplicialModuleMap)
    -- make
    a new map of simplicial modules, induced from an existing one
  * id _ SimplicialModule -- the identity map
    of a simplicial module
  * face/degeneracy maps of a simplicial module
    -- access the face and degeneracy maps of a simplicial
    module
  * SimplicialModuleMap Array -- shift a
    simplicial module or simplicial module map
  * isWellDefined(SimplicialModuleMap)
    -- whether a map of simplicial
    modules is well-defined

Important computations creating new simplicial module maps
==========================================================

  * SimplicialModule ** Matrix -- create
    the tensor product of a simplicial module and a map of modules

Canonical maps between simplicial modules
=========================================

Some simplicial modules come with canonical maps. These are best accessed using
inducedMap (Macaulay2Doc).

  * kernel(SimplicialModuleMap) -- make
    the kernel of a map of simplicial modules
  * cokernel(SimplicialModuleMap), --
    make the cokernel of a map of simplicial modules
  * image(SimplicialModuleMap) -- make
    the image of a map of simplicial modules
  * coimage(SimplicialModuleMap), --
    make the coimage of a map of simplicial modules
  * inducedMap(SimplicialModule,SimplicialModule)
    -- make the map of
    simplicial modules induced at each term by the identity map

Random maps of simplicial modules
=================================

The method randomSimplicialMap(SimplicialModule,SimplicialModule),
allows one to
construct random simplicial module maps, random morphisms between simplicial
modules, and random null homotopies between simplicial modules.

  * isCommutative(SimplicialModuleMap)
    -- whether a simplicial module map
    commutes with the face/degeneracy maps
  * isSimplicialMorphism(SimplicialModuleMap)
    -- whether a simplicial
    module map is a morphism of simplicial modules

Elementary operations on simplicial module maps
===============================================

  * arithmetic with simplicial module maps
    -- perform arithmetic operations on simplicial module maps
  * SimplicialModuleMap + SimplicialModuleMap
    -- perform arithmetic operations on simplicial module maps
  * SimplicialModuleMap | SimplicialModuleMap
    -- join or concatenate maps horizontally
  * SimplicialModuleMap || SimplicialModuleMap
    -- join or concatenate maps vertically
  * SimplicialModuleMap ++ SimplicialModuleMap
    -- direct sum of simplicial module maps
  * SimplicialModuleMap ** SimplicialModuleMap
    -- the map of simplicial modules between tensor
    simplicial modules
  * SimplicialModuleMap _ Array -- the
    composition with the canonical inclusion or projection map
  * SimplicialModuleMap ^ Array -- the
    composition with the canonical inclusion or projection map
  * truncate(List,SimplicialModuleMap)
    -- truncation of a simplicial
    module map at a specified degree or set of degrees
  * RingMap SimplicialModuleMap -- apply a
    ring map to a map of simplicial modules
  * RingMap ** SimplicialModuleMap,
    -- tensor a map of simplicial modules along a ring map

See also
========

  * Making simplicial modules -- information
    about the basic constructors
  * Basic invariants and properties, --
    information about accessing basic features
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1326:0.

Making simplicial modules -- information about the basic constructors
*********************************************************************

Basic constructors
==================

  * simplicialModule(HashTable,HashTable,HashTable,ZZ),
    -- make a simplicial module
  * simplicialModule(Module,ZZ), --
    make a simplicial module associated to a complex concentrated in degree 0
  * SimplicialModule Array -- shift a simplicial
    module or simplicial module map
  * isWellDefined(SimplicialModule),
    -- whether a simplicial module is well-defined

Important computations creating new simplicial modules
======================================================

  * simplicialModule(Complex),
    -- compute the image of a non-negatively graded complex under the Dold-Kan
    correspondence
  * simplicialTensor(Complex,Complex) -- compute the
    simplicial tensor product and cache direct sum indices for easy access

More advanced constructors
==========================

  * SimplicialModule ++ SimplicialModule
    -- direct sum of simplicial modules
  * SimplicialModule ** SimplicialModule
    -- tensor product of simplicial modules
  * RingMap SimplicialModule -- apply a ring
    map to a simplicial module
  * RingMap ** SimplicialModule, --
    tensor a simplicial module along a ring map
  * minimalPresentation(SimplicialModule)
    -- minimal presentation of all
    terms in a simplicial module
  * truncate(List,SimplicialModule)
    -- truncation of a simplicial module
    at a specified degree or set of degrees

Extracting simplicial modules from simplicial maps
==================================================

  * source(SimplicialModuleMap) -- get
    the source of a map of simplicial modules
  * target(SimplicialModuleMap) -- get
    the target of a map of simplicial modules
  * kernel(SimplicialModuleMap) -- make
    the kernel of a map of simplicial modules
  * cokernel(SimplicialModuleMap), --
    make the cokernel of a map of simplicial modules
  * image(SimplicialModuleMap) -- make
    the image of a map of simplicial modules
  * coimage(SimplicialModuleMap), --
    make the coimage of a map of simplicial modules

See also
========

  * Making maps between simplicial modules
    -- information about the basic constructors
  * Basic invariants and properties, --
    information about accessing basic features
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:292:0.

map(...,Degree=>...) -- specify the degree of a map of simplicial modules
*************************************************************************

Description
===========

When constructing a map of simplicial modules, this option specifies the degree
$d$ of the resulting map.  A map of degree $d$ consists of component maps $f_i
: C_i \to D_{i+d}$. By default, the degree is 0.

See also
========

  * map(SimplicialModule,SimplicialModule,HashTable)
    -- make a map of
    simplicial modules

Functions with optional argument named Degree:
==============================================

  * "basis(...,Degree=>...)" -- see basis (Macaulay2Doc), -- basis
    or generating set of all or part of a ring, ideal or module
  * "cohomology(...,Degree=>...)" -- see cohomology (Macaulay2Doc),
    -- general cohomology functor
  * inducedMap(...,Degree=>...) (Macaulay2Doc)
    -- specify the
    degree of a map
  * map(...,Degree=>...) (Macaulay2Doc)
    -- specify the degree
    of a map
  * "matrix(...,Degree=>...)" -- see matrix(List) (Macaulay2Doc),
    -- create a matrix from a doubly-nested
    list of ring elements or matrices
  * "randomComplexMap(...,Degree=>...)" -- see
    randomComplexMap(Complex,Complex) (Complexes),
    -- a random map of
    chain complexes
  * "randomSimplicialMap(...,Degree=>...)" -- see
    randomSimplicialMap(SimplicialModule,SimplicialModule),
    -- a random
    map of simplicial modules

Further information
===================

  * Default value: null
  * Function: map (Macaulay2Doc), -- make a map
  * Option key: Degree (Macaulay2Doc), -- an optional argument
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:4140:0.

map(...,DegreeLift=>...) -- unused option for maps of simplicial modules
************************************************************************

Description
===========

This option is inherited from the core map method and is not used for maps of
simplicial modules.

Functions with optional argument named DegreeLift:
==================================================

  * "map(...,DegreeLift=>...)" -- see map(Ring,Ring,Matrix) (Macaulay2Doc),
    -- make a ring map
  * monoid(...,DegreeLift=>...) (Macaulay2Doc)
    -- specify
    maps between degree groups
  * "newRing(...,DegreeLift=>...)" -- see newRing (Macaulay2Doc),
    -- make a copy of a ring, with some features changed
  * "symmetricAlgebra(...,DegreeLift=>...)" -- see symmetricAlgebra (Macaulay2Doc),
    -- the symmetric algebra of a module
  * "tensor(Monoid,Monoid,DegreeLift=>...)" -- see tensor(Monoid,Monoid) (Macaulay2Doc),
    -- tensor product of monoids
  * "tensor(Ring,Ring,DegreeLift=>...)" -- see tensor(Monoid,Monoid) (Macaulay2Doc),
    -- tensor product of monoids

Further information
===================

  * Default value: null
  * Function: map (Macaulay2Doc), -- make a map
  * Option key: DegreeLift (Macaulay2Doc), -- an optional
    argument
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:4151:0.

map(...,DegreeMap=>...) -- unused option for maps of simplicial modules
***********************************************************************

Description
===========

This option is inherited from the core map method and is not used for maps of
simplicial modules.

Functions with optional argument named DegreeMap:
=================================================

  * "map(...,DegreeMap=>...)" -- see map(Ring,Ring,Matrix) (Macaulay2Doc),
    -- make a ring map
  * "monoid(...,DegreeMap=>...)" -- see monoid(...,DegreeLift=>...) (Macaulay2Doc),
    -- specify
    maps between degree groups
  * "newRing(...,DegreeMap=>...)" -- see newRing (Macaulay2Doc),
    -- make a copy of a ring, with some features changed
  * "symmetricAlgebra(...,DegreeMap=>...)" -- see symmetricAlgebra (Macaulay2Doc),
    -- the symmetric algebra of a module
  * "tensor(Monoid,Monoid,DegreeMap=>...)" -- see tensor(Monoid,Monoid) (Macaulay2Doc),
    -- tensor product of monoids
  * "tensor(Ring,Ring,DegreeMap=>...)" -- see tensor(Monoid,Monoid) (Macaulay2Doc),
    -- tensor product of monoids

Further information
===================

  * Default value: null
  * Function: map (Macaulay2Doc), -- make a map
  * Option key: DegreeMap (Macaulay2Doc), -- an optional
    argument
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:4162:0.

map(SimplicialModule,SimplicialModule,HashTable) -- make a map of simplicial modules
************************************************************************************

  * Function: map (Macaulay2Doc),
  * Usage:
        f = map(D, C, H)
  * Inputs:
      * C, a Simplicial Module,
      * D, a Simplicial Module,
      * H, a hash table (Macaulay2Doc), whose keys are
        integers, and whose values are the maps between the corresponding terms
  * Optional inputs (Macaulay2Doc),:
      * Degree, => ..., default
        value null, specify the degree of a map of simplicial modules
      * DegreeLift, => ...,
        default value null, unused option for maps of simplicial modules
      * DegreeMap, => ...,
        default value null, unused option for maps of simplicial modules
  * Outputs:
      * f, a Map of Simplicial Modules,

Description
===========

A map of simplicial modules $f : C \rightarrow D$ of degree $d$ is a sequence
of maps $f_i : C_i \rightarrow D_{d+i}$. No relationship between the maps $f_i$
and the face/degeneracy maps of either $C$ or $D$ is assumed.

We construct a map of simplicial modules by specifying the individual maps
between the terms. Note that this constructor is typically used more behind the
scenes, and using the simplicialModule, or basic map
constructors is typically much more efficient than typing the maps by hand.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[a,b,c];                                                      |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(F = freeResolution coker                           |
|                                                                             |
|      1      4      10      20                                               |
|o2 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|matrix{{a^2-b^2,b^3-c^3,c^4}}, Degeneracy => true)                           |
+-----------------------------------------------------------------------------+
|i3 : D = simplicialModule(G = freeResolution coker vars R, Degeneracy =>     |
|                                                                             |
|      1      4      10      20                                               |
|o3 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o3 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|true)                                                                        |
+-----------------------------------------------------------------------------+
|i4 : H = hashTable { 0 => map(D_0, C_0, 1),                                  |
|         1 => map(D_1, C_1, {{1, 0, 0, 0}, {0, a, 0, 0}, {0, -b, b^2, 0},    |
|         2 => map(D_2, C_2, {{1, 0, 0, 0, 0, 0, 0, 0, 0, 0},                 |
|                              {0, a, 0, 0, 0, 0,0, 0, 0, 0},                 |
|                              {0, -b, b^2, 0, 0, 0, 0, 0, 0, 0},             |
|                              {0, 0,-c^2, c^3, 0, 0, 0, 0, 0, 0},            |
|                              {0, 0, 0, 0, a, 0, 0, 0, 0,0},                 |
|                              {0, 0, 0, 0, -b, b^2, 0, 0, 0, 0},             |
|                              {0, 0, 0, 0, 0,-c^2, c^3, 0, 0, 0},            |
|                              {0, 0, 0, 0, 0, 0, 0, a*b^2, 0, 0},            |
|                              {0, 0, 0, 0, 0, 0, 0, -a*c^2, a*c^3, 0},       |
|                              {0, 0, 0, 0, 0, 0, 0, b*c^2, -b*c^3, b^2*c^    |
|         3 => map(D_3, C_3, {{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0    |
|                             {0, a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0    |
|                             {0, -b, b^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0,    |
|                             {0, 0, -c^2, c^3, 0, 0, 0, 0, 0, 0,0, 0, 0,     |
|                             {0, 0, 0, 0, a, 0, 0, 0, 0,0, 0, 0, 0, 0, 0,    |
|                             {0, 0, 0, 0, -b, b^2, 0,0, 0, 0, 0, 0, 0, 0,    |
|                             {0, 0, 0, 0, 0,-c^2, c^3, 0, 0, 0, 0, 0, 0,     |
|                             {0, 0,0, 0, 0, 0, 0, a, 0, 0, 0, 0, 0, 0, 0,    |
|                             {0,0, 0, 0, 0, 0, 0, -b, b^2, 0, 0, 0, 0, 0,    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, -c^2, c^3, 0, 0, 0,    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a*b^2, 0, 0,0    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-a*c^2, a*c^3,    |
|                             {0, 0, 0, 0, 0, 0,0, 0, 0, 0, b*c^2, -b*c^3,    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a*b^    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -a*c    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, b*c^2    |
|                             {0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0,    |
|                             {0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0    |
|                             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0,    |
|                     }                                                       |
|                                                                             |
|o4 = HashTable{0 => | 1 |                                                    |
|               1 => {0} | 1 0  0   0  |                                      |
|                    {1} | 0 a  0   0  |                                      |
|                    {1} | 0 -b b2  0  |                                      |
|                    {1} | 0 0  -c2 c3 |                                      |
|               2 => {0} | 1 0  0   0  0  0   0  0    0    0    |             |
|                    {1} | 0 a  0   0  0  0   0  0    0    0    |             |
|                    {1} | 0 -b b2  0  0  0   0  0    0    0    |             |
|                    {1} | 0 0  -c2 c3 0  0   0  0    0    0    |             |
|                    {1} | 0 0  0   0  a  0   0  0    0    0    |             |
|                    {1} | 0 0  0   0  -b b2  0  0    0    0    |             |
|                    {1} | 0 0  0   0  0  -c2 c3 0    0    0    |             |
|                    {2} | 0 0  0   0  0  0   0  ab2  0    0    |             |
|                    {2} | 0 0  0   0  0  0   0  -ac2 ac3  0    |             |
|                    {2} | 0 0  0   0  0  0   0  bc2  -bc3 b2c3 |             |
|               3 => {0} | 1 0  0   0  0  0   0  0  0   0  0    0    0        |
|                    {1} | 0 a  0   0  0  0   0  0  0   0  0    0    0        |
|                    {1} | 0 -b b2  0  0  0   0  0  0   0  0    0    0        |
|                    {1} | 0 0  -c2 c3 0  0   0  0  0   0  0    0    0        |
|                    {1} | 0 0  0   0  a  0   0  0  0   0  0    0    0        |
|                    {1} | 0 0  0   0  -b b2  0  0  0   0  0    0    0        |
|                    {1} | 0 0  0   0  0  -c2 c3 0  0   0  0    0    0        |
|                    {1} | 0 0  0   0  0  0   0  a  0   0  0    0    0        |
|                    {1} | 0 0  0   0  0  0   0  -b b2  0  0    0    0        |
|                    {1} | 0 0  0   0  0  0   0  0  -c2 c3 0    0    0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  ab2  0    0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  -ac2 ac3  0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  bc2  -bc3 b2c3     |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0        |
|                    {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0        |
|                    {3} | 0 0  0   0  0  0   0  0  0   0  0    0    0        |
|                                                                             |
|o4 : HashTable                                                               |
|-----------------------------------------------------------------------------|
| {0, 0, -c^2, c^3}}),                                                        |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|3}}),                                                                        |
|, 0,0, 0, 0, 0},                                                             |
|,0, 0, 0, 0, 0},                                                             |
| 0, 0, 0, 0, 0, 0},                                                          |
|0, 0, 0, 0, 0, 0, 0},                                                        |
| 0, 0, 0, 0, 0},                                                             |
| 0, 0, 0, 0, 0, 0},                                                          |
|0, 0, 0, 0, 0, 0, 0},                                                        |
| 0, 0, 0, 0, 0},                                                             |
| 0, 0, 0, 0, 0,0},                                                           |
| 0, 0, 0,0, 0, 0, 0},                                                        |
|, 0, 0, 0, 0, 0, 0},                                                         |
| 0, 0, 0, 0, 0, 0, 0,0},                                                     |
| b^2*c^3, 0, 0, 0, 0, 0, 0, 0},                                              |
|2, 0, 0, 0, 0,0, 0},                                                         |
|^2,a*c^3, 0, 0, 0, 0, 0},                                                    |
|, -b*c^3, b^2*c^3, 0, 0, 0, 0},                                              |
| 0, a*b^2, 0, 0, 0},                                                         |
| 0, -a*c^2, a*c^3,0, 0},                                                     |
|, 0, b*c^2, -b*c^3, b^2*c^3, 0},                                             |
| 0, 0, 0, 0, a*b^2*c^3}})                                                    |
|                                                                             |
|                                                                             |
|                                     }                                       |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|0    0    0    0    0    0    0     |                                        |
|ab2  0    0    0    0    0    0     |                                        |
|-ac2 ac3  0    0    0    0    0     |                                        |
|bc2  -bc3 b2c3 0    0    0    0     |                                        |
|0    0    0    ab2  0    0    0     |                                        |
|0    0    0    -ac2 ac3  0    0     |                                        |
|0    0    0    bc2  -bc3 b2c3 0     |                                        |
|0    0    0    0    0    0    ab2c3 |                                        |
+-----------------------------------------------------------------------------+
|i5 : f = map(D, C, H)                                                        |
|                                                                             |
|          1             1                                                    |
|o5 = 0 : R  <--------- R  : 0                                                |
|               | 1 |                                                         |
|                                                                             |
|          4                           4                                      |
|     1 : R  <----------------------- R  : 1                                  |
|               {0} | 1 0  0   0  |                                           |
|               {1} | 0 a  0   0  |                                           |
|               {1} | 0 -b b2  0  |                                           |
|               {1} | 0 0  -c2 c3 |                                           |
|                                                                             |
|          10                                                    10           |
|     2 : R   <------------------------------------------------ R   : 2       |
|                {0} | 1 0  0   0  0  0   0  0    0    0    |                 |
|                {1} | 0 a  0   0  0  0   0  0    0    0    |                 |
|                {1} | 0 -b b2  0  0  0   0  0    0    0    |                 |
|                {1} | 0 0  -c2 c3 0  0   0  0    0    0    |                 |
|                {1} | 0 0  0   0  a  0   0  0    0    0    |                 |
|                {1} | 0 0  0   0  -b b2  0  0    0    0    |                 |
|                {1} | 0 0  0   0  0  -c2 c3 0    0    0    |                 |
|                {2} | 0 0  0   0  0  0   0  ab2  0    0    |                 |
|                {2} | 0 0  0   0  0  0   0  -ac2 ac3  0    |                 |
|                {2} | 0 0  0   0  0  0   0  bc2  -bc3 b2c3 |                 |
|                                                                             |
|          20                                                                 |
|     3 : R   <---------------------------------------------------------------|
|                {0} | 1 0  0   0  0  0   0  0  0   0  0    0    0    0    0  |
|                {1} | 0 a  0   0  0  0   0  0  0   0  0    0    0    0    0  |
|                {1} | 0 -b b2  0  0  0   0  0  0   0  0    0    0    0    0  |
|                {1} | 0 0  -c2 c3 0  0   0  0  0   0  0    0    0    0    0  |
|                {1} | 0 0  0   0  a  0   0  0  0   0  0    0    0    0    0  |
|                {1} | 0 0  0   0  -b b2  0  0  0   0  0    0    0    0    0  |
|                {1} | 0 0  0   0  0  -c2 c3 0  0   0  0    0    0    0    0  |
|                {1} | 0 0  0   0  0  0   0  a  0   0  0    0    0    0    0  |
|                {1} | 0 0  0   0  0  0   0  -b b2  0  0    0    0    0    0  |
|                {1} | 0 0  0   0  0  0   0  0  -c2 c3 0    0    0    0    0  |
|                {2} | 0 0  0   0  0  0   0  0  0   0  ab2  0    0    0    0  |
|                {2} | 0 0  0   0  0  0   0  0  0   0  -ac2 ac3  0    0    0  |
|                {2} | 0 0  0   0  0  0   0  0  0   0  bc2  -bc3 b2c3 0    0  |
|                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    ab2  0  |
|                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    -ac2 ac3|
|                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    bc2  -bc|
|                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0  |
|                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0  |
|                {2} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0  |
|                {3} | 0 0  0   0  0  0   0  0  0   0  0    0    0    0    0  |
|                                                                             |
|o5 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                 20                                          |
|------------------------------- R   : 3                                      |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|  0    0    0    0    0     |                                                |
|3 b2c3 0    0    0    0     |                                                |
|  0    ab2  0    0    0     |                                                |
|  0    -ac2 ac3  0    0     |                                                |
|  0    bc2  -bc3 b2c3 0     |                                                |
|  0    0    0    0    ab2c3 |                                                |
+-----------------------------------------------------------------------------+
|i6 : assert isWellDefined f                                                  |
+-----------------------------------------------------------------------------+
|i7 : assert isHomogeneous f                                                  |
+-----------------------------------------------------------------------------+
|i8 : assert(degree f == 0)                                                   |
+-----------------------------------------------------------------------------+
|i9 : assert isSimplicialMorphism f                                           |
+-----------------------------------------------------------------------------+

The keys in the hash table index the terms in the source of the map.  If a key
is missing, that map is taken to be the zero map. We illustrate this by
constructing the 0 map as follows:

+---------------------------------+
|i10 : h = map(C, C, hashTable {})|
|                                 |
|o10 = 0                          |
|                                 |
|o10 : SimplicialModuleMap        |
+---------------------------------+
|i11 : h == 0                     |
|                                 |
|o11 = true                       |
+---------------------------------+

This is the primary constructor used by all of the more user friendly methods
for constructing a simplicial module map.

Caveat
======

This constructor minimizes computation and does very little error checking. To
verify that a simplicial module map is well constructed, use
isWellDefined(SimplicialModuleMap).

See also
========

  * SimplicialModuleMap -- the class of all maps
    between simplicial modules
  * isWellDefined(SimplicialModuleMap)
    -- whether a map of simplicial
    modules is well-defined
  * isHomogeneous(SimplicialModuleMap)
    -- whether a map of simplicial
    modules is homogeneous
  * degree(SimplicialModuleMap) -- get
    the degree of a map of simplicial modules
  * isSimplicialMorphism(SimplicialModuleMap)
    -- whether a simplicial
    module map is a morphism of simplicial modules
  * isCommutative(SimplicialModuleMap)
    -- whether a simplicial module map
    commutes with the face/degeneracy maps
  * source(SimplicialModuleMap) -- get
    the source of a map of simplicial modules
  * target(SimplicialModuleMap) -- get
    the target of a map of simplicial modules

Ways to use this method:
========================

  * map(SimplicialModule,SimplicialModule,HashTable)
    -- make a map of
    simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1450:0.

map(SimplicialModule,SimplicialModule,SimplicialModuleMap) -- make a new map of simplicial modules, induced from an existing one
********************************************************************************************************************************

  * Function: map (Macaulay2Doc),
  * Usage:
        g = map(D, C, f)
  * Inputs:
      * C, a Simplicial Module,
      * D, a Simplicial Module,
      * f, a Map of Simplicial Modules, regarded as
        providing matrices which induce maps between the terms of $C$ and $D$
  * Optional inputs (Macaulay2Doc),:
      * Degree, => ..., default
        value null, specify the degree of a map of simplicial modules
      * DegreeLift, => ...,
        default value null, unused option for maps of simplicial modules
      * DegreeMap, => ...,
        default value null, unused option for maps of simplicial modules
  * Outputs:
      * g, a Map of Simplicial Modules,

Description
===========

A map of simplicial modules $f : C' \rightarrow D'$ is a sequence of maps $f_i
: C'_i \rightarrow D'_{d'+i}$. The new map $g : C \rightarrow D$ is the
sequence of maps $g_i : C_i \rightarrow D_{d+i}$ induced by the matrix of
$f_i$.

One use for this function is to get the new map of simplicial modules induced
by forgetting the underlying complexes of the source and target, assuming that
the source and target are obtained as Dold-Kan images.

+--------------------------------------------------------------------------+
|i1 : R = ZZ/101[a,b,c];                                                   |
+--------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution coker vars R, Degeneracy => true)|
|                                                                          |
|      1      4      10      20                                            |
|o2 = R  <-- R  <-- R   <-- R  <-- ...                                     |
|                                                                          |
|     0      1      2       3                                              |
|                                                                          |
|o2 : SimplicialModule                                                     |
+--------------------------------------------------------------------------+
|i3 : f = map(forgetComplex C, forgetComplex C, id_C)                      |
|                                                                          |
|          1             1                                                 |
|o3 = 0 : R  <--------- R  : 0                                             |
|               | 1 |                                                      |
|                                                                          |
|          4                       4                                       |
|     1 : R  <------------------- R  : 1                                   |
|               {0} | 1 0 0 0 |                                            |
|               {1} | 0 1 0 0 |                                            |
|               {1} | 0 0 1 0 |                                            |
|               {1} | 0 0 0 1 |                                            |
|                                                                          |
|          10                                   10                         |
|     2 : R   <------------------------------- R   : 2                     |
|                {0} | 1 0 0 0 0 0 0 0 0 0 |                               |
|                {1} | 0 1 0 0 0 0 0 0 0 0 |                               |
|                {1} | 0 0 1 0 0 0 0 0 0 0 |                               |
|                {1} | 0 0 0 1 0 0 0 0 0 0 |                               |
|                {1} | 0 0 0 0 1 0 0 0 0 0 |                               |
|                {1} | 0 0 0 0 0 1 0 0 0 0 |                               |
|                {1} | 0 0 0 0 0 0 1 0 0 0 |                               |
|                {2} | 0 0 0 0 0 0 0 1 0 0 |                               |
|                {2} | 0 0 0 0 0 0 0 0 1 0 |                               |
|                {2} | 0 0 0 0 0 0 0 0 0 1 |                               |
|                                                                          |
|          20                                                       20     |
|     3 : R   <--------------------------------------------------- R   : 3 |
|                {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |           |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |           |
|                                                                          |
|o3 : SimplicialModuleMap                                                  |
+--------------------------------------------------------------------------+
|i4 : assert isWellDefined f                                               |
+--------------------------------------------------------------------------+
|i5 : assert(degree f == 0)                                                |
+--------------------------------------------------------------------------+
|i6 : assert isCommutative f                                               |
+--------------------------------------------------------------------------+
|i7 : assert isSimplicialMorphism f                                        |
+--------------------------------------------------------------------------+
|i8 : normalize f --notice how the normalization is not already pruned     |
|                                                                          |
|          1             1                                                 |
|o8 = 0 : R  <--------- R  : 0                                             |
|               | 1 |                                                      |
|                                                                          |
|     1 : image {0} | 0 0 0 | <----------------- image {0} | 0 0 0 | : 1   |
|               {1} | 1 0 0 |    {1} | 1 0 0 |         {1} | 1 0 0 |       |
|               {1} | 0 1 0 |    {1} | 0 1 0 |         {1} | 0 1 0 |       |
|               {1} | 0 0 1 |    {1} | 0 0 1 |         {1} | 0 0 1 |       |
|                                                                          |
|     2 : image {0} | 0 0 0 | <----------------- image {0} | 0 0 0 | : 2   |
|               {1} | 0 0 0 |    {2} | 1 0 0 |         {1} | 0 0 0 |       |
|               {1} | 0 0 0 |    {2} | 0 1 0 |         {1} | 0 0 0 |       |
|               {1} | 0 0 0 |    {2} | 0 0 1 |         {1} | 0 0 0 |       |
|               {1} | 0 0 0 |                          {1} | 0 0 0 |       |
|               {1} | 0 0 0 |                          {1} | 0 0 0 |       |
|               {1} | 0 0 0 |                          {1} | 0 0 0 |       |
|               {2} | 1 0 0 |                          {2} | 1 0 0 |       |
|               {2} | 0 1 0 |                          {2} | 0 1 0 |       |
|               {2} | 0 0 1 |                          {2} | 0 0 1 |       |
|                                                                          |
|     3 : image {0} | 0 | <------------- image {0} | 0 | : 3               |
|               {1} | 0 |    {3} | 1 |         {1} | 0 |                   |
|               {1} | 0 |                      {1} | 0 |                   |
|               {1} | 0 |                      {1} | 0 |                   |
|               {1} | 0 |                      {1} | 0 |                   |
|               {1} | 0 |                      {1} | 0 |                   |
|               {1} | 0 |                      {1} | 0 |                   |
|               {1} | 0 |                      {1} | 0 |                   |
|               {1} | 0 |                      {1} | 0 |                   |
|               {1} | 0 |                      {1} | 0 |                   |
|               {2} | 0 |                      {2} | 0 |                   |
|               {2} | 0 |                      {2} | 0 |                   |
|               {2} | 0 |                      {2} | 0 |                   |
|               {2} | 0 |                      {2} | 0 |                   |
|               {2} | 0 |                      {2} | 0 |                   |
|               {2} | 0 |                      {2} | 0 |                   |
|               {2} | 0 |                      {2} | 0 |                   |
|               {2} | 0 |                      {2} | 0 |                   |
|               {2} | 0 |                      {2} | 0 |                   |
|               {3} | 1 |                      {3} | 1 |                   |
|                                                                          |
|o8 : ComplexMap                                                           |
+--------------------------------------------------------------------------+
|i9 : normalize id_C                                                       |
|                                                                          |
|          1             1                                                 |
|o9 = 0 : R  <--------- R  : 0                                             |
|               | 1 |                                                      |
|                                                                          |
|          3                     3                                         |
|     1 : R  <----------------- R  : 1                                     |
|               {1} | 1 0 0 |                                              |
|               {1} | 0 1 0 |                                              |
|               {1} | 0 0 1 |                                              |
|                                                                          |
|          3                     3                                         |
|     2 : R  <----------------- R  : 2                                     |
|               {2} | 1 0 0 |                                              |
|               {2} | 0 1 0 |                                              |
|               {2} | 0 0 1 |                                              |
|                                                                          |
|          1                 1                                             |
|     3 : R  <------------- R  : 3                                         |
|               {3} | 1 |                                                  |
|                                                                          |
|o9 : ComplexMap                                                           |
+--------------------------------------------------------------------------+
|i10 : prune normalize f == normalize id_C                                 |
|                                                                          |
|o10 = true                                                                |
+--------------------------------------------------------------------------+

See also
========

  * SimplicialModuleMap -- the class of all maps
    between simplicial modules
  * isWellDefined(SimplicialModuleMap)
    -- whether a map of simplicial
    modules is well-defined
  * degree(SimplicialModuleMap) -- get
    the degree of a map of simplicial modules
  * isSimplicialMorphism(SimplicialModuleMap)
    -- whether a simplicial
    module map is a morphism of simplicial modules
  * isCommutative(SimplicialModuleMap)
    -- whether a simplicial module map
    commutes with the face/degeneracy maps
  * SimplicialModuleMap Array -- shift a
    simplicial module or simplicial module map

Ways to use this method:
========================

  * map(SimplicialModule,SimplicialModule,SimplicialModuleMap)
    -- make
    a new map of simplicial modules, induced from an existing one
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1550:0.

map(SimplicialModule,SimplicialModule,ZZ) -- make the zero map or identity between simplicial modules
*****************************************************************************************************

  * Function: map (Macaulay2Doc),
  * Usage:
        f = map(D, C, 0)
        f = map(C, C, 1)
  * Inputs:
      * C, a Simplicial Module,
      * D, a Simplicial Module,
      * 0, an integer (Macaulay2Doc), or 1
  * Optional inputs (Macaulay2Doc),:
      * Degree, => ..., default
        value null, specify the degree of a map of simplicial modules
      * DegreeLift, => ...,
        default value null, unused option for maps of simplicial modules
      * DegreeMap, => ...,
        default value null, unused option for maps of simplicial modules
  * Outputs:
      * f, a Map of Simplicial Modules, the zero
        map from $C$ to $D$ or the identity map from $C$ to $C$

Description
===========

A map of simplicial modules $f : C \rightarrow D$ of degree $d$ is a sequence
of maps $f_i : C_i \rightarrow D_{d+i}$.

We construct the zero map between two simplicial modules.

+-----------------------------------------------------------------------------+
|i1 : R = QQ[a,b,c]                                                           |
|                                                                             |
|o1 = R                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution coker vars R, Degeneracy => true)   |
|                                                                             |
|      1      4      10      20                                               |
|o2 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o2 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i3 : D = simplicialModule(freeResolution coker matrix{{a^2, b^2, c^2}},      |
|                                                                             |
|      1      4      10      20                                               |
|o3 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o3 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i4 : f = map(D, C, 0)                                                        |
|                                                                             |
|o4 = 0                                                                       |
|                                                                             |
|o4 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+
|i5 : assert isWellDefined f                                                  |
+-----------------------------------------------------------------------------+
|i6 : assert isSimplicialMorphism f                                           |
+-----------------------------------------------------------------------------+
|i7 : g = map(C, C, 0, Degree => 13)                                          |
|                                                                             |
|o7 = 0                                                                       |
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+
|i8 : assert isWellDefined g                                                  |
+-----------------------------------------------------------------------------+
|i9 : assert(degree g == 13)                                                  |
+-----------------------------------------------------------------------------+
|i10 : assert not isSimplicialMorphism g                                      |
+-----------------------------------------------------------------------------+
|i11 : assert isCommutative g                                                 |
+-----------------------------------------------------------------------------+
|i12 : assert isHomogeneous g                                                 |
+-----------------------------------------------------------------------------+
|i13 : assert(source g == C)                                                  |
+-----------------------------------------------------------------------------+
|i14 : assert(target g == C)                                                  |
+-----------------------------------------------------------------------------+

Using this function to create the identity map is the same as using id _ SimplicialModule.

+-----------------------------------+
|i15 : assert(map(C, C, 1) === id_C)|
+-----------------------------------+

See also
========

  * SimplicialModuleMap -- the class of all maps
    between simplicial modules
  * isWellDefined(SimplicialModuleMap)
    -- whether a map of simplicial
    modules is well-defined
  * isHomogeneous(SimplicialModuleMap)
    -- whether a map of simplicial
    modules is homogeneous
  * degree(SimplicialModuleMap) -- get
    the degree of a map of simplicial modules
  * isSimplicialMorphism(SimplicialModuleMap)
    -- whether a simplicial
    module map is a morphism of simplicial modules
  * isCommutative(SimplicialModuleMap)
    -- whether a simplicial module map
    commutes with the face/degeneracy maps
  * source(SimplicialModuleMap) -- get
    the source of a map of simplicial modules
  * target(SimplicialModuleMap) -- get
    the target of a map of simplicial modules
  * id _ SimplicialModule -- the identity map
    of a simplicial module

Ways to use this method:
========================

  * map(SimplicialModule,SimplicialModule,ZZ)
    -- make the zero map or
    identity between simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1506:0.

minimalPresentation(...,Exclude=>...) -- exclude option for minimal presentation
********************************************************************************

Description
===========

This option is inherited from the core minimalPresentation (Macaulay2Doc),
method.

See also
========

  * minimalPresentation(SimplicialModule)
    -- minimal presentation of all
    terms in a simplicial module

Functions with optional argument named Exclude:
===============================================

  * "copyDirectory(...,Exclude=>...)" -- see
    copyDirectory(String,String) (Macaulay2Doc),
  * "findFiles(...,Exclude=>...)" -- see findFiles (Macaulay2Doc),
    -- find files recursively
  * "minimalPresentation(...,Exclude=>...)" -- see
    minimalPresentation(Ring) (Macaulay2Doc), --
    compute a minimal presentation of a quotient ring
  * "prune(...,Exclude=>...)" -- see minimalPresentation(Ring) (Macaulay2Doc),
    -- compute a minimal
    presentation of a quotient ring
  * "symlinkDirectory(...,Exclude=>...)" -- see
    symlinkDirectory(String,String) (Macaulay2Doc),
    -- make symbolic links
    for all files in a directory tree

Further information
===================

  * Default value: {}
  * Function: minimalPresentation (Macaulay2Doc), --
    compute a minimal presentation
  * Option key: Exclude (Macaulay2Doc), -- an optional argument
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:4125:0.

minimalPresentation(SimplicialModule) -- minimal presentation of all terms in a simplicial module
*************************************************************************************************

  * Function: minimalPresentation (Macaulay2Doc),
  * Usage:
        D = minimalPresentation C
        D = prune C
        h = minimalPresentation f
        h = prune f
  * Inputs:
      * C, a Simplicial Module, or $f$ a Map of Simplicial Modules,
  * Optional inputs (Macaulay2Doc),:
      * Exclude,
        => ..., default value {}, unused
  * Outputs:
      * D, a Simplicial Module, isomorphic to the
        input, where each term is replaced by a minimally presented model, or
        $h$ a Map of Simplicial Modules, where the
        source and target are minimally presented
  * Consequences:
      * The isomorphism $g : D \to C$ is available as g = D.cache.pruningMap.
        The inverse isomorphism can be obtained as g^-1

Description
===========

This is frequently useful to make the output of certain operations readable or
understandable.  This operation is functorial, applying both to simplicial
modules and simplicial module maps.

In particular, images/kernels/cokernels of maps need to be pruned to be
understood.  For instance, this is useful for recognizing when terms given by
subquotient modules are actually zero.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a,b,c,d,e];                                                  |
+-----------------------------------------------------------------------------+
|i2 : I = ideal(a,b) * ideal(c,d,e)                                           |
|                                                                             |
|o2 = ideal (a*c, a*d, a*e, b*c, b*d, b*e)                                    |
|                                                                             |
|o2 : Ideal of S                                                              |
+-----------------------------------------------------------------------------+
|i3 : F = simplicialModule((dual freeResolution I)[-4], 2, Degeneracy => true)|
|                                                                             |
|      1      6      20                                                       |
|o3 = S  <-- S  <-- S  <-- ...                                                |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : C = HH F                                                                |
|                                                                             |
|o4 = cokernel {-5} | -e d -c -b a | <-- subquotient ({-5} | 1 0 0  0 0  0 0  |
|                                                     {-4} | 0 d c  0 b  0 0  |
|     0                                               {-4} | 0 e 0  c 0  b 0  |
|                                                     {-4} | 0 0 -e d 0  0 b  |
|                                                     {-4} | 0 0 0  0 -e d -c |
|                                                     {-4} | 0 0 0  0 0  0 0  |
|                                                                             |
|                                        1                                    |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o4 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|0 0  0 0 |, {-5} | -e d -c -b a 0 0  0  0 0  0  0  0  0 |) <-- subquotient   |
|a 0  0 0 |  {-4} | 0  0 0  0  0 d -c -b a 0  0  0  0  0 |                    |
|0 a  0 0 |  {-4} | 0  0 0  0  0 e 0  0  0 -c -b a  0  0 |                    |
|0 0  a 0 |  {-4} | 0  0 0  0  0 0 e  0  0 -d 0  0  -b a |                    |
|0 0  0 a |  {-4} | 0  0 0  0  0 0 0  e  0 0  -d 0  c  0 |                    |
|e -d c b |  {-4} | 0  0 0  0  0 0 0  0  e 0  0  -d 0  c |                    |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                               2             |
|-----------------------------------------------------------------------------|
|({-5} | 1 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 0 0 0 0  0  0  0  0  |
| {-4} | 0 d c  0 b  0 0  a 0  0 0 0 0  0 0  0 0  0 0  0 0 0 0 0  0  0  0  0  |
| {-4} | 0 e 0  c 0  b 0  0 a  0 0 0 0  0 0  0 0  0 0  0 0 0 0 0  0  0  0  0  |
| {-4} | 0 0 -e d 0  0 b  0 0  a 0 0 0  0 0  0 0  0 0  0 0 0 0 0  0  0  0  0  |
| {-4} | 0 0 0  0 -e d -c 0 0  0 a 0 0  0 0  0 0  0 0  0 0 0 0 0  0  0  0  0  |
| {-4} | 0 0 0  0 0  0 0  e -d c b 0 0  0 0  0 0  0 0  0 0 0 0 0  0  0  0  0  |
| {-4} | 0 0 0  0 0  0 0  0 0  0 0 d c  0 b  0 0  a 0  0 0 0 0 0  0  0  0  0  |
| {-4} | 0 0 0  0 0  0 0  0 0  0 0 e 0  c 0  b 0  0 a  0 0 0 0 0  0  0  0  0  |
| {-4} | 0 0 0  0 0  0 0  0 0  0 0 0 -e d 0  0 b  0 0  a 0 0 0 0  0  0  0  0  |
| {-4} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 -e d -c 0 0  0 a 0 0 0  0  0  0  0  |
| {-4} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  e -d c b 0 0 0  0  0  0  0  |
| {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 0 c b a  0  0  0  0  |
| {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 0 d 0 0  b  0  a  0  |
| {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 0 0 d 0  -c 0  0  0  |
| {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 0 0 0 -d 0  0  c  0  |
| {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 0 e 0 0  0  b  0  a  |
| {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 0 0 e 0  0  -c 0  0  |
| {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 0 0 0 -e 0  0  0  c  |
| {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 0 0 0 0  e  -d 0  0  |
| {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 0 0 0 0  0  0  -e d  |
|-----------------------------------------------------------------------------|
||, {-5} | -e d -c -b a 0 0  0  0 0  0  0  0  0 0 0  0  0 0  0  0  0  0 0  0  |
||  {-4} | 0  0 0  0  0 d -c -b a 0  0  0  0  0 0 0  0  0 0  0  0  0  0 0  0  |
||  {-4} | 0  0 0  0  0 e 0  0  0 -c -b a  0  0 0 0  0  0 0  0  0  0  0 0  0  |
||  {-4} | 0  0 0  0  0 0 e  0  0 -d 0  0  -b a 0 0  0  0 0  0  0  0  0 0  0  |
||  {-4} | 0  0 0  0  0 0 0  e  0 0  -d 0  c  0 0 0  0  0 0  0  0  0  0 0  0  |
||  {-4} | 0  0 0  0  0 0 0  0  e 0  0  -d 0  c 0 0  0  0 0  0  0  0  0 0  0  |
||  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 d -c -b a 0  0  0  0  0 0  0  |
||  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 e 0  0  0 -c -b a  0  0 0  0  |
||  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 e  0  0 -d 0  0  -b a 0  0  |
||  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  e  0 0  -d 0  c  0 0  0  |
||  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  e 0  0  -d 0  c 0  0  |
||  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 0  0  0  0  0 -b a  |
||  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 0  0  0  0  0 0  0  |
||  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 0  0  0  0  0 -d 0  |
||  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 0  0  0  0  0 0  -d |
||  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 0  0  0  0  0 0  0  |
||  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 0  0  0  0  0 -e 0  |
||  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 0  0  0  0  0 0  -e |
||  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 0  0  0  0  0 0  0  |
||  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 0  0  0  0  0 0  0  |
|-----------------------------------------------------------------------------|
|0  0  0  0 |)<-- ...                                                         |
|0  0  0  0 |                                                                 |
|0  0  0  0 |                                                                 |
|0  0  0  0 |                                                                 |
|0  0  0  0 |                                                                 |
|0  0  0  0 |                                                                 |
|0  0  0  0 |                                                                 |
|0  0  0  0 |                                                                 |
|0  0  0  0 |                                                                 |
|0  0  0  0 |                                                                 |
|0  0  0  0 |                                                                 |
|0  0  0  0 |                                                                 |
|-b a  0  0 |                                                                 |
|c  0  0  0 |                                                                 |
|0  c  0  0 |                                                                 |
|0  0  -b a |                                                                 |
|0  0  c  0 |                                                                 |
|0  0  0  c |                                                                 |
|-e 0  d  0 |                                                                 |
|0  -e 0  d |                                                                 |
+-----------------------------------------------------------------------------+
|i5 : D = prune C                                                             |
|                                                                             |
|o5 = cokernel {-5} | e d c b a | <-- cokernel {-5} | e d c b a 0 0 0 | <--   |
|                                              {-3} | 0 0 0 0 0 e d c |       |
|     0                                                                       |
|                                     1                                       |
|                                                                             |
|                                                                             |
|                                                                             |
|o5 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|cokernel {-5} | e d c b a 0 0 0 0 0 0 0 0 |<-- ...                           |
|         {-3} | 0 0 0 0 0 e d c 0 0 0 0 0 |                                  |
|         {-3} | 0 0 0 0 0 0 0 0 e d c 0 0 |                                  |
|         {-2} | 0 0 0 0 0 0 0 0 0 0 0 b a |                                  |
|                                                                             |
|2                                                                            |
+-----------------------------------------------------------------------------+
|i6 : g = D.cache.pruningMap                                                  |
|                                                                             |
|o6 = 0 : cokernel {-5} | -e d -c -b a | <-------------- cokernel {-5}        |
|                                           {-5} | 1 |                        |
|                                                                             |
|     1 : subquotient ({-5} | 1 0 0  0 0  0 0  0 0  0 0 |, {-5} | -e d        |
|                      {-4} | 0 d c  0 b  0 0  a 0  0 0 |  {-4} | 0  0        |
|                      {-4} | 0 e 0  c 0  b 0  0 a  0 0 |  {-4} | 0  0        |
|                      {-4} | 0 0 -e d 0  0 b  0 0  a 0 |  {-4} | 0  0        |
|                      {-4} | 0 0 0  0 -e d -c 0 0  0 a |  {-4} | 0  0        |
|                      {-4} | 0 0 0  0 0  0 0  e -d c b |  {-4} | 0  0        |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|     2 : subquotient ({-5} | 1 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0         |
|                      {-4} | 0 d c  0 b  0 0  a 0  0 0 0 0  0 0  0 0         |
|                      {-4} | 0 e 0  c 0  b 0  0 a  0 0 0 0  0 0  0 0         |
|                      {-4} | 0 0 -e d 0  0 b  0 0  a 0 0 0  0 0  0 0         |
|                      {-4} | 0 0 0  0 -e d -c 0 0  0 a 0 0  0 0  0 0         |
|                      {-4} | 0 0 0  0 0  0 0  e -d c b 0 0  0 0  0 0         |
|                      {-4} | 0 0 0  0 0  0 0  0 0  0 0 d c  0 b  0 0         |
|                      {-4} | 0 0 0  0 0  0 0  0 0  0 0 e 0  c 0  b 0         |
|                      {-4} | 0 0 0  0 0  0 0  0 0  0 0 0 -e d 0  0 b         |
|                      {-4} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 -e d -c        |
|                      {-4} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0         |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0         |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0         |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0         |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0         |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0         |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0         |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0         |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0         |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0         |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o6 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|| e d c b a | : 0                                                            |
|                                                                             |
|                                                                             |
|-c -b a 0 0  0  0 0  0  0  0  0 |) <---------------- cokernel {-5} |         |
|0  0  0 d -c -b a 0  0  0  0  0 |     {-5} | 1 0 |            {-3} |         |
|0  0  0 e 0  0  0 -c -b a  0  0 |     {-3} | 0 0 |                           |
|0  0  0 0 e  0  0 -d 0  0  -b a |     {-3} | 0 0 |                           |
|0  0  0 0 0  e  0 0  -d 0  c  0 |     {-3} | 0 0 |                           |
|0  0  0 0 0  0  e 0  0  -d 0  c |     {-3} | 0 0 |                           |
|                                      {-3} | 0 0 |                           |
|                                      {-3} | 0 0 |                           |
|                                      {-3} | 0 0 |                           |
|                                      {-3} | 0 0 |                           |
|                                      {-3} | 0 0 |                           |
|                                      {-3} | 0 1 |                           |
|                                                                             |
|0 0  0 0 0 0 0  0  0  0  0 |, {-5} | -e d -c -b a 0 0  0  0 0  0  0          |
|0 0  0 0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 d -c -b a 0  0  0          |
|0 0  0 0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 e 0  0  0 -c -b a          |
|0 0  0 0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 e  0  0 -d 0  0          |
|0 0  0 0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 0  e  0 0  -d 0          |
|0 0  0 0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 0  0  e 0  0  -d         |
|a 0  0 0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0          |
|0 a  0 0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0          |
|0 0  a 0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0          |
|0 0  0 a 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0          |
|e -d c b 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0          |
|0 0  0 0 c b a  0  0  0  0 |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0          |
|0 0  0 0 d 0 0  b  0  a  0 |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0          |
|0 0  0 0 0 d 0  -c 0  0  0 |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0          |
|0 0  0 0 0 0 -d 0  0  c  0 |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0          |
|0 0  0 0 e 0 0  0  b  0  a |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0          |
|0 0  0 0 0 e 0  0  -c 0  0 |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0          |
|0 0  0 0 0 0 -e 0  0  0  c |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0          |
|0 0  0 0 0 0 0  e  -d 0  0 |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0          |
|0 0  0 0 0 0 0  0  0  -e d |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|e d c b a 0 0 0 | : 1                                                        |
|0 0 0 0 0 e d c |                                                            |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|0  0 0 0  0  0 0  0  0  0  0 0  0  0  0  0  0 |) <--------------------       |
|0  0 0 0  0  0 0  0  0  0  0 0  0  0  0  0  0 |     {-5} | 1 0 0 0 |         |
|0  0 0 0  0  0 0  0  0  0  0 0  0  0  0  0  0 |     {-3} | 0 0 0 0 |         |
|-b a 0 0  0  0 0  0  0  0  0 0  0  0  0  0  0 |     {-3} | 0 0 0 0 |         |
|c  0 0 0  0  0 0  0  0  0  0 0  0  0  0  0  0 |     {-3} | 0 0 0 0 |         |
|0  c 0 0  0  0 0  0  0  0  0 0  0  0  0  0  0 |     {-3} | 0 0 0 0 |         |
|0  0 d -c -b a 0  0  0  0  0 0  0  0  0  0  0 |     {-3} | 0 0 0 0 |         |
|0  0 e 0  0  0 -c -b a  0  0 0  0  0  0  0  0 |     {-3} | 0 0 0 0 |         |
|0  0 0 e  0  0 -d 0  0  -b a 0  0  0  0  0  0 |     {-3} | 0 0 0 0 |         |
|0  0 0 0  e  0 0  -d 0  c  0 0  0  0  0  0  0 |     {-3} | 0 0 0 0 |         |
|0  0 0 0  0  e 0  0  -d 0  c 0  0  0  0  0  0 |     {-3} | 0 0 0 0 |         |
|0  0 0 0  0  0 0  0  0  0  0 -b a  0  0  0  0 |     {-3} | 0 1 0 0 |         |
|0  0 0 0  0  0 0  0  0  0  0 0  0  -b a  0  0 |     {-3} | 0 0 0 0 |         |
|0  0 0 0  0  0 0  0  0  0  0 -d 0  c  0  0  0 |     {-3} | 0 0 0 0 |         |
|0  0 0 0  0  0 0  0  0  0  0 0  -d 0  c  0  0 |     {-3} | 0 0 0 0 |         |
|0  0 0 0  0  0 0  0  0  0  0 0  0  0  0  -b a |     {-3} | 0 0 0 0 |         |
|0  0 0 0  0  0 0  0  0  0  0 -e 0  0  0  c  0 |     {-3} | 0 0 0 0 |         |
|0  0 0 0  0  0 0  0  0  0  0 0  -e 0  0  0  c |     {-3} | 0 0 0 0 |         |
|0  0 0 0  0  0 0  0  0  0  0 0  0  -e 0  d  0 |     {-3} | 0 0 0 0 |         |
|0  0 0 0  0  0 0  0  0  0  0 0  0  0  -e 0  d |     {-3} | 0 0 0 0 |         |
|                                                    {-3} | 0 0 0 0 |         |
|                                                    {-3} | 0 0 1 0 |         |
|                                                    {-2} | 0 0 0 1 |         |
|                                                    {-2} | 0 0 0 0 |         |
|                                                    {-2} | 0 0 0 0 |         |
|                                                    {-2} | 0 0 0 0 |         |
|                                                    {-2} | 0 0 0 0 |         |
|                                                    {-2} | 0 0 0 0 |         |
|                                                    {-2} | 0 0 0 0 |         |
|-----------------------------------------------------------------------------|
|cokernel {-5} | e d c b a 0 0 0 0 0 0 0 0 | : 2                              |
|         {-3} | 0 0 0 0 0 e d c 0 0 0 0 0 |                                  |
|         {-3} | 0 0 0 0 0 0 0 0 e d c 0 0 |                                  |
|         {-2} | 0 0 0 0 0 0 0 0 0 0 0 b a |                                  |
+-----------------------------------------------------------------------------+
|i7 : assert isWellDefined g                                                  |
+-----------------------------------------------------------------------------+
|i8 : assert isSimplicialMorphism g                                           |
+-----------------------------------------------------------------------------+
|i9 : assert (target g == C)                                                  |
+-----------------------------------------------------------------------------+
|i10 : assert (source g == D)                                                 |
+-----------------------------------------------------------------------------+
|i11 : g^-1                                                                   |
|                                                                             |
|o11 = 0 : cokernel {-5} | e d c b a |                                        |
|                                                                             |
|                                                                             |
|      1 : cokernel {-5} | e d c b a 0                                        |
|                   {-3} | 0 0 0 0 0 e                                        |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      2 : cokernel {-5} | e d c b a 0                                        |
|                   {-3} | 0 0 0 0 0 e                                        |
|                   {-3} | 0 0 0 0 0 0                                        |
|                   {-2} | 0 0 0 0 0 0                                        |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o11 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|<-------------- cokernel {-5} | -e d -c -b a | : 0                           |
|   {-5} | 1 |                                                                |
|                                                                             |
|0 0 | <---------------------------------- subquotient ({-5} | 1 0 0  0 0  0 0|
|d c |    {-5} | 1 0 0 0 0 0 0 0 0 0 0 |                {-4} | 0 d c  0 b  0 0|
|         {-3} | 0 0 0 0 0 0 0 0 0 0 1 |                {-4} | 0 e 0  c 0  b 0|
|                                                       {-4} | 0 0 -e d 0  0 b|
|                                                       {-4} | 0 0 0  0 -e d -|
|                                                       {-4} | 0 0 0  0 0  0 0|
|                                                                             |
|0 0 0 0 0 0 0 | <------------------------------------------------------------|
|d c 0 0 0 0 0 |    {-5} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 e d c 0 0 |    {-3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 b a |    {-3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|                   {-2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|  0 0  0 0 |, {-5} | -e d -c -b a 0 0  0  0 0  0  0  0  0 |) : 1             |
|  a 0  0 0 |  {-4} | 0  0 0  0  0 d -c -b a 0  0  0  0  0 |                  |
|  0 a  0 0 |  {-4} | 0  0 0  0  0 e 0  0  0 -c -b a  0  0 |                  |
|  0 0  a 0 |  {-4} | 0  0 0  0  0 0 e  0  0 -d 0  0  -b a |                  |
|c 0 0  0 a |  {-4} | 0  0 0  0  0 0 0  e  0 0  -d 0  c  0 |                  |
|  e -d c b |  {-4} | 0  0 0  0  0 0 0  0  e 0  0  -d 0  c |                  |
|                                                                             |
|-------- subquotient ({-5} | 1 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 |
| 0 0 |                {-4} | 0 d c  0 b  0 0  a 0  0 0 0 0  0 0  0 0  0 0  0 |
| 0 0 |                {-4} | 0 e 0  c 0  b 0  0 a  0 0 0 0  0 0  0 0  0 0  0 |
| 0 0 |                {-4} | 0 0 -e d 0  0 b  0 0  a 0 0 0  0 0  0 0  0 0  0 |
| 0 0 |                {-4} | 0 0 0  0 -e d -c 0 0  0 a 0 0  0 0  0 0  0 0  0 |
|                      {-4} | 0 0 0  0 0  0 0  e -d c b 0 0  0 0  0 0  0 0  0 |
|                      {-4} | 0 0 0  0 0  0 0  0 0  0 0 d c  0 b  0 0  a 0  0 |
|                      {-4} | 0 0 0  0 0  0 0  0 0  0 0 e 0  c 0  b 0  0 a  0 |
|                      {-4} | 0 0 0  0 0  0 0  0 0  0 0 0 -e d 0  0 b  0 0  a |
|                      {-4} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 -e d -c 0 0  0 |
|                      {-4} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  e -d c |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 |
|                      {-3} | 0 0 0  0 0  0 0  0 0  0 0 0 0  0 0  0 0  0 0  0 |
|-----------------------------------------------------------------------------|
|0 0 0 0  0  0  0  0 |, {-5} | -e d -c -b a 0 0  0  0 0  0  0  0  0 0 0  0  0 |
|0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 d -c -b a 0  0  0  0  0 0 0  0  0 |
|0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 e 0  0  0 -c -b a  0  0 0 0  0  0 |
|0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 e  0  0 -d 0  0  -b a 0 0  0  0 |
|0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 0  e  0 0  -d 0  c  0 0 0  0  0 |
|0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 0  0  e 0  0  -d 0  c 0 0  0  0 |
|0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 d -c -b a |
|0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 e 0  0  0 |
|0 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 e  0  0 |
|a 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  e  0 |
|b 0 0 0  0  0  0  0 |  {-4} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  e |
|0 c b a  0  0  0  0 |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 |
|0 d 0 0  b  0  a  0 |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 |
|0 0 d 0  -c 0  0  0 |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 |
|0 0 0 -d 0  0  c  0 |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 |
|0 e 0 0  0  b  0  a |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 |
|0 0 e 0  0  -c 0  0 |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 |
|0 0 0 -e 0  0  0  c |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 |
|0 0 0 0  e  -d 0  0 |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 |
|0 0 0 0  0  0  -e d |  {-3} | 0  0 0  0  0 0 0  0  0 0  0  0  0  0 0 0  0  0 |
|-----------------------------------------------------------------------------|
|0  0  0  0  0 0  0  0  0  0  0 |) : 2                                        |
|0  0  0  0  0 0  0  0  0  0  0 |                                             |
|0  0  0  0  0 0  0  0  0  0  0 |                                             |
|0  0  0  0  0 0  0  0  0  0  0 |                                             |
|0  0  0  0  0 0  0  0  0  0  0 |                                             |
|0  0  0  0  0 0  0  0  0  0  0 |                                             |
|0  0  0  0  0 0  0  0  0  0  0 |                                             |
|-c -b a  0  0 0  0  0  0  0  0 |                                             |
|-d 0  0  -b a 0  0  0  0  0  0 |                                             |
|0  -d 0  c  0 0  0  0  0  0  0 |                                             |
|0  0  -d 0  c 0  0  0  0  0  0 |                                             |
|0  0  0  0  0 -b a  0  0  0  0 |                                             |
|0  0  0  0  0 0  0  -b a  0  0 |                                             |
|0  0  0  0  0 -d 0  c  0  0  0 |                                             |
|0  0  0  0  0 0  -d 0  c  0  0 |                                             |
|0  0  0  0  0 0  0  0  0  -b a |                                             |
|0  0  0  0  0 -e 0  0  0  c  0 |                                             |
|0  0  0  0  0 0  -e 0  0  0  c |                                             |
|0  0  0  0  0 0  0  -e 0  d  0 |                                             |
|0  0  0  0  0 0  0  0  -e 0  d |                                             |
+-----------------------------------------------------------------------------+
|i12 : assert(g*g^-1 == 1 and g^-1*g == 1)                                    |
+-----------------------------------------------------------------------------+

The image of a map of simplicial modules also becomes more understandable via
pruning.

+-----------------------------------------------------------------------------+
|i13 : S = ZZ/101[a,b,c];                                                     |
+-----------------------------------------------------------------------------+
|i14 : I = ideal(a^2,b^2,c^2);                                                |
|                                                                             |
|o14 : Ideal of S                                                             |
+-----------------------------------------------------------------------------+
|i15 : J = I + ideal(a*b*c);                                                  |
|                                                                             |
|o15 : Ideal of S                                                             |
+-----------------------------------------------------------------------------+
|i16 : FI = simplicialModule(freeResolution I, Degeneracy => true)            |
|                                                                             |
|       1      4      10      20                                              |
|o16 = S  <-- S  <-- S   <-- S  <-- ...                                       |
|                                                                             |
|      0      1      2       3                                                |
|                                                                             |
|o16 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i17 : FJ = simplicialModule(freeResolution J, Degeneracy => true)            |
|                                                                             |
|       1      5      15      34                                              |
|o17 = S  <-- S  <-- S   <-- S  <-- ...                                       |
|                                                                             |
|      0      1      2       3                                                |
|                                                                             |
|o17 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i18 : f = randomSimplicialMap(FJ, FI ** S^{-1}, Cycle => true)               |
|                                                                             |
|           1                       1                                         |
|o18 = 0 : S  <------------------- S  : 0                                     |
|                | 32a-36b-30c |                                              |
|                                                                             |
|           5                                                                 |
|      1 : S  <----------------------------------------------------------- ima|
|                {0} | 32a-36b-30c 0           0           0           |      |
|                {2} | 0           32a-36b-30c 0           0           |      |
|                {2} | 0           0           32a-36b-30c 0           |      |
|                {2} | 0           0           0           32a-36b-30c |      |
|                {3} | 0           0           0           0           |      |
|                                                                             |
|           15                                                                |
|      2 : S   <--------------------------------------------------------------|
|                 {0} | 32a-36b-30c 0           0           0           0     |
|                 {2} | 0           32a-36b-30c 0           0           0     |
|                 {2} | 0           0           32a-36b-30c 0           0     |
|                 {2} | 0           0           0           32a-36b-30c 0     |
|                 {3} | 0           0           0           0           0     |
|                 {2} | 0           0           0           0           32a-36|
|                 {2} | 0           0           0           0           0     |
|                 {2} | 0           0           0           0           0     |
|                 {3} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|           34                                                                |
|      3 : S   <--------------------------------------------------------------|
|                 {0} | 32a-36b-30c 0           0           0           0     |
|                 {2} | 0           32a-36b-30c 0           0           0     |
|                 {2} | 0           0           32a-36b-30c 0           0     |
|                 {2} | 0           0           0           32a-36b-30c 0     |
|                 {3} | 0           0           0           0           0     |
|                 {2} | 0           0           0           0           32a-36|
|                 {2} | 0           0           0           0           0     |
|                 {2} | 0           0           0           0           0     |
|                 {3} | 0           0           0           0           0     |
|                 {2} | 0           0           0           0           0     |
|                 {2} | 0           0           0           0           0     |
|                 {2} | 0           0           0           0           0     |
|                 {3} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {5} | 0           0           0           0           0     |
|                 {5} | 0           0           0           0           0     |
|                 {5} | 0           0           0           0           0     |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
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|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o18 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|ge {1} | 1 0 0 0 | : 1                                                       |
|   {1} | 0 0 0 0 |                                                           |
|   {3} | 0 1 0 0 |                                                           |
|   {3} | 0 0 1 0 |                                                           |
|   {3} | 0 0 0 1 |                                                           |
|                                                                             |
|                                                                             |
|                                                                             |
|-------------------------------------------------------------------- image {1|
|      0           0           0           0          0           |         {1|
|      0           0           0           0          0           |         {3|
|      0           0           0           0          0           |         {3|
|      0           0           0           0          0           |         {3|
|      0           0           0           0          0           |         {1|
|b-30c 0           0           0           0          0           |         {3|
|      32a-36b-30c 0           0           0          0           |         {3|
|      0           32a-36b-30c 0           0          0           |         {3|
|      0           0           0           0          0           |         {1|
|      0           0           32a-36b+42c -19c       19c         |         {3|
|      0           0           29b+10c     19b-29c    -19b+22c    |         {3|
|      0           0           -29a+29c    -19a-24c   19a+8c      |         {3|
|      0           0           -10b        32a-7b-30c -22b        |         {3|
|      0           0           -10a-29b    29a+24b    -22a-8b     |         {3|
|      0           0           -29a        24a        24a-36b-30c |         {3|
|                                                                           {5|
|                                                                           {5|
|                                                                           {5|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|b-30c 0           0           0           0           0           0          |
|      32a-36b-30c 0           0           0           0           0          |
|      0           32a-36b-30c 0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           32a-36b-30c 0           0           0          |
|      0           0           0           32a-36b-30c 0           0          |
|      0           0           0           0           32a-36b-30c 0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           32a-36b+42c|
|      0           0           0           0           0           29b+10c    |
|      0           0           0           0           0           -29a+29c   |
|      0           0           0           0           0           -10b       |
|      0           0           0           0           0           -10a-29b   |
|      0           0           0           0           0           -29a       |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|                                                                             |
|                                                                             |
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|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|} | 1 0 0 0 0 0 0 0 0 0 | : 2                                                |
|} | 0 0 0 0 0 0 0 0 0 0 |                                                    |
|} | 0 1 0 0 0 0 0 0 0 0 |                                                    |
|} | 0 0 1 0 0 0 0 0 0 0 |                                                    |
|} | 0 0 0 1 0 0 0 0 0 0 |                                                    |
|} | 0 0 0 0 0 0 0 0 0 0 |                                                    |
|} | 0 0 0 0 1 0 0 0 0 0 |                                                    |
|} | 0 0 0 0 0 1 0 0 0 0 |                                                    |
|} | 0 0 0 0 0 0 1 0 0 0 |                                                    |
|} | 0 0 0 0 0 0 0 0 0 0 |                                                    |
|} | 0 0 0 0 0 0 0 0 0 0 |                                                    |
|} | 0 0 0 0 0 0 0 0 0 0 |                                                    |
|} | 0 0 0 0 0 0 0 0 0 0 |                                                    |
|} | 0 0 0 0 0 0 0 0 0 0 |                                                    |
|} | 0 0 0 0 0 0 0 0 0 0 |                                                    |
|} | 0 0 0 0 0 0 0 0 0 0 |                                                    |
|} | 0 0 0 0 0 0 0 1 0 0 |                                                    |
|} | 0 0 0 0 0 0 0 0 1 0 |                                                    |
|} | 0 0 0 0 0 0 0 0 0 1 |                                                    |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| -19c       19c         0           0          0           0           0     |
| 19b-29c    -19b+22c    0           0          0           0           0     |
| -19a-24c   19a+8c      0           0          0           0           0     |
| 32a-7b-30c -22b        0           0          0           0           0     |
| 29a+24b    -22a-8b     0           0          0           0           0     |
| 24a        24a-36b-30c 0           0          0           0           0     |
| 0          0           32a-36b+42c -19c       19c         0           0     |
| 0          0           29b+10c     19b-29c    -19b+22c    0           0     |
| 0          0           -29a+29c    -19a-24c   19a+8c      0           0     |
| 0          0           -10b        32a-7b-30c -22b        0           0     |
| 0          0           -10a-29b    29a+24b    -22a-8b     0           0     |
| 0          0           -29a        24a        24a-36b-30c 0           0     |
| 0          0           0           0          0           32a-36b+42c -19c  |
| 0          0           0           0          0           29b+10c     19b-29|
| 0          0           0           0          0           -29a+29c    -19a-2|
| 0          0           0           0          0           -10b        32a-7b|
| 0          0           0           0          0           -10a-29b    29a+24|
| 0          0           0           0          0           -29a        24a   |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
|                                                                             |
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|                                                                             |
|-----------------------------------------------------------------------------|
|--------------------------------------------- image {1} | 1 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {1} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {3} | 0 1 0 0 0 0 0 0 0 0|
|     0           0                        |         {3} | 0 0 1 0 0 0 0 0 0 0|
|     0           0                        |         {3} | 0 0 0 1 0 0 0 0 0 0|
|     0           0                        |         {1} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {3} | 0 0 0 0 1 0 0 0 0 0|
|     0           0                        |         {3} | 0 0 0 0 0 1 0 0 0 0|
|     0           0                        |         {3} | 0 0 0 0 0 0 1 0 0 0|
|     0           0                        |         {1} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {3} | 0 0 0 0 0 0 0 1 0 0|
|     0           0                        |         {3} | 0 0 0 0 0 0 0 0 1 0|
|     0           0                        |         {3} | 0 0 0 0 0 0 0 0 0 1|
|     0           0                        |         {1} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {3} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {3} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {3} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {3} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {3} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {3} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {5} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {5} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {5} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {1} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {3} | 0 0 0 0 0 0 0 0 0 0|
|     0           0                        |         {3} | 0 0 0 0 0 0 0 0 0 0|
|     19c         0                        |         {3} | 0 0 0 0 0 0 0 0 0 0|
|c    -19b+22c    0                        |         {3} | 0 0 0 0 0 0 0 0 0 0|
|4c   19a+8c      0                        |         {3} | 0 0 0 0 0 0 0 0 0 0|
|-30c -22b        0                        |         {3} | 0 0 0 0 0 0 0 0 0 0|
|b    -22a-8b     0                        |         {5} | 0 0 0 0 0 0 0 0 0 0|
|     24a-36b-30c 0                        |         {5} | 0 0 0 0 0 0 0 0 0 0|
|     0           19a2+19b2+32ac-36bc+42c2 |         {5} | 0 0 0 0 0 0 0 0 0 0|
|     0           -22a2-32ab+7b2+30bc-10c2 |         {1} | 0 0 0 0 0 0 0 0 0 0|
|     0           24a2-36ab-24b2-30ac-29c2 |         {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {5} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {5} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {5} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {1} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {5} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {5} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {5} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {5} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {5} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {5} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {5} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {5} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {5} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {7} | 0 0 0 0 0 0 0 0 0 0|
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 | : 3                                                    |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|1 0 0 0 0 0 0 0 0 0 |                                                        |
|0 1 0 0 0 0 0 0 0 0 |                                                        |
|0 0 1 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 1 0 0 0 0 0 0 |                                                        |
|0 0 0 0 1 0 0 0 0 0 |                                                        |
|0 0 0 0 0 1 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 1 0 0 0 |                                                        |
|0 0 0 0 0 0 0 1 0 0 |                                                        |
|0 0 0 0 0 0 0 0 1 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 1 |                                                        |
+-----------------------------------------------------------------------------+
|i19 : C = image f                                                            |
|                                                                             |
|o19 = image | 32a-36b-30c | <-- image {0} | 32a-36b-30c 0                    |
|                                      {2} | 0           32a-36b-30c          |
|      0                               {2} | 0           0                    |
|                                      {2} | 0           0                    |
|                                      {3} | 0           0                    |
|                                                                             |
|                                1                                            |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o19 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|0           0           | <-- image {0} | 32a-36b-30c 0           0          |
|0           0           |           {2} | 0           32a-36b-30c 0          |
|32a-36b-30c 0           |           {2} | 0           0           32a-36b-30c|
|0           32a-36b-30c |           {2} | 0           0           0          |
|0           0           |           {3} | 0           0           0          |
|                                    {2} | 0           0           0          |
|                                    {2} | 0           0           0          |
|                                    {2} | 0           0           0          |
|                                    {3} | 0           0           0          |
|                                    {4} | 0           0           0          |
|                                    {4} | 0           0           0          |
|                                    {4} | 0           0           0          |
|                                    {4} | 0           0           0          |
|                                    {4} | 0           0           0          |
|                                    {4} | 0           0           0          |
|                                                                             |
|                              2                                              |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|32a-36b-30c 0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           32a-36b-30c 0           0           0           0                |
|0           0           32a-36b-30c 0           0           0                |
|0           0           0           32a-36b-30c 0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           32a-36b+42c -19c             |
|0           0           0           0           29b+10c     19b-29c          |
|0           0           0           0           -29a+29c    -19a-24c         |
|0           0           0           0           -10b        32a-7b-30c       |
|0           0           0           0           -10a-29b    29a+24b          |
|0           0           0           0           -29a        24a              |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|0           | <-- image {0} | 32a-36b-30c 0           0           0          |
|0           |           {2} | 0           32a-36b-30c 0           0          |
|0           |           {2} | 0           0           32a-36b-30c 0          |
|0           |           {2} | 0           0           0           32a-36b-30c|
|0           |           {3} | 0           0           0           0          |
|0           |           {2} | 0           0           0           0          |
|0           |           {2} | 0           0           0           0          |
|0           |           {2} | 0           0           0           0          |
|0           |           {3} | 0           0           0           0          |
|19c         |           {2} | 0           0           0           0          |
|-19b+22c    |           {2} | 0           0           0           0          |
|19a+8c      |           {2} | 0           0           0           0          |
|-22b        |           {3} | 0           0           0           0          |
|-22a-8b     |           {4} | 0           0           0           0          |
|24a-36b-30c |           {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {4} | 0           0           0           0          |
|                        {5} | 0           0           0           0          |
|                        {5} | 0           0           0           0          |
|                        {5} | 0           0           0           0          |
|                                                                             |
|                  3                                                          |
|-----------------------------------------------------------------------------|
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|32a-36b-30c 0           0           0           0           0                |
|0           32a-36b-30c 0           0           0           0                |
|0           0           32a-36b-30c 0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           32a-36b-30c 0           0                |
|0           0           0           0           32a-36b-30c 0                |
|0           0           0           0           0           32a-36b-30c      |
|0           0           0           0           0           0                |
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|0           0           0           0           0           0                |
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|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
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|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|-----------------------------------------------------------------------------|
|0           0          0           0           0          0                  |
|0           0          0           0           0          0                  |
|0           0          0           0           0          0                  |
|0           0          0           0           0          0                  |
|0           0          0           0           0          0                  |
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|0           0          0           0           0          0                  |
|0           0          0           0           0          0                  |
|32a-36b+42c -19c       19c         0           0          0                  |
|29b+10c     19b-29c    -19b+22c    0           0          0                  |
|-29a+29c    -19a-24c   19a+8c      0           0          0                  |
|-10b        32a-7b-30c -22b        0           0          0                  |
|-10a-29b    29a+24b    -22a-8b     0           0          0                  |
|-29a        24a        24a-36b-30c 0           0          0                  |
|0           0          0           32a-36b+42c -19c       19c                |
|0           0          0           29b+10c     19b-29c    -19b+22c           |
|0           0          0           -29a+29c    -19a-24c   19a+8c             |
|0           0          0           -10b        32a-7b-30c -22b               |
|0           0          0           -10a-29b    29a+24b    -22a-8b            |
|0           0          0           -29a        24a        24a-36b-30c        |
|0           0          0           0           0          0                  |
|0           0          0           0           0          0                  |
|0           0          0           0           0          0                  |
|0           0          0           0           0          0                  |
|0           0          0           0           0          0                  |
|0           0          0           0           0          0                  |
|0           0          0           0           0          0                  |
|0           0          0           0           0          0                  |
|0           0          0           0           0          0                  |
|-----------------------------------------------------------------------------|
|0           0          0           0                        |<-- ...         |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|0           0          0           0                        |                |
|32a-36b+42c -19c       19c         0                        |                |
|29b+10c     19b-29c    -19b+22c    0                        |                |
|-29a+29c    -19a-24c   19a+8c      0                        |                |
|-10b        32a-7b-30c -22b        0                        |                |
|-10a-29b    29a+24b    -22a-8b     0                        |                |
|-29a        24a        24a-36b-30c 0                        |                |
|0           0          0           19a2+19b2+32ac-36bc+42c2 |                |
|0           0          0           -22a2-32ab+7b2+30bc-10c2 |                |
|0           0          0           24a2-36ab-24b2-30ac-29c2 |                |
+-----------------------------------------------------------------------------+
|i20 : D = prune C                                                            |
|                                                                             |
|       1      4      10      20                                              |
|o20 = S  <-- S  <-- S   <-- S  <-- ...                                       |
|                                                                             |
|      0      1      2       3                                                |
|                                                                             |
|o20 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i21 : g = D.cache.pruningMap                                                 |
|                                                                             |
|                                                1                            |
|o21 = 0 : image | 32a-36b-30c | <------------- S  : 0                        |
|                                   {1} | 1 |                                 |
|                                                                             |
|                                                                             |
|      1 : image {0} | 32a-36b-30c 0           0           0                  |
|                {2} | 0           32a-36b-30c 0           0                  |
|                {2} | 0           0           32a-36b-30c 0                  |
|                {2} | 0           0           0           32a-36b-30c        |
|                {3} | 0           0           0           0                  |
|                                                                             |
|                                                                             |
|      2 : image {0} | 32a-36b-30c 0           0           0                  |
|                {2} | 0           32a-36b-30c 0           0                  |
|                {2} | 0           0           32a-36b-30c 0                  |
|                {2} | 0           0           0           32a-36b-30c        |
|                {3} | 0           0           0           0                  |
|                {2} | 0           0           0           0                  |
|                {2} | 0           0           0           0                  |
|                {2} | 0           0           0           0                  |
|                {3} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                                                                             |
|                                                                             |
|      3 : image {0} | 32a-36b-30c 0           0           0                  |
|                {2} | 0           32a-36b-30c 0           0                  |
|                {2} | 0           0           32a-36b-30c 0                  |
|                {2} | 0           0           0           32a-36b-30c        |
|                {3} | 0           0           0           0                  |
|                {2} | 0           0           0           0                  |
|                {2} | 0           0           0           0                  |
|                {2} | 0           0           0           0                  |
|                {3} | 0           0           0           0                  |
|                {2} | 0           0           0           0                  |
|                {2} | 0           0           0           0                  |
|                {2} | 0           0           0           0                  |
|                {3} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {4} | 0           0           0           0                  |
|                {5} | 0           0           0           0                  |
|                {5} | 0           0           0           0                  |
|                {5} | 0           0           0           0                  |
|                                                                             |
|o21 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                        4                                                    |
|| <------------------- S  : 1                                                |
||    {1} | 1 0 0 0 |                                                         |
||    {3} | 0 1 0 0 |                                                         |
||    {3} | 0 0 1 0 |                                                         |
||    {3} | 0 0 0 1 |                                                         |
|                                                                             |
|                                                                             |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|32a-36b-30c 0           0           0                                        |
|0           32a-36b-30c 0           0                                        |
|0           0           32a-36b-30c 0                                        |
|0           0           0           0                                        |
|0           0           0           32a-36b+42c                              |
|0           0           0           29b+10c                                  |
|0           0           0           -29a+29c                                 |
|0           0           0           -10b                                     |
|0           0           0           -10a-29b                                 |
|0           0           0           -29a                                     |
|                                                                             |
|                                                                             |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|32a-36b-30c 0           0           0                                        |
|0           32a-36b-30c 0           0                                        |
|0           0           32a-36b-30c 0                                        |
|0           0           0           0                                        |
|0           0           0           32a-36b-30c                              |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|0           0           0           0                                        |
|-----------------------------------------------------------------------------|
|                                                           10                |
|0          0           | <------------------------------- S   : 2            |
|0          0           |    {1} | 1 0 0 0 0 0 0 0 0 0 |                      |
|0          0           |    {3} | 0 1 0 0 0 0 0 0 0 0 |                      |
|0          0           |    {3} | 0 0 1 0 0 0 0 0 0 0 |                      |
|0          0           |    {3} | 0 0 0 1 0 0 0 0 0 0 |                      |
|0          0           |    {3} | 0 0 0 0 1 0 0 0 0 0 |                      |
|0          0           |    {3} | 0 0 0 0 0 1 0 0 0 0 |                      |
|0          0           |    {3} | 0 0 0 0 0 0 1 0 0 0 |                      |
|0          0           |    {5} | 0 0 0 0 0 0 0 1 0 0 |                      |
|-19c       19c         |    {5} | 0 0 0 0 0 0 0 0 1 0 |                      |
|19b-29c    -19b+22c    |    {5} | 0 0 0 0 0 0 0 0 0 1 |                      |
|-19a-24c   19a+8c      |                                                     |
|32a-7b-30c -22b        |                                                     |
|29a+24b    -22a-8b     |                                                     |
|24a        24a-36b-30c |                                                     |
|                                                                             |
|                                                                             |
|0           0           0           0          0           0                 |
|0           0           0           0          0           0                 |
|0           0           0           0          0           0                 |
|0           0           0           0          0           0                 |
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|0           0           0           0          0           0                 |
|0           0           0           0          0           0                 |
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|32a-36b-30c 0           0           0          0           0                 |
|0           32a-36b-30c 0           0          0           0                 |
|0           0           0           0          0           0                 |
|0           0           32a-36b+42c -19c       19c         0                 |
|0           0           29b+10c     19b-29c    -19b+22c    0                 |
|0           0           -29a+29c    -19a-24c   19a+8c      0                 |
|0           0           -10b        32a-7b-30c -22b        0                 |
|0           0           -10a-29b    29a+24b    -22a-8b     0                 |
|0           0           -29a        24a        24a-36b-30c 0                 |
|0           0           0           0          0           32a-36b+42c       |
|0           0           0           0          0           29b+10c           |
|0           0           0           0          0           -29a+29c          |
|0           0           0           0          0           -10b              |
|0           0           0           0          0           -10a-29b          |
|0           0           0           0          0           -29a              |
|0           0           0           0          0           0                 |
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|0           0           0           0          0           0                 |
|0           0           0           0          0           0                 |
|0           0           0           0          0           0                 |
|-----------------------------------------------------------------------------|
|                                                                             |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|-19c       19c         0           0          0                              |
|19b-29c    -19b+22c    0           0          0                              |
|-19a-24c   19a+8c      0           0          0                              |
|32a-7b-30c -22b        0           0          0                              |
|29a+24b    -22a-8b     0           0          0                              |
|24a        24a-36b-30c 0           0          0                              |
|0          0           32a-36b+42c -19c       19c                            |
|0          0           29b+10c     19b-29c    -19b+22c                       |
|0          0           -29a+29c    -19a-24c   19a+8c                         |
|0          0           -10b        32a-7b-30c -22b                           |
|0          0           -10a-29b    29a+24b    -22a-8b                        |
|0          0           -29a        24a        24a-36b-30c                    |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|0          0           0           0          0                              |
|-----------------------------------------------------------------------------|
|                                                                             |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|0                        |                                                   |
|19a2+19b2+32ac-36bc+42c2 |                                                   |
|-22a2-32ab+7b2+30bc-10c2 |                                                   |
|24a2-36ab-24b2-30ac-29c2 |                                                   |
|-----------------------------------------------------------------------------|
|                                                      20                     |
|<--------------------------------------------------- S   : 3                 |
|   {1} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {3} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {3} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {3} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {3} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {3} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {3} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {3} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {3} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                           |
|   {3} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                           |
|   {5} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                           |
|   {5} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                           |
|   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                           |
|   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                           |
|   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                           |
|   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                           |
|   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                           |
|   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                           |
|   {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                           |
|   {7} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                           |
+-----------------------------------------------------------------------------+
|i22 : assert isWellDefined g                                                 |
+-----------------------------------------------------------------------------+
|i23 : assert isSimplicialMorphism g                                          |
+-----------------------------------------------------------------------------+
|i24 : assert (target g == C)                                                 |
+-----------------------------------------------------------------------------+
|i25 : assert (source g == D)                                                 |
+-----------------------------------------------------------------------------+
|i26 : g^-1                                                                   |
|                                                                             |
|           1                                                                 |
|o26 = 0 : S  <------------- image | 32a-36b-30c | : 0                        |
|                {1} | 1 |                                                    |
|                                                                             |
|           4                                                                 |
|      1 : S  <------------------- image {0} | 32a-36b-30c 0           0      |
|                {1} | 1 0 0 0 |         {2} | 0           32a-36b-30c 0      |
|                {3} | 0 1 0 0 |         {2} | 0           0           32a-   |
|                {3} | 0 0 1 0 |         {2} | 0           0           0      |
|                {3} | 0 0 0 1 |         {3} | 0           0           0      |
|                                                                             |
|           10                                                                |
|      2 : S   <------------------------------- image {0} | 32a-36b-30c 0     |
|                 {1} | 1 0 0 0 0 0 0 0 0 0 |         {2} | 0           32a   |
|                 {3} | 0 1 0 0 0 0 0 0 0 0 |         {2} | 0           0     |
|                 {3} | 0 0 1 0 0 0 0 0 0 0 |         {2} | 0           0     |
|                 {3} | 0 0 0 1 0 0 0 0 0 0 |         {3} | 0           0     |
|                 {3} | 0 0 0 0 1 0 0 0 0 0 |         {2} | 0           0     |
|                 {3} | 0 0 0 0 0 1 0 0 0 0 |         {2} | 0           0     |
|                 {3} | 0 0 0 0 0 0 1 0 0 0 |         {2} | 0           0     |
|                 {5} | 0 0 0 0 0 0 0 1 0 0 |         {3} | 0           0     |
|                 {5} | 0 0 0 0 0 0 0 0 1 0 |         {4} | 0           0     |
|                 {5} | 0 0 0 0 0 0 0 0 0 1 |         {4} | 0           0     |
|                                                     {4} | 0           0     |
|                                                     {4} | 0           0     |
|                                                     {4} | 0           0     |
|                                                     {4} | 0           0     |
|                                                                             |
|           20                                                                |
|      3 : S   <--------------------------------------------------- image {   |
|                 {1} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {   |
|                 {3} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {   |
|                 {3} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {   |
|                 {3} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {   |
|                 {3} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {   |
|                 {3} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {   |
|                 {3} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |         {   |
|                 {3} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |         {   |
|                 {3} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |         {   |
|                 {3} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |         {   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |         {   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |         {   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |         {   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |         {   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |         {   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |         {   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |         {   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |         {   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |         {   |
|                 {7} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |         {   |
|                                                                         {   |
|                                                                         {   |
|                                                                         {   |
|                                                                         {   |
|                                                                         {   |
|                                                                         {   |
|                                                                         {   |
|                                                                         {   |
|                                                                         {   |
|                                                                         {   |
|                                                                         {   |
|                                                                         {   |
|                                                                         {   |
|                                                                             |
|o26 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|        0           | : 1                                                    |
|        0           |                                                        |
|36b-30c 0           |                                                        |
|        32a-36b-30c |                                                        |
|        0           |                                                        |
|                                                                             |
|                                                                             |
|         0           0           0           0           0           0       |
|-36b-30c 0           0           0           0           0           0       |
|         32a-36b-30c 0           0           0           0           0       |
|         0           32a-36b-30c 0           0           0           0       |
|         0           0           0           0           0           0       |
|         0           0           32a-36b-30c 0           0           0       |
|         0           0           0           32a-36b-30c 0           0       |
|         0           0           0           0           32a-36b-30c 0       |
|         0           0           0           0           0           0       |
|         0           0           0           0           0           32a-36b+|
|         0           0           0           0           0           29b+10c |
|         0           0           0           0           0           -29a+29c|
|         0           0           0           0           0           -10b    |
|         0           0           0           0           0           -10a-29b|
|         0           0           0           0           0           -29a    |
|                                                                             |
|                                                                             |
|0} | 32a-36b-30c 0           0           0           0           0           |
|2} | 0           32a-36b-30c 0           0           0           0           |
|2} | 0           0           32a-36b-30c 0           0           0           |
|2} | 0           0           0           32a-36b-30c 0           0           |
|3} | 0           0           0           0           0           0           |
|2} | 0           0           0           0           32a-36b-30c 0           |
|2} | 0           0           0           0           0           32a-36b-30c |
|2} | 0           0           0           0           0           0           |
|3} | 0           0           0           0           0           0           |
|2} | 0           0           0           0           0           0           |
|2} | 0           0           0           0           0           0           |
|2} | 0           0           0           0           0           0           |
|3} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|4} | 0           0           0           0           0           0           |
|5} | 0           0           0           0           0           0           |
|5} | 0           0           0           0           0           0           |
|5} | 0           0           0           0           0           0           |
|-----------------------------------------------------------------------------|
|    0          0           | : 2                                             |
|    0          0           |                                                 |
|    0          0           |                                                 |
|    0          0           |                                                 |
|    0          0           |                                                 |
|    0          0           |                                                 |
|    0          0           |                                                 |
|    0          0           |                                                 |
|    0          0           |                                                 |
|42c -19c       19c         |                                                 |
|    19b-29c    -19b+22c    |                                                 |
|    -19a-24c   19a+8c      |                                                 |
|    32a-7b-30c -22b        |                                                 |
|    29a+24b    -22a-8b     |                                                 |
|    24a        24a-36b-30c |                                                 |
|                                                                             |
|                                                                             |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|32a-36b-30c 0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           32a-36b-30c 0           0           0           0                |
|0           0           32a-36b-30c 0           0           0                |
|0           0           0           32a-36b-30c 0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           32a-36b+42c -19c             |
|0           0           0           0           29b+10c     19b-29c          |
|0           0           0           0           -29a+29c    -19a-24c         |
|0           0           0           0           -10b        32a-7b-30c       |
|0           0           0           0           -10a-29b    29a+24b          |
|0           0           0           0           -29a        24a              |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|0           0           0           0           0           0                |
|-----------------------------------------------------------------------------|
|0           0           0          0           0           0                 |
|0           0           0          0           0           0                 |
|0           0           0          0           0           0                 |
|0           0           0          0           0           0                 |
|0           0           0          0           0           0                 |
|0           0           0          0           0           0                 |
|0           0           0          0           0           0                 |
|0           0           0          0           0           0                 |
|0           0           0          0           0           0                 |
|0           0           0          0           0           0                 |
|0           0           0          0           0           0                 |
|0           0           0          0           0           0                 |
|0           0           0          0           0           0                 |
|19c         0           0          0           0           0                 |
|-19b+22c    0           0          0           0           0                 |
|19a+8c      0           0          0           0           0                 |
|-22b        0           0          0           0           0                 |
|-22a-8b     0           0          0           0           0                 |
|24a-36b-30c 0           0          0           0           0                 |
|0           32a-36b+42c -19c       19c         0           0                 |
|0           29b+10c     19b-29c    -19b+22c    0           0                 |
|0           -29a+29c    -19a-24c   19a+8c      0           0                 |
|0           -10b        32a-7b-30c -22b        0           0                 |
|0           -10a-29b    29a+24b    -22a-8b     0           0                 |
|0           -29a        24a        24a-36b-30c 0           0                 |
|0           0           0          0           32a-36b+42c -19c              |
|0           0           0          0           29b+10c     19b-29c           |
|0           0           0          0           -29a+29c    -19a-24c          |
|0           0           0          0           -10b        32a-7b-30c        |
|0           0           0          0           -10a-29b    29a+24b           |
|0           0           0          0           -29a        24a               |
|0           0           0          0           0           0                 |
|0           0           0          0           0           0                 |
|0           0           0          0           0           0                 |
|-----------------------------------------------------------------------------|
|0           0                        | : 3                                   |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|0           0                        |                                       |
|19c         0                        |                                       |
|-19b+22c    0                        |                                       |
|19a+8c      0                        |                                       |
|-22b        0                        |                                       |
|-22a-8b     0                        |                                       |
|24a-36b-30c 0                        |                                       |
|0           19a2+19b2+32ac-36bc+42c2 |                                       |
|0           -22a2-32ab+7b2+30bc-10c2 |                                       |
|0           24a2-36ab-24b2-30ac-29c2 |                                       |
+-----------------------------------------------------------------------------+
|i27 : assert(g*g^-1 == 1 and g^-1*g == 1)                                    |
+-----------------------------------------------------------------------------+

One can directly prune the map of simplicial modules $f$.

+-----------------------------------------------------------------------------+
|i28 : h = prune f                                                            |
|                                                                             |
|           1                       1                                         |
|o28 = 0 : S  <------------------- S  : 0                                     |
|                | 32a-36b-30c |                                              |
|                                                                             |
|           5                                                               4 |
|      1 : S  <----------------------------------------------------------- S  |
|                {0} | 32a-36b-30c 0           0           0           |      |
|                {2} | 0           32a-36b-30c 0           0           |      |
|                {2} | 0           0           32a-36b-30c 0           |      |
|                {2} | 0           0           0           32a-36b-30c |      |
|                {3} | 0           0           0           0           |      |
|                                                                             |
|           15                                                                |
|      2 : S   <--------------------------------------------------------------|
|                 {0} | 32a-36b-30c 0           0           0           0     |
|                 {2} | 0           32a-36b-30c 0           0           0     |
|                 {2} | 0           0           32a-36b-30c 0           0     |
|                 {2} | 0           0           0           32a-36b-30c 0     |
|                 {3} | 0           0           0           0           0     |
|                 {2} | 0           0           0           0           32a-36|
|                 {2} | 0           0           0           0           0     |
|                 {2} | 0           0           0           0           0     |
|                 {3} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                                                                             |
|           34                                                                |
|      3 : S   <--------------------------------------------------------------|
|                 {0} | 32a-36b-30c 0           0           0           0     |
|                 {2} | 0           32a-36b-30c 0           0           0     |
|                 {2} | 0           0           32a-36b-30c 0           0     |
|                 {2} | 0           0           0           32a-36b-30c 0     |
|                 {3} | 0           0           0           0           0     |
|                 {2} | 0           0           0           0           32a-36|
|                 {2} | 0           0           0           0           0     |
|                 {2} | 0           0           0           0           0     |
|                 {3} | 0           0           0           0           0     |
|                 {2} | 0           0           0           0           0     |
|                 {2} | 0           0           0           0           0     |
|                 {2} | 0           0           0           0           0     |
|                 {3} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {4} | 0           0           0           0           0     |
|                 {5} | 0           0           0           0           0     |
|                 {5} | 0           0           0           0           0     |
|                 {5} | 0           0           0           0           0     |
|                                                                             |
|o28 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|: 1                                                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                      10     |
|-------------------------------------------------------------------- S   : 2 |
|      0           0           0           0          0           |           |
|      0           0           0           0          0           |           |
|      0           0           0           0          0           |           |
|      0           0           0           0          0           |           |
|      0           0           0           0          0           |           |
|b-30c 0           0           0           0          0           |           |
|      32a-36b-30c 0           0           0          0           |           |
|      0           32a-36b-30c 0           0          0           |           |
|      0           0           0           0          0           |           |
|      0           0           32a-36b+42c -19c       19c         |           |
|      0           0           29b+10c     19b-29c    -19b+22c    |           |
|      0           0           -29a+29c    -19a-24c   19a+8c      |           |
|      0           0           -10b        32a-7b-30c -22b        |           |
|      0           0           -10a-29b    29a+24b    -22a-8b     |           |
|      0           0           -29a        24a        24a-36b-30c |           |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|b-30c 0           0           0           0           0           0          |
|      32a-36b-30c 0           0           0           0           0          |
|      0           32a-36b-30c 0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           32a-36b-30c 0           0           0          |
|      0           0           0           32a-36b-30c 0           0          |
|      0           0           0           0           32a-36b-30c 0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           32a-36b+42c|
|      0           0           0           0           0           29b+10c    |
|      0           0           0           0           0           -29a+29c   |
|      0           0           0           0           0           -10b       |
|      0           0           0           0           0           -10a-29b   |
|      0           0           0           0           0           -29a       |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|      0           0           0           0           0           0          |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| -19c       19c         0           0          0           0           0     |
| 19b-29c    -19b+22c    0           0          0           0           0     |
| -19a-24c   19a+8c      0           0          0           0           0     |
| 32a-7b-30c -22b        0           0          0           0           0     |
| 29a+24b    -22a-8b     0           0          0           0           0     |
| 24a        24a-36b-30c 0           0          0           0           0     |
| 0          0           32a-36b+42c -19c       19c         0           0     |
| 0          0           29b+10c     19b-29c    -19b+22c    0           0     |
| 0          0           -29a+29c    -19a-24c   19a+8c      0           0     |
| 0          0           -10b        32a-7b-30c -22b        0           0     |
| 0          0           -10a-29b    29a+24b    -22a-8b     0           0     |
| 0          0           -29a        24a        24a-36b-30c 0           0     |
| 0          0           0           0          0           32a-36b+42c -19c  |
| 0          0           0           0          0           29b+10c     19b-29|
| 0          0           0           0          0           -29a+29c    -19a-2|
| 0          0           0           0          0           -10b        32a-7b|
| 0          0           0           0          0           -10a-29b    29a+24|
| 0          0           0           0          0           -29a        24a   |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
| 0          0           0           0          0           0           0     |
|-----------------------------------------------------------------------------|
|                                               20                            |
|--------------------------------------------- S   : 3                        |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     0           0                        |                                  |
|     19c         0                        |                                  |
|c    -19b+22c    0                        |                                  |
|4c   19a+8c      0                        |                                  |
|-30c -22b        0                        |                                  |
|b    -22a-8b     0                        |                                  |
|     24a-36b-30c 0                        |                                  |
|     0           19a2+19b2+32ac-36bc+42c2 |                                  |
|     0           -22a2-32ab+7b2+30bc-10c2 |                                  |
|     0           24a2-36ab-24b2-30ac-29c2 |                                  |
+-----------------------------------------------------------------------------+
|i29 : assert(source h === prune source f)                                    |
+-----------------------------------------------------------------------------+
|i30 : assert(target h === prune target f)                                    |
+-----------------------------------------------------------------------------+

See also
========

  * Making simplicial modules -- information
    about the basic constructors
  * minimalPresentation(Module) (Macaulay2Doc)
    -- minimal presentation of a
    module
  * randomSimplicialMap
    -- a random
    map of simplicial modules
  * isSimplicialMorphism,
    -- whether a simplicial module map is a morphism of simplicial modules

Ways to use this method:
========================

  * minimalPresentation(SimplicialModule)
    -- minimal presentation of all
    terms in a simplicial module
  * "minimalPresentation(SimplicialModuleMap)"
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1255:0.

naiveNorm(SimplicialModule,ZZ) -- compute the naive normalization of a simplicial object
****************************************************************************************

  * Function: naiveNorm,
  * Usage:
        naiveNorm(S, n)
  * Inputs:
      * S, a Simplicial Module, A simplicial module or
        Complex (Complexes),
      * n, an integer (Macaulay2Doc), and an integer n specifying the
        top degree for normalization.
  * Outputs:
      * a complex (Complexes), The naive normalization complex
        of S, where the differential is the alternating sum of face maps.

Description
===========

This function computes the naive normalization of a simplicial object S. The
naive normalization is a complex built from the modules of the simplicial
object, with a differential that is the alternating sum of the face maps. In
general the naive normalization is homotopy equivalent to the normalization
(see normalize (SchurFunctors)), but is much bigger in
general.

+---------------------------------------------------+
|i1 : Q = ZZ/101[a..d]                              |
|                                                   |
|o1 = Q                                             |
|                                                   |
|o1 : PolynomialRing                                |
+---------------------------------------------------+
|i2 : K = koszulComplex vars Q                      |
|                                                   |
|      1      4      6      4      1                |
|o2 = Q  <-- Q  <-- Q  <-- Q  <-- Q                 |
|                                                   |
|     0      1      2      3      4                 |
|                                                   |
|o2 : Complex                                       |
+---------------------------------------------------+
|i3 : S = simplicialModule(K, Degeneracy => true)   |
|                                                   |
|      1      5      15      35      70             |
|o3 = Q  <-- Q  <-- Q   <-- Q   <-- Q  <-- ...      |
|                                                   |
|     0      1      2       3       4               |
|                                                   |
|o3 : SimplicialModule                              |
+---------------------------------------------------+
|i4 : nK = naiveNorm(S)                             |
|                                                   |
|      1      5      15      35      70             |
|o4 = Q  <-- Q  <-- Q   <-- Q   <-- Q               |
|                                                   |
|     0      1      2       3       4               |
|                                                   |
|o4 : Complex                                       |
+---------------------------------------------------+
|i5 : isWellDefined nK                              |
|                                                   |
|o5 = true                                          |
+---------------------------------------------------+
|i6 : prune HH nK                                   |
|                                                   |
|                                                 46|
|o6 = cokernel | d c b a | <-- 0 <-- 0 <-- 0 <-- Q  |
|                                                   |
|     0                        1     2     3     4  |
|                                                   |
|o6 : Complex                                       |
+---------------------------------------------------+

Note that in the above, the naive normalization will always be an infinite
complex, so there will always be extraneous homology at the tail end. Note in
this case that the homology of the naive normalization is precisely the
homology of K, as it should be (in fact, it is homotopy equivalent to K).

See also
========

  * normalize (SchurFunctors) -- sort each column of a
    Filling, with Koszul sign

Ways to use this method:
========================

  * "naiveNorm(Complex,ZZ)"
  * "naiveNorm(SimplicialModule)"
  * naiveNorm(SimplicialModule,ZZ),
    -- compute the naive normalization of a simplicial object
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3438:0.

normalize(SimplicialModule,ZZ) -- normalization functor from simplicial modules to nonnegatively-graded chain complexes
***********************************************************************************************************************

  * Function: normalize (SchurFunctors),
  * Usage:
        normalize(S, d)
        normalize S
  * Inputs:
      * S, a Simplicial Module, or
        SimplicialModuleMap, the object to be normalized.
      * d, an integer (Macaulay2Doc), An integer specifying the
        degree up to which the normalization should be computed.
  * Optional inputs (Macaulay2Doc),:
      * CheckSum => a Boolean value (Macaulay2Doc), default
        value true, Default value is true. If true and S has more than one
        component, computes the direct sum of normalized components.
      * CheckComplex => a Boolean value (Macaulay2Doc), default
        value true, Default value is true. If true and S contains a cached
        simplicial module or map, it outputs this cached value.
  * Outputs:
      * a complex (Complexes), or ComplexMap (Complexes),
        the normalized nonnegatively-graded chain
        complex resulting from the normalization process.

Description
===========

This function computes the normalization functor from the category of
simplicial modules to the category of nonnegatively-graded chain complexes. It
is implemented in a functorial way, applying both to simplicial modules and
simplicial module maps.

The normalization of a simplicial module $S$ is by definition equal to the
complex $N (S)$ with: $$N(S)_n := \bigcap_{i=1}^n \ker d^S_{(n,i)},$$ with
differential induced by the face map $d_{(n,0)}^S$. As currently implemented,
the normalization does not prune the output; the user should use prune to
obtain the best looking output.

The normalize command is implemented so that it first checks whether a
simplicial module has been obtained as the Dold-Kan image of some complex. If
it has, then it returns that complex without doing any additional computation.
If the user prefers that the normalization is computed by definition instead of
accessing this cached value, use the option CheckComplex => false.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[x_1..x_3];                                                   |
+-----------------------------------------------------------------------------+
|i2 : K = koszulComplex vars R                                                |
|                                                                             |
|      1      3      3      1                                                 |
|o2 = R  <-- R  <-- R  <-- R                                                  |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o2 : Complex                                                                 |
+-----------------------------------------------------------------------------+
|i3 : S = simplicialModule(K,10, Degeneracy => true)                          |
|                                                                             |
|      1      4      10      20      35      56      84      120      165     |
|o3 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R   <-- R    <-- R    <-- |
|                                                                             |
|     0      1      2       3       4       5       6       7        8        |
|                                                                             |
|o3 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
| 220      286                                                                |
|R    <-- R   <-- ...                                                         |
|                                                                             |
|9        10                                                                  |
+-----------------------------------------------------------------------------+
|i4 : keys S                                                                  |
|                                                                             |
|o4 = {complexLength, module, ss, dd, ring, complex, topDegree, cache}        |
|                                                                             |
|o4 : List                                                                    |
+-----------------------------------------------------------------------------+
|i5 : K == normalize S                                                        |
|                                                                             |
|o5 = true                                                                    |
+-----------------------------------------------------------------------------+
|i6 : Kn = normalize(S, CheckComplex => false)                                |
|                                                                             |
|      1                                                                      |
|o6 = R  <-- image {0} | 0 0 0 | <-- image {0} | 0 0 0 | <-- image {0} | 0 |  |
|                  {1} | 1 0 0 |           {1} | 0 0 0 |           {1} | 0 |  |
|     0            {1} | 0 1 0 |           {1} | 0 0 0 |           {1} | 0 |  |
|                  {1} | 0 0 1 |           {1} | 0 0 0 |           {1} | 0 |  |
|                                          {1} | 0 0 0 |           {1} | 0 |  |
|            1                             {1} | 0 0 0 |           {1} | 0 |  |
|                                          {1} | 0 0 0 |           {1} | 0 |  |
|                                          {2} | 1 0 0 |           {1} | 0 |  |
|                                          {2} | 0 1 0 |           {1} | 0 |  |
|                                          {2} | 0 0 1 |           {1} | 0 |  |
|                                                                  {2} | 0 |  |
|                                    2                             {2} | 0 |  |
|                                                                  {2} | 0 |  |
|                                                                  {2} | 0 |  |
|                                                                  {2} | 0 |  |
|                                                                  {2} | 0 |  |
|                                                                  {2} | 0 |  |
|                                                                  {2} | 0 |  |
|                                                                  {2} | 0 |  |
|                                                                  {3} | 1 |  |
|                                                                             |
|                                                            3                |
|                                                                             |
|o6 : Complex                                                                 |
+-----------------------------------------------------------------------------+
|i7 : Kn.dd                                                                   |
|                                                                             |
|          1                                                                  |
|o7 = 0 : R  <------------------- image {0} | 0 0 0 | : 1                     |
|               | x_1 x_2 x_3 |         {1} | 1 0 0 |                         |
|                                       {1} | 0 1 0 |                         |
|                                       {1} | 0 0 1 |                         |
|                                                                             |
|     1 : image {0} | 0 0 0 | <-------------------------- image {0} | 0 0 0 | |
|               {1} | 1 0 0 |    {1} | -x_2 -x_3 0    |         {1} | 0 0 0 | |
|               {1} | 0 1 0 |    {1} | x_1  0    -x_3 |         {1} | 0 0 0 | |
|               {1} | 0 0 1 |    {1} | 0    x_1  x_2  |         {1} | 0 0 0 | |
|                                                               {1} | 0 0 0 | |
|                                                               {1} | 0 0 0 | |
|                                                               {1} | 0 0 0 | |
|                                                               {2} | 1 0 0 | |
|                                                               {2} | 0 1 0 | |
|                                                               {2} | 0 0 1 | |
|                                                                             |
|     2 : image {0} | 0 0 0 | <---------------- image {0} | 0 | : 3           |
|               {1} | 0 0 0 |    {2} | x_3  |         {1} | 0 |               |
|               {1} | 0 0 0 |    {2} | -x_2 |         {1} | 0 |               |
|               {1} | 0 0 0 |    {2} | x_1  |         {1} | 0 |               |
|               {1} | 0 0 0 |                         {1} | 0 |               |
|               {1} | 0 0 0 |                         {1} | 0 |               |
|               {1} | 0 0 0 |                         {1} | 0 |               |
|               {2} | 1 0 0 |                         {1} | 0 |               |
|               {2} | 0 1 0 |                         {1} | 0 |               |
|               {2} | 0 0 1 |                         {1} | 0 |               |
|                                                     {2} | 0 |               |
|                                                     {2} | 0 |               |
|                                                     {2} | 0 |               |
|                                                     {2} | 0 |               |
|                                                     {2} | 0 |               |
|                                                     {2} | 0 |               |
|                                                     {2} | 0 |               |
|                                                     {2} | 0 |               |
|                                                     {2} | 0 |               |
|                                                     {3} | 1 |               |
|                                                                             |
|o7 : ComplexMap                                                              |
|-----------------------------------------------------------------------------|
|: 2                                                                          |
+-----------------------------------------------------------------------------+
|i8 : K == Kn                                                                 |
|                                                                             |
|o8 = false                                                                   |
+-----------------------------------------------------------------------------+
|i9 : K == prune Kn                                                           |
|                                                                             |
|o9 = true                                                                    |
+-----------------------------------------------------------------------------+

For computational efficiency, this method also checks if the simplicial module
has been constructed as a direct sum of simplicial modules. If it has, then it
returns the direct sum of the normalizations of each component. If the user
prefers that this shortcut is not taken, use CheckSum => false.

+-----------------------------------------------------------------------------+
|i10 : S10 = directSum toList(10: forgetComplex S)                            |
|                                                                             |
|       10      40      100      200      350      560      840      1200     |
|o10 = R   <-- R   <-- R    <-- R    <-- R    <-- R    <-- R    <-- R     <-- |
|                                                                             |
|      0       1       2        3        4        5        6        7         |
|                                                                             |
|o10 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
| 1650      2200      2860                                                    |
|R     <-- R     <-- R    <-- ...                                             |
|                                                                             |
|8         9         10                                                       |
+-----------------------------------------------------------------------------+
|i11 : elapsedTime prune normalize S10                                        |
| -- 4.86821s elapsed                                                         |
|                                                                             |
|       10      30      30      10                                            |
|o11 = R   <-- R   <-- R   <-- R                                              |
|                                                                             |
|      0       1       2       3                                              |
|                                                                             |
|o11 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i12 : elapsedTime prune normalize(S10, CheckSum => false) --about 3-4 times  |
| -- 9.62336s elapsed                                                         |
|                                                                             |
|       10      30      30      10                                            |
|o12 = R   <-- R   <-- R   <-- R                                              |
|                                                                             |
|      0       1       2       3                                              |
|                                                                             |
|o12 : Complex                                                                |
|-----------------------------------------------------------------------------|
|slower; becomes significant for larger ranks                                 |
+-----------------------------------------------------------------------------+

The user may also specify the top homological degree to compute the
normalization up to. Note that this can help speed up computational time; if
the user knows the normalization should have a shorter length, then they should
specify this upper bound in the syntax:

+---------------------------------------------------------------------------+
|i13 : elapsedTime prune normalize(S10, 3, CheckSum => false) --MUCH FASTER!|
| -- .0453822s elapsed                                                      |
|                                                                           |
|       10      30      30      10                                          |
|o13 = R   <-- R   <-- R   <-- R                                            |
|                                                                           |
|      0       1       2       3                                            |
|                                                                           |
|o13 : Complex                                                              |
+---------------------------------------------------------------------------+

Again, this method is functorial, and when combined with other methods in this
package can be a particularly powerful way of obtaining nontrivial morphisms of
complexes. We use this method to obtain the image of the inclusion
$$\bigwedge^3 K \to \bigwedge^2 K \otimes K$$ for a Koszul complex K.
Constructing this map directly using the naive definitions of tensor
products/exterior powers of complexes is not possible in full generality.
Taking advantage of the Dold-Kan correspondence and simplicial methods allows
us to obtain this inclusion explicitly.

+-----------------------------------------------------------------------------+
|i14 : Q = ZZ/3[a,b];                                                         |
+-----------------------------------------------------------------------------+
|i15 : K = koszulComplex vars Q                                               |
|                                                                             |
|       1      2      1                                                       |
|o15 = Q  <-- Q  <-- Q                                                        |
|                                                                             |
|      0      1      2                                                        |
|                                                                             |
|o15 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i16 : SK = simplicialModule(K,6) --want top degree 6 since the resulting     |
|                                                                             |
|       1      3      6      10      15      21      28                       |
|o16 = Q  <-- Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- Q  <-- ...                |
|                                                                             |
|      0      1      2      3       4       5       6                         |
|                                                                             |
|o16 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|complexes should have length 6                                               |
+-----------------------------------------------------------------------------+
|i17 : w21K = extPower(2, SK) ** SK                                           |
|                                                                             |
|             9      90      450      1575      4410      10584               |
|o17 = 0 <-- Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q     <-- ...        |
|                                                                             |
|      0     1      2       3        4         5         6                    |
|                                                                             |
|o17 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i18 : w3K = extPower(3, SK)                                                  |
|                                                                             |
|             1      20      120      455      1330      3276                 |
|o18 = 0 <-- Q  <-- Q   <-- Q    <-- Q    <-- Q     <-- Q    <-- ...          |
|                                                                             |
|      0     1      2       3        4        5         6                     |
|                                                                             |
|o18 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i19 : H = hashTable for i from 0 to 6 list i => dual wedgeProduct(2,1, dual  |
|-----------------------------------------------------------------------------|
|SK_i);                                                                       |
+-----------------------------------------------------------------------------+
|i20 : inclusion = map(w21K, w3K, H);                                         |
+-----------------------------------------------------------------------------+
|i21 : inc = prune normalize(inclusion,3);                                    |
+-----------------------------------------------------------------------------+
|i22 : isWellDefined inc                                                      |
|                                                                             |
|o22 = true                                                                   |
+-----------------------------------------------------------------------------+
|i23 : isCommutative inc                                                      |
|                                                                             |
|o23 = true                                                                   |
+-----------------------------------------------------------------------------+

Taking the cokernel of the above morphism yields a canonically defined Schur
functor $\mathbb{S}^{(2,1)} (K)$; constructing this complex using the classical
Schur complex definition will yield a complex that does not have finite length
homology (and hence isn't even homotopically equivalent).

+-----------------------------------------------------------------------------+
|i24 : S21K = prune coker inc --this is only a snapshot; should go up to      |
|                                                                             |
|       8      54      144                                                    |
|o24 = Q  <-- Q   <-- Q                                                       |
|                                                                             |
|      1      2       3                                                       |
|                                                                             |
|o24 : Complex                                                                |
|-----------------------------------------------------------------------------|
|degree 6                                                                     |
+-----------------------------------------------------------------------------+
|i25 : (minimize S21K).dd                                                     |
|                                                                             |
|           2                          5                                      |
|o25 = 1 : Q  <---------------------- Q  : 2                                  |
|                {1} | a 0 b 0 -b |                                           |
|                {1} | 0 a a b a  |                                           |
|                                                                             |
|           5                                                                 |
|      2 : Q  <---------------------------------------------------------------|
|                {2} | -b 0  b  0  -b 0 0 b2  0 b2  0 0   0 0 -ab b2  0 -b2 0 |
|                {2} | -a -b -a -b 0  0 0 ab  0 ab  0 b2  0 0 -a2 -ab 0 0   0 |
|                {2} | a  0  0  b  -a b 0 -ab 0 -ab 0 0   0 0 a2  -ab 0 ab  0 |
|                {2} | 0  a  0  -a 0  a 0 0   0 0   0 -ab 0 0 0   -a2 0 -a2 0 |
|                {2} | 0  0  a  b  a  b 0 0   0 0   0 0   0 0 0   0   0 0   0 |
|                                                                             |
|o25 : ComplexMap                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|-b2 b2  0   0 0 0 0 0 b2  -b2 0   b2  b2  0 ab  0   0 0 0 0 b2  0   0 0 0 0 -|
|-ab -ab -b2 0 0 0 0 0 ab  0   b2  0   0   0 -a2 -ab 0 0 0 0 -ab -b2 0 0 0 0 -|
|ab  -ab 0   0 0 0 0 0 -ab ab  0   -ab -ab 0 0   ab  0 0 0 0 0   b2  0 0 0 0 a|
|0   -a2 ab  0 0 0 0 0 0   -a2 -ab a2  a2  0 0   -a2 0 0 0 0 0   -ab 0 0 0 0 0|
|0   0   0   0 0 0 0 0 0   0   0   0   0   0 a2  ab  0 0 0 0 ab  b2  0 0 0 0 0|
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|ab b2  b2  0   0 0 0 0 -b2 b2  ab2  0    -ab2 0 ab2  b3   ab2  b3   0 b3   0 |
|a2 -ab -ab -b2 0 0 0 0 0   0   a2b  ab2  -a2b 0 a2b  ab2  0    0    0 0    0 |
|2  -ab -ab 0   0 0 0 0 ab  -ab -a2b 0    a2b  0 -a2b -ab2 -a2b -ab2 0 -ab2 0 |
|   -a2 -a2 ab  0 0 0 0 -a2 a2  0    -a2b 0    0 0    0    a3   a2b  0 a2b  0 |
|   0   0   0   0 0 0 0 0   0   0    0    0    0 0    0    0    0    0 0    0 |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|b3   ab2  0 0 b3   0 0 ab2  0 b3   0 0 0 0 0 -ab2 -b3  0 0 0 0 ab2  b3   ab2 |
|ab2  0    0 0 0    0 0 0    0 0    0 0 0 0 0 0    0    0 0 0 0 0    0    0   |
|-ab2 -a2b 0 0 -ab2 0 0 -a2b 0 -ab2 0 0 0 0 0 a2b  ab2  0 0 0 0 -a2b -ab2 -a2b|
|0    a3   0 0 a2b  0 0 a3   0 a2b  0 0 0 0 0 -a3  -a2b 0 0 0 0 a3   a2b  a3  |
|0    0    0 0 0    0 0 0    0 0    0 0 0 0 0 0    0    0 0 0 0 0    0    0   |
|-----------------------------------------------------------------------------|
|                                       101                                   |
|------------------------------------- Q    : 3                               |
| b3   0 0 ab3   ab3   0 0 0 0 0 0 |                                          |
| 0    0 0 0     a2b2  0 0 0 0 0 0 |                                          |
| -ab2 0 0 -a2b2 -a2b2 0 0 0 0 0 0 |                                          |
| a2b  0 0 a3b   0     0 0 0 0 0 0 |                                          |
| 0    0 0 0     0     0 0 0 0 0 0 |                                          |
+-----------------------------------------------------------------------------+
|i26 : prune HH S21K --free part is because we truncated                      |
|                                                                             |
|                                                             98              |
|o26 = cokernel {1} | b a 0 0 | <-- cokernel {2} | b a | <-- Q                |
|               {1} | 0 0 b a |                                               |
|                                   2                        3                |
|      1                                                                      |
|                                                                             |
|o26 : Complex                                                                |
+-----------------------------------------------------------------------------+

See also
========

  * directSum(SimplicialModule), --
    direct sum of simplicial modules
  * naiveNorm -- compute the naive
    normalization of a simplicial object
  * extPower -- compute the Dold-Puppe extension of the
    exterior power functor to simplicial modules
  * simplicialTensor -- compute the simplicial tensor
    product and cache direct sum indices for easy access

Ways to use this method:
========================

  * "normalize(SimplicialModule)"
  * normalize(SimplicialModule,ZZ),
    -- normalization functor from simplicial modules to nonnegatively-graded
    chain complexes
  * "normalize(SimplicialModuleMap)"
  * "normalize(SimplicialModuleMap,ZZ)"
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3847:0.

randomSimplicialMap(SimplicialModule,SimplicialModule) -- a random map of simplicial modules
********************************************************************************************

  * Function: randomSimplicialMap,
  * Usage:
        f = randomSimplicialMap(C,D)
  * Inputs:
      * C, a Simplicial Module,
      * D, a Simplicial Module,
  * Optional inputs (Macaulay2Doc),:
      * Boundary => a Boolean value (Macaulay2Doc), default
        value false, whether the constructed f is a simplicial null homotopy
      * Cycle => a Boolean value (Macaulay2Doc), default value
        false, whether the constructed f commutes with the face/degeneracy maps
      * Degree => an integer (Macaulay2Doc), default value 0, the
        degree of the constructed map of simplicial modules
      * InternalDegree => a list (Macaulay2Doc), default value
        null, or an integer (Macaulay2Doc),
  * Outputs:
      * f, a Map of Simplicial Modules,

Description
===========

A random simplicial module map $f : C \to D$ is obtained from choosing a random
map of the underlying normalizations, which uses the randomComplexMap (Complexes),
command.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a..c]                                                        |
|                                                                             |
|o1 = S                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution coker matrix{{a*b, a*c, b*c}}, 3,   |
|                                                                             |
|      1      4      9      16                                                |
|o2 = S  <-- S  <-- S  <-- S  <-- ...                                         |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i3 : D = simplicialModule(freeResolution coker vars S, Degeneracy => true)   |
|                                                                             |
|      1      4      10      20                                               |
|o3 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : f = randomSimplicialMap(D,C)                                            |
|                                                                             |
|          1              1                                                   |
|o4 = 0 : S  <---------- S  : 0                                               |
|               | 24 |                                                        |
|                                                                             |
|          4                                                         4        |
|     1 : S  <----------------------------------------------------- S  : 1    |
|               {0} | 24 0            0            0            |             |
|               {1} | 0  -36a-30b-29c -29a-24b-38c -39a-18b-13c |             |
|               {1} | 0  19a+19b-10c  -16a+39b+21c -43a-15b-28c |             |
|               {1} | 0  -29a-8b-22c  34a+19b-47c  -47a+38b+2c  |             |
|                                                                             |
|          10                                                                 |
|     2 : S   <---------------------------------------------------------------|
|                {0} | 24 0            0            0            0            |
|                {1} | 0  -36a-30b-29c -29a-24b-38c -39a-18b-13c 0            |
|                {1} | 0  19a+19b-10c  -16a+39b+21c -43a-15b-28c 0            |
|                {1} | 0  -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0            |
|                {1} | 0  0            0            0            -36a-30b-29c |
|                {1} | 0  0            0            0            19a+19b-10c  |
|                {1} | 0  0            0            0            -29a-8b-22c  |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                                                                             |
|          20                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {0} | 24 0            0            0            0            |
|                {1} | 0  -36a-30b-29c -29a-24b-38c -39a-18b-13c 0            |
|                {1} | 0  19a+19b-10c  -16a+39b+21c -43a-15b-28c 0            |
|                {1} | 0  -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0            |
|                {1} | 0  0            0            0            -36a-30b-29c |
|                {1} | 0  0            0            0            19a+19b-10c  |
|                {1} | 0  0            0            0            -29a-8b-22c  |
|                {1} | 0  0            0            0            0            |
|                {1} | 0  0            0            0            0            |
|                {1} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {3} | 0  0            0            0            0            |
|                                                                             |
|o4 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                                         9                   |
|------------------------------------------------------- S  : 2               |
|0            0            0            0            |                        |
|0            0            0            0            |                        |
|0            0            0            0            |                        |
|0            0            0            0            |                        |
|-29a-24b-38c -39a-18b-13c 0            0            |                        |
|-16a+39b+21c -43a-15b-28c 0            0            |                        |
|34a+19b-47c  -47a+38b+2c  0            0            |                        |
|0            0            16a+22b+45c  7a+15b-23c   |                        |
|0            0            -34a-48b-47c 39a+43b-17c  |                        |
|0            0            47a+19b-16c  -11a+48b+36c |                        |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|-29a-24b-38c -39a-18b-13c 0            0            0            0           |
|-16a+39b+21c -43a-15b-28c 0            0            0            0           |
|34a+19b-47c  -47a+38b+2c  0            0            0            0           |
|0            0            -36a-30b-29c -29a-24b-38c -39a-18b-13c 0           |
|0            0            19a+19b-10c  -16a+39b+21c -43a-15b-28c 0           |
|0            0            -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0           |
|0            0            0            0            0            16a+22b+45c |
|0            0            0            0            0            -34a-48b-47c|
|0            0            0            0            0            47a+19b-16c |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|-----------------------------------------------------------------------------|
|                                                                       16    |
|--------------------------------------------------------------------- S   : 3|
| 0            0            0            0            0            |          |
| 0            0            0            0            0            |          |
| 0            0            0            0            0            |          |
| 0            0            0            0            0            |          |
| 0            0            0            0            0            |          |
| 0            0            0            0            0            |          |
| 0            0            0            0            0            |          |
| 0            0            0            0            0            |          |
| 0            0            0            0            0            |          |
| 0            0            0            0            0            |          |
| 7a+15b-23c   0            0            0            0            |          |
| 39a+43b-17c  0            0            0            0            |          |
| -11a+48b+36c 0            0            0            0            |          |
| 0            16a+22b+45c  7a+15b-23c   0            0            |          |
| 0            -34a-48b-47c 39a+43b-17c  0            0            |          |
| 0            47a+19b-16c  -11a+48b+36c 0            0            |          |
| 0            0            0            16a+22b+45c  7a+15b-23c   |          |
| 0            0            0            -34a-48b-47c 39a+43b-17c  |          |
| 0            0            0            47a+19b-16c  -11a+48b+36c |          |
| 0            0            0            0            0            |          |
+-----------------------------------------------------------------------------+
|i5 : assert isWellDefined f                                                  |
+-----------------------------------------------------------------------------+
|i6 : assert not isCommutative f                                              |
+-----------------------------------------------------------------------------+

When the random element is chosen with option Cycle => true, the associated map
of simplicial modules commutes with the face/degeneracy map.

+-----------------------------------------------------------------------------+
|i7 : g = randomSimplicialMap(D,C, Cycle => true)                             |
|                                                                             |
|          1               1                                                  |
|o7 = 0 : S  <----------- S  : 0                                              |
|               | -50 |                                                       |
|                                                                             |
|          4                                             4                    |
|     1 : S  <----------------------------------------- S  : 1                |
|               {0} | -50 0       0        0        |                         |
|               {1} | 0   40b-35c -46b+18c 28b+3c   |                         |
|               {1} | 0   11a-11c 46a+22c  -28a-37c |                         |
|               {1} | 0   35a+11b 33a-22b  -3a-13b  |                         |
|                                                                             |
|          10                                                                 |
|     2 : S   <---------------------------------------------------------------|
|                {0} | -50 0       0        0        0       0        0       |
|                {1} | 0   40b-35c -46b+18c 28b+3c   0       0        0       |
|                {1} | 0   11a-11c 46a+22c  -28a-37c 0       0        0       |
|                {1} | 0   35a+11b 33a-22b  -3a-13b  0       0        0       |
|                {1} | 0   0       0        0        40b-35c -46b+18c 28b+3c  |
|                {1} | 0   0       0        0        11a-11c 46a+22c  -28a-37c|
|                {1} | 0   0       0        0        35a+11b 33a-22b  -3a-13b |
|                {2} | 0   0       0        0        0       0        0       |
|                {2} | 0   0       0        0        0       0        0       |
|                {2} | 0   0       0        0        0       0        0       |
|                                                                             |
|          20                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {0} | -50 0       0        0        0       0        0       |
|                {1} | 0   40b-35c -46b+18c 28b+3c   0       0        0       |
|                {1} | 0   11a-11c 46a+22c  -28a-37c 0       0        0       |
|                {1} | 0   35a+11b 33a-22b  -3a-13b  0       0        0       |
|                {1} | 0   0       0        0        40b-35c -46b+18c 28b+3c  |
|                {1} | 0   0       0        0        11a-11c 46a+22c  -28a-37c|
|                {1} | 0   0       0        0        35a+11b 33a-22b  -3a-13b |
|                {1} | 0   0       0        0        0       0        0       |
|                {1} | 0   0       0        0        0       0        0       |
|                {1} | 0   0       0        0        0       0        0       |
|                {2} | 0   0       0        0        0       0        0       |
|                {2} | 0   0       0        0        0       0        0       |
|                {2} | 0   0       0        0        0       0        0       |
|                {2} | 0   0       0        0        0       0        0       |
|                {2} | 0   0       0        0        0       0        0       |
|                {2} | 0   0       0        0        0       0        0       |
|                {2} | 0   0       0        0        0       0        0       |
|                {2} | 0   0       0        0        0       0        0       |
|                {2} | 0   0       0        0        0       0        0       |
|                {3} | 0   0       0        0        0       0        0       |
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                9                                            |
|------------------------------ S  : 2                                        |
| 0            0            |                                                 |
| 0            0            |                                                 |
| 0            0            |                                                 |
| 0            0            |                                                 |
| 0            0            |                                                 |
| 0            0            |                                                 |
| 0            0            |                                                 |
| 46b-49c      -28a-46b+17c |                                                 |
| -30b-35c     -3a+b        |                                                 |
| -38a-22b-11c -47a+22b     |                                                 |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
| 0       0        0        0            0            0            0          |
| 0       0        0        0            0            0            0          |
| 0       0        0        0            0            0            0          |
| 0       0        0        0            0            0            0          |
| 0       0        0        0            0            0            0          |
| 0       0        0        0            0            0            0          |
| 0       0        0        0            0            0            0          |
| 40b-35c -46b+18c 28b+3c   0            0            0            0          |
| 11a-11c 46a+22c  -28a-37c 0            0            0            0          |
| 35a+11b 33a-22b  -3a-13b  0            0            0            0          |
| 0       0        0        46b-49c      -28a-46b+17c 0            0          |
| 0       0        0        -30b-35c     -3a+b        0            0          |
| 0       0        0        -38a-22b-11c -47a+22b     0            0          |
| 0       0        0        0            0            46b-49c      -28a-46b+17|
| 0       0        0        0            0            -30b-35c     -3a+b      |
| 0       0        0        0            0            -38a-22b-11c -47a+22b   |
| 0       0        0        0            0            0            0          |
| 0       0        0        0            0            0            0          |
| 0       0        0        0            0            0            0          |
| 0       0        0        0            0            0            0          |
|-----------------------------------------------------------------------------|
|                                 16                                          |
|------------------------------- S   : 3                                      |
|  0            0            |                                                |
|  0            0            |                                                |
|  0            0            |                                                |
|  0            0            |                                                |
|  0            0            |                                                |
|  0            0            |                                                |
|  0            0            |                                                |
|  0            0            |                                                |
|  0            0            |                                                |
|  0            0            |                                                |
|  0            0            |                                                |
|  0            0            |                                                |
|  0            0            |                                                |
|c 0            0            |                                                |
|  0            0            |                                                |
|  0            0            |                                                |
|  46b-49c      -28a-46b+17c |                                                |
|  -30b-35c     -3a+b        |                                                |
|  -38a-22b-11c -47a+22b     |                                                |
|  0            0            |                                                |
+-----------------------------------------------------------------------------+
|i8 : assert isWellDefined g                                                  |
+-----------------------------------------------------------------------------+
|i9 : assert isCommutative g                                                  |
+-----------------------------------------------------------------------------+
|i10 : assert isSimplicialMorphism g                                          |
+-----------------------------------------------------------------------------+

When the random element is chosen with option Boundary => true, the associated
map of simplicial modules is a simplicial null homotopy.

+-----------------------------------------------------------------------------+
|i11 : h = randomSimplicialMap(D,C, Boundary => true)                         |
|                                                                             |
|           1         1                                                       |
|o11 = 0 : S  <----- S  : 0                                                   |
|                0                                                            |
|                                                                             |
|           4                                           4                     |
|      1 : S  <--------------------------------------- S  : 1                 |
|                {0} | 0 0       0        0        |                          |
|                {1} | 0 23b+7c  -29b+47c 37b+13c  |                          |
|                {1} | 0 -23a-2c 29a-15c  -37a+10c |                          |
|                {1} | 0 -7a+2b  -47a+15b -13a-10b |                          |
|                                                                             |
|           10                                                                |
|      2 : S   <--------------------------------------------------------------|
|                 {0} | 0 0       0        0        0       0        0        |
|                 {1} | 0 23b+7c  -29b+47c 37b+13c  0       0        0        |
|                 {1} | 0 -23a-2c 29a-15c  -37a+10c 0       0        0        |
|                 {1} | 0 -7a+2b  -47a+15b -13a-10b 0       0        0        |
|                 {1} | 0 0       0        0        23b+7c  -29b+47c 37b+13c  |
|                 {1} | 0 0       0        0        -23a-2c 29a-15c  -37a+10c |
|                 {1} | 0 0       0        0        -7a+2b  -47a+15b -13a-10b |
|                 {2} | 0 0       0        0        0       0        0        |
|                 {2} | 0 0       0        0        0       0        0        |
|                 {2} | 0 0       0        0        0       0        0        |
|                                                                             |
|           20                                                                |
|      3 : S   <--------------------------------------------------------------|
|                 {0} | 0 0       0        0        0       0        0        |
|                 {1} | 0 23b+7c  -29b+47c 37b+13c  0       0        0        |
|                 {1} | 0 -23a-2c 29a-15c  -37a+10c 0       0        0        |
|                 {1} | 0 -7a+2b  -47a+15b -13a-10b 0       0        0        |
|                 {1} | 0 0       0        0        23b+7c  -29b+47c 37b+13c  |
|                 {1} | 0 0       0        0        -23a-2c 29a-15c  -37a+10c |
|                 {1} | 0 0       0        0        -7a+2b  -47a+15b -13a-10b |
|                 {1} | 0 0       0        0        0       0        0        |
|                 {1} | 0 0       0        0        0       0        0        |
|                 {1} | 0 0       0        0        0       0        0        |
|                 {2} | 0 0       0        0        0       0        0        |
|                 {2} | 0 0       0        0        0       0        0        |
|                 {2} | 0 0       0        0        0       0        0        |
|                 {2} | 0 0       0        0        0       0        0        |
|                 {2} | 0 0       0        0        0       0        0        |
|                 {2} | 0 0       0        0        0       0        0        |
|                 {2} | 0 0       0        0        0       0        0        |
|                 {2} | 0 0       0        0        0       0        0        |
|                 {2} | 0 0       0        0        0       0        0        |
|                 {3} | 0 0       0        0        0       0        0        |
|                                                                             |
|o11 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                             9                                               |
|--------------------------- S  : 2                                           |
|0          0            |                                                    |
|0          0            |                                                    |
|0          0            |                                                    |
|0          0            |                                                    |
|0          0            |                                                    |
|0          0            |                                                    |
|0          0            |                                                    |
|29b-48c    -37a-29b-18c |                                                    |
|24b+7c     -13a-36b     |                                                    |
|30a+15b-2c -28a-15b     |                                                    |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|0       0        0        0          0            0          0            0  |
|0       0        0        0          0            0          0            0  |
|0       0        0        0          0            0          0            0  |
|0       0        0        0          0            0          0            0  |
|0       0        0        0          0            0          0            0  |
|0       0        0        0          0            0          0            0  |
|0       0        0        0          0            0          0            0  |
|23b+7c  -29b+47c 37b+13c  0          0            0          0            0  |
|-23a-2c 29a-15c  -37a+10c 0          0            0          0            0  |
|-7a+2b  -47a+15b -13a-10b 0          0            0          0            0  |
|0       0        0        29b-48c    -37a-29b-18c 0          0            0  |
|0       0        0        24b+7c     -13a-36b     0          0            0  |
|0       0        0        30a+15b-2c -28a-15b     0          0            0  |
|0       0        0        0          0            29b-48c    -37a-29b-18c 0  |
|0       0        0        0          0            24b+7c     -13a-36b     0  |
|0       0        0        0          0            30a+15b-2c -28a-15b     0  |
|0       0        0        0          0            0          0            29b|
|0       0        0        0          0            0          0            24b|
|0       0        0        0          0            0          0            30a|
|0       0        0        0          0            0          0            0  |
|-----------------------------------------------------------------------------|
|                          16                                                 |
|------------------------ S   : 3                                             |
|        0            |                                                       |
|        0            |                                                       |
|        0            |                                                       |
|        0            |                                                       |
|        0            |                                                       |
|        0            |                                                       |
|        0            |                                                       |
|        0            |                                                       |
|        0            |                                                       |
|        0            |                                                       |
|        0            |                                                       |
|        0            |                                                       |
|        0            |                                                       |
|        0            |                                                       |
|        0            |                                                       |
|        0            |                                                       |
|-48c    -37a-29b-18c |                                                       |
|+7c     -13a-36b     |                                                       |
|+15b-2c -28a-15b     |                                                       |
|        0            |                                                       |
+-----------------------------------------------------------------------------+
|i12 : assert isWellDefined h                                                 |
+-----------------------------------------------------------------------------+
|i13 : assert isCommutative h                                                 |
+-----------------------------------------------------------------------------+
|i14 : assert isSimplicialMorphism h                                          |
+-----------------------------------------------------------------------------+
|i15 : assert isNullHomotopic normalize h                                     |
+-----------------------------------------------------------------------------+
|i16 : nullHomotopy normalize h                                               |
|                                                                             |
|           3         1                                                       |
|o16 = 1 : S  <----- S  : 0                                                   |
|                0                                                            |
|                                                                             |
|           3                           3                                     |
|      2 : S  <----------------------- S  : 1                                 |
|                {2} | -23 29  -37 |                                          |
|                {2} | -7  -47 -13 |                                          |
|                {2} | 2   15  -10 |                                          |
|                                                                             |
|           1                      2                                          |
|      3 : S  <------------------ S  : 2                                      |
|                {3} | 30 -18 |                                               |
|                                                                             |
|o16 : ComplexMap                                                             |
+-----------------------------------------------------------------------------+

When the degree of the random element is chosen with a specific degree, the
associated map of simplicial modules will be a well-defined degree 0 simplicial
morphism mapping to the Dold-Kan image of the shift of the normalization. Thus,
even when specifying nonzero degree this constructor will still yield a
simplicial morphism.

+-----------------------------------------------------------------------------+
|i17 : p = randomSimplicialMap(D, C, Cycle => true, Degree => -1)             |
|                                                                             |
|           1                                                                 |
|o17 = 0 : S  <---------------------------------------------------------------|
|                | 19a2-30ab+36b2+40ac+4bc-32c2 36a2+42ab-33b2+13ac-30bc+9c2 2|
|                                                                             |
|           4                                                                 |
|      1 : S  <---------------------------------------------------------------|
|                {0} | 19a2-30ab+36b2+40ac+4bc-32c2 36a2+42ab-33b2+13ac-30bc+9|
|                {1} | 0                            0                         |
|                {1} | 0                            0                         |
|                {1} | 0                            0                         |
|                                                                             |
|           10                                                                |
|      2 : S   <--------------------------------------------------------------|
|                 {0} | 19a2-30ab+36b2+40ac+4bc-32c2 36a2+42ab-33b2+13ac-30bc+|
|                 {1} | 0                            0                        |
|                 {1} | 0                            0                        |
|                 {1} | 0                            0                        |
|                 {1} | 0                            0                        |
|                 {1} | 0                            0                        |
|                 {1} | 0                            0                        |
|                 {2} | 0                            0                        |
|                 {2} | 0                            0                        |
|                 {2} | 0                            0                        |
|                                                                             |
|           20                                                                |
|      3 : S   <--------------------------------------------------------------|
|                 {0} | 19a2-30ab+36b2+40ac+4bc-32c2 36a2+42ab-33b2+13ac-30bc+|
|                 {1} | 0                            0                        |
|                 {1} | 0                            0                        |
|                 {1} | 0                            0                        |
|                 {1} | 0                            0                        |
|                 {1} | 0                            0                        |
|                 {1} | 0                            0                        |
|                 {1} | 0                            0                        |
|                 {1} | 0                            0                        |
|                 {1} | 0                            0                        |
|                 {2} | 0                            0                        |
|                 {2} | 0                            0                        |
|                 {2} | 0                            0                        |
|                 {2} | 0                            0                        |
|                 {2} | 0                            0                        |
|                 {2} | 0                            0                        |
|                 {2} | 0                            0                        |
|                 {2} | 0                            0                        |
|                 {2} | 0                            0                        |
|                 {3} | 0                            0                        |
|                                                                             |
|o17 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                                  3                                          |
|-------------------------------- S  : 0                                      |
|0a2+12ab-44b2-13ac-16bc+26c2 |                                               |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|c2 20a2+12ab-44b2-13ac-16bc+26c2 0                            0              |
|   0                             -10ab-31b2-20ac+31bc-39c2    -20a2+24ab-48b2|
|   0                             -26a2-11ab+33b2+34bc+4c2     33ab-33b2-49ac-|
|   0                             39a2+27ab+32b2-22ac-9bc-32c2 43a2-8ab-11b2+3|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|9c2 20a2+12ab-44b2-13ac-16bc+26c2 0                            0             |
|    0                             -10ab-31b2-20ac+31bc-39c2    -20a2+24ab-48b|
|    0                             -26a2-11ab+33b2+34bc+4c2     33ab-33b2-49ac|
|    0                             39a2+27ab+32b2-22ac-9bc-32c2 43a2-8ab-11b2+|
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|9c2 20a2+12ab-44b2-13ac-16bc+26c2 0                            0             |
|    0                             -10ab-31b2-20ac+31bc-39c2    -20a2+24ab-48b|
|    0                             -26a2-11ab+33b2+34bc+4c2     33ab-33b2-49ac|
|    0                             39a2+27ab+32b2-22ac-9bc-32c2 43a2-8ab-11b2+|
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|    0                             0                            0             |
|-----------------------------------------------------------------------------|
|                     5                                                       |
|------------------- S  : 1                                                   |
|                |                                                            |
|-30ac-15bc+39c2 |                                                            |
|19bc+17c2       |                                                            |
|6ac-8bc         |                                                            |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                 0                            0                              |
|2-30ac-15bc+39c2 0                            0                              |
|-19bc+17c2       0                            0                              |
|36ac-8bc         0                            0                              |
|                 -10ab-31b2-20ac+31bc-39c2    -20a2+24ab-48b2-30ac-15bc+39c2 |
|                 -26a2-11ab+33b2+34bc+4c2     33ab-33b2-49ac-19bc+17c2       |
|                 39a2+27ab+32b2-22ac-9bc-32c2 43a2-8ab-11b2+36ac-8bc         |
|                 0                            0                              |
|                 0                            0                              |
|                 0                            0                              |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                 0                            0                              |
|2-30ac-15bc+39c2 0                            0                              |
|-19bc+17c2       0                            0                              |
|36ac-8bc         0                            0                              |
|                 -10ab-31b2-20ac+31bc-39c2    -20a2+24ab-48b2-30ac-15bc+39c2 |
|                 -26a2-11ab+33b2+34bc+4c2     33ab-33b2-49ac-19bc+17c2       |
|                 39a2+27ab+32b2-22ac-9bc-32c2 43a2-8ab-11b2+36ac-8bc         |
|                 0                            0                              |
|                 0                            0                              |
|                 0                            0                              |
|                 0                            0                              |
|                 0                            0                              |
|                 0                            0                              |
|                 0                            0                              |
|                 0                            0                              |
|                 0                            0                              |
|                 0                            0                              |
|                 0                            0                              |
|                 0                            0                              |
|                 0                            0                              |
|-----------------------------------------------------------------------------|
|     7                                                                       |
|--- S  : 2                                                                   |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
|                                                                             |
|                                                                 9           |
|--------------------------------------------------------------- S  : 3       |
|0                            0                              |                |
|0                            0                              |                |
|0                            0                              |                |
|0                            0                              |                |
|0                            0                              |                |
|0                            0                              |                |
|0                            0                              |                |
|-10ab-31b2-20ac+31bc-39c2    -20a2+24ab-48b2-30ac-15bc+39c2 |                |
|-26a2-11ab+33b2+34bc+4c2     33ab-33b2-49ac-19bc+17c2       |                |
|39a2+27ab+32b2-22ac-9bc-32c2 43a2-8ab-11b2+36ac-8bc         |                |
|0                            0                              |                |
|0                            0                              |                |
|0                            0                              |                |
|0                            0                              |                |
|0                            0                              |                |
|0                            0                              |                |
|0                            0                              |                |
|0                            0                              |                |
|0                            0                              |                |
|0                            0                              |                |
+-----------------------------------------------------------------------------+
|i18 : assert isWellDefined p                                                 |
+-----------------------------------------------------------------------------+
|i19 : assert isCommutative p                                                 |
+-----------------------------------------------------------------------------+
|i20 : assert isSimplicialMorphism p                                          |
+-----------------------------------------------------------------------------+

Given an internal degree, the random element is constructed as maps of modules
with this degree.

+---------------------------------------------------------------------------+
|i21 : q = randomSimplicialMap(D, C, Boundary => true, InternalDegree => 2);|
+---------------------------------------------------------------------------+
|i22 : assert isCommutative q                                               |
+---------------------------------------------------------------------------+
|i23 : assert isSimplicialMorphism q                                        |
+---------------------------------------------------------------------------+
|i24 : source q === C                                                       |
|                                                                           |
|o24 = false                                                                |
+---------------------------------------------------------------------------+
|i25 : target q === D                                                       |
|                                                                           |
|o25 = false                                                                |
+---------------------------------------------------------------------------+
|i26 : assert isNullHomotopic normalize q                                   |
+---------------------------------------------------------------------------+

Ways to use this method:
========================

  * randomSimplicialMap(SimplicialModule,SimplicialModule)
    -- a random
    map of simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3041:0.

ring(SimplicialModule) -- access the ring of a simplicial module or a simplicial module map
*******************************************************************************************

  * Function: ring (Macaulay2Doc),
  * Usage:
        ring C
  * Inputs:
      * C, a Simplicial Module, or a
        SimplicialModuleMap,
  * Outputs:
      * a ring (Macaulay2Doc),

Description
===========

Every simplicial module or simplicial module map has a base ring.  This
function accesses that information.

+-----------------------------------------------------+
|i1 : S = ZZ/101[a,b,c,d];                            |
+-----------------------------------------------------+
|i2 : C = simplicialModule freeResolution coker vars S|
|                                                     |
|      1      5      15      35      70               |
|o2 = S  <-- S  <-- S   <-- S   <-- S  <-- ...        |
|                                                     |
|     0      1      2       3       4                 |
|                                                     |
|o2 : SimplicialModule                                |
+-----------------------------------------------------+
|i3 : ring C                                          |
|                                                     |
|o3 = S                                               |
|                                                     |
|o3 : PolynomialRing                                  |
+-----------------------------------------------------+
|i4 : assert(ring C === S)                            |
+-----------------------------------------------------+
|i5 : ring id_C                                       |
|                                                     |
|o5 = S                                               |
|                                                     |
|o5 : PolynomialRing                                  |
+-----------------------------------------------------+
|i6 : assert(ring id_C === S)                         |
+-----------------------------------------------------+

See also
========

  * Basic invariants and properties, --
    information about accessing basic features
  * ring (Macaulay2Doc) -- get the associated ring of an object

Ways to use this method:
========================

  * ring(SimplicialModule) -- access the
    ring of a simplicial module or a simplicial module map
  * "ring(SimplicialModuleMap)"
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:363:0.

RingMap ** SimplicialModule -- tensor a simplicial module along a ring map
**************************************************************************

  * Operator: ** (Macaulay2Doc),
  * Usage:
        phi ** C
        tensor(phi, C)
        S ** C
        C ** S
  * Inputs:
      * phi, a ring map (Macaulay2Doc), whose source is a ring
        $R$ and whose target is a ring $S$
      * C, a Simplicial Module, over the ring $R$
  * Outputs:
      * a Simplicial Module, over the ring $S$

Description
===========

These methods implement the base change of rings.  As input, one can either
give a ring map $\phi$, or the ring $S$ (when there is a canonical map from $R$
to $S$).

We illustrate the tensor product of a simplicial module along a ring map.

+-----------------------------------------------------------------------------+
|i1 : R = QQ[x,y,z];                                                          |
+-----------------------------------------------------------------------------+
|i2 : S = QQ[s,t];                                                            |
+-----------------------------------------------------------------------------+
|i3 : phi = map(S, R, {s, s+t, t})                                            |
|                                                                             |
|o3 = map (S, R, {s, s + t, t})                                               |
|                                                                             |
|o3 : RingMap S <-- R                                                         |
+-----------------------------------------------------------------------------+
|i4 : I = ideal(x^3, x^2*y, x*y^4, y*z^5)                                     |
|                                                                             |
|             3   2      4     5                                              |
|o4 = ideal (x , x y, x*y , y*z )                                             |
|                                                                             |
|o4 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i5 : C = simplicialModule(freeResolution I, 3, Degeneracy => true)           |
|                                                                             |
|      1      5      13      26                                               |
|o5 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o5 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i6 : D = phi ** C                                                            |
|                                                                             |
|      1      5      13      26                                               |
|o6 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o6 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i7 : assert isWellDefined D                                                  |
+-----------------------------------------------------------------------------+
|i8 : dd^D                                                                    |
|                                                                             |
|               1                                                      5      |
|o8 = (0, 0) : S  <-------------------------------------------------- S  : (1,|
|                    | 1 s3 s3+s2t s5+4s4t+6s3t2+4s2t3+st4 st5+t6 |           |
|                                                                             |
|               1                     5                                       |
|     (0, 1) : S  <----------------- S  : (1, 1)                              |
|                    | 1 0 0 0 0 |                                            |
|                                                                             |
|               5                                                             |
|     (1, 0) : S  <-----------------------------------------------------------|
|                    {0} | 1 s3 s3+s2t s5+4s4t+6s3t2+4s2t3+st4 st5+t6 0 0 0 0 |
|                    {3} | 0 0  0      0                       0      1 0 0 0 |
|                    {3} | 0 0  0      0                       0      0 1 0 0 |
|                    {5} | 0 0  0      0                       0      0 0 1 0 |
|                    {6} | 0 0  0      0                       0      0 0 0 1 |
|                                                                             |
|               5                                         13                  |
|     (1, 1) : S  <------------------------------------- S   : (2, 1)         |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 |                        |
|                    {3} | 0 1 0 0 0 1 0 0 0 0 0 0 0 |                        |
|                    {3} | 0 0 1 0 0 0 1 0 0 0 0 0 0 |                        |
|                    {5} | 0 0 0 1 0 0 0 1 0 0 0 0 0 |                        |
|                    {6} | 0 0 0 0 1 0 0 0 1 0 0 0 0 |                        |
|                                                                             |
|               5                                         13                  |
|     (1, 2) : S  <------------------------------------- S   : (2, 2)         |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 |                        |
|                    {3} | 0 1 0 0 0 0 0 0 0 0 0 0 0 |                        |
|                    {3} | 0 0 1 0 0 0 0 0 0 0 0 0 0 |                        |
|                    {5} | 0 0 0 1 0 0 0 0 0 0 0 0 0 |                        |
|                    {6} | 0 0 0 0 1 0 0 0 0 0 0 0 0 |                        |
|                                                                             |
|               13                                                            |
|     (2, 0) : S   <----------------------------------------------------------|
|                     {0}  | 1 s3 s3+s2t s5+4s4t+6s3t2+4s2t3+st4 st5+t6 0 0 0 |
|                     {3}  | 0 0  0      0                       0      1 0 0 |
|                     {3}  | 0 0  0      0                       0      0 1 0 |
|                     {5}  | 0 0  0      0                       0      0 0 1 |
|                     {6}  | 0 0  0      0                       0      0 0 0 |
|                     {3}  | 0 0  0      0                       0      0 0 0 |
|                     {3}  | 0 0  0      0                       0      0 0 0 |
|                     {5}  | 0 0  0      0                       0      0 0 0 |
|                     {6}  | 0 0  0      0                       0      0 0 0 |
|                     {4}  | 0 0  0      0                       0      0 0 0 |
|                     {6}  | 0 0  0      0                       0      0 0 0 |
|                     {8}  | 0 0  0      0                       0      0 0 0 |
|                     {10} | 0 0  0      0                       0      0 0 0 |
|                                                                             |
|               13                                                            |
|     (2, 1) : S   <----------------------------------------------------------|
|                     {0}  | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {5}  | 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {5}  | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {4}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0|
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0|
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0|
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1|
|                                                                             |
|               13                                                            |
|     (2, 2) : S   <----------------------------------------------------------|
|                     {0}  | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {5}  | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {5}  | 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {4}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0|
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0|
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0|
|                                                                             |
|               13                                                            |
|     (2, 3) : S   <----------------------------------------------------------|
|                     {0}  | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {5}  | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {5}  | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {4}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|                                                                             |
|o8 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
| 0)                                                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                 13                          |
|----------------------------------------------- S   : (2, 0)                 |
|0    0                0   0                 |                                |
|-s-t 0                0   0                 |                                |
|s    -s3-3s2t-3st2-t3 -t5 0                 |                                |
|0    s                0   -t5               |                                |
|0    0                s2  s4+3s3t+3s2t2+st3 |                                |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0    0                0   0                 0    0                0|
|0 0 0 0 0 -s-t 0                0   0                 0    0                0|
|0 0 0 0 0 s    -s3-3s2t-3st2-t3 -t5 0                 0    0                0|
|0 0 0 0 0 0    s                0   -t5               0    0                0|
|1 0 0 0 0 0    0                s2  s4+3s3t+3s2t2+st3 0    0                0|
|0 1 0 0 0 0    0                0   0                 -s-t 0                0|
|0 0 1 0 0 0    0                0   0                 s    -s3-3s2t-3st2-t3 -|
|0 0 0 1 0 0    0                0   0                 0    s                0|
|0 0 0 0 1 0    0                0   0                 0    0                s|
|0 0 0 0 0 0    0                0   0                 0    0                0|
|0 0 0 0 0 0    0                0   0                 0    0                0|
|0 0 0 0 0 0    0                0   0                 0    0                0|
|0 0 0 0 0 0    0                0   0                 0    0                0|
|                                                                             |
|        26                                                                   |
|------ S   : (3, 1)                                                          |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
|                                                                             |
|        26                                                                   |
|------ S   : (3, 2)                                                          |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
|                                                                             |
|        26                                                                   |
|------ S   : (3, 3)                                                          |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
|-----------------------------------------------------------------------------|
|                                                   26                        |
|------------------------------------------------- S   : (3, 0)               |
|   0                 0 0 0 0 0                |                              |
|   0                 0 0 0 0 0                |                              |
|   0                 0 0 0 0 0                |                              |
|   0                 0 0 0 0 0                |                              |
|   0                 0 0 0 0 0                |                              |
|   0                 0 0 0 0 0                |                              |
|t5 0                 0 0 0 0 0                |                              |
|   -t5               0 0 0 0 0                |                              |
|2  s4+3s3t+3s2t2+st3 0 0 0 0 0                |                              |
|   0                 1 0 0 0 0                |                              |
|   0                 0 1 0 0 t5               |                              |
|   0                 0 0 1 0 -s3-3s2t-3st2-t3 |                              |
|   0                 0 0 0 1 s                |                              |
+-----------------------------------------------------------------------------+
|i9 : ss^D                                                                    |
|                                                                             |
|               5                 1                                           |
|o9 = (1, 0) : S  <------------- S  : (0, 0)                                  |
|                    {0} | 1 |                                                |
|                    {3} | 0 |                                                |
|                    {3} | 0 |                                                |
|                    {5} | 0 |                                                |
|                    {6} | 0 |                                                |
|                                                                             |
|               13                          5                                 |
|     (2, 0) : S   <---------------------- S  : (1, 0)                        |
|                     {0}  | 1 0 0 0 0 |                                      |
|                     {3}  | 0 0 0 0 0 |                                      |
|                     {3}  | 0 0 0 0 0 |                                      |
|                     {5}  | 0 0 0 0 0 |                                      |
|                     {6}  | 0 0 0 0 0 |                                      |
|                     {3}  | 0 1 0 0 0 |                                      |
|                     {3}  | 0 0 1 0 0 |                                      |
|                     {5}  | 0 0 0 1 0 |                                      |
|                     {6}  | 0 0 0 0 1 |                                      |
|                     {4}  | 0 0 0 0 0 |                                      |
|                     {6}  | 0 0 0 0 0 |                                      |
|                     {8}  | 0 0 0 0 0 |                                      |
|                     {10} | 0 0 0 0 0 |                                      |
|                                                                             |
|               13                          5                                 |
|     (2, 1) : S   <---------------------- S  : (1, 1)                        |
|                     {0}  | 1 0 0 0 0 |                                      |
|                     {3}  | 0 1 0 0 0 |                                      |
|                     {3}  | 0 0 1 0 0 |                                      |
|                     {5}  | 0 0 0 1 0 |                                      |
|                     {6}  | 0 0 0 0 1 |                                      |
|                     {3}  | 0 0 0 0 0 |                                      |
|                     {3}  | 0 0 0 0 0 |                                      |
|                     {5}  | 0 0 0 0 0 |                                      |
|                     {6}  | 0 0 0 0 0 |                                      |
|                     {4}  | 0 0 0 0 0 |                                      |
|                     {6}  | 0 0 0 0 0 |                                      |
|                     {8}  | 0 0 0 0 0 |                                      |
|                     {10} | 0 0 0 0 0 |                                      |
|                                                                             |
|               26                                          13                |
|     (3, 0) : S   <-------------------------------------- S   : (2, 0)       |
|                     {0}  | 1 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {5}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 1 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 0 1 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {5}  | 0 0 0 1 0 0 0 0 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 1 0 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 0 0 0 0 1 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 0 0 0 0 0 1 0 0 0 0 0 0 |                      |
|                     {5}  | 0 0 0 0 0 0 0 1 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 1 0 0 0 0 |                      |
|                     {4}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {4}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {4}  | 0 0 0 0 0 0 0 0 0 1 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 1 0 0 |                      |
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 1 0 |                      |
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 1 |                      |
|                     {11} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                                                                             |
|               26                                          13                |
|     (3, 1) : S   <-------------------------------------- S   : (2, 1)       |
|                     {0}  | 1 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 1 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 0 1 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {5}  | 0 0 0 1 0 0 0 0 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 1 0 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {5}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 0 0 0 0 1 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 0 0 0 0 0 1 0 0 0 0 0 0 |                      |
|                     {5}  | 0 0 0 0 0 0 0 1 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 1 0 0 0 0 |                      |
|                     {4}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {4}  | 0 0 0 0 0 0 0 0 0 1 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 1 0 0 |                      |
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 1 0 |                      |
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 1 |                      |
|                     {4}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {11} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                                                                             |
|               26                                          13                |
|     (3, 2) : S   <-------------------------------------- S   : (2, 2)       |
|                     {0}  | 1 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 1 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 0 1 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {5}  | 0 0 0 1 0 0 0 0 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 1 0 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 0 0 0 0 1 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 0 0 0 0 0 1 0 0 0 0 0 0 |                      |
|                     {5}  | 0 0 0 0 0 0 0 1 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 1 0 0 0 0 |                      |
|                     {3}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {3}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {5}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {4}  | 0 0 0 0 0 0 0 0 0 1 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 1 0 0 |                      |
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 1 0 |                      |
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 1 |                      |
|                     {4}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {4}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                     {11} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                                                                             |
|o9 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+

If a ring is used rather than a ring map, then the implicit map from the
underlying ring of the complex to the given ring is used.

+--------------------------------------+
|i10 : A = R/(x^2+y^2+z^2);            |
+--------------------------------------+
|i11 : C ** A                          |
|                                      |
|       1      5      13      26       |
|o11 = A  <-- A  <-- A   <-- A  <-- ...|
|                                      |
|      0      1      2       3         |
|                                      |
|o11 : SimplicialModule                |
+--------------------------------------+
|i12 : assert(map(A,R) ** C == C ** A) |
+--------------------------------------+

The commutativity of tensor product is witnessed as follows.

+------------------------------+
|i13 : assert(D == C ** phi)   |
+------------------------------+
|i14 : assert(C ** A == A ** C)|
+------------------------------+

When the modules in the complex are not free modules, this is different than
the image of a complex under a ring map.

+-----------------------------------------------------------------------------+
|i15 : use R                                                                  |
|                                                                             |
|o15 = R                                                                      |
|                                                                             |
|o15 : PolynomialRing                                                         |
+-----------------------------------------------------------------------------+
|i16 : I = ideal(x*y, x*z, y*z);                                              |
|                                                                             |
|o16 : Ideal of R                                                             |
+-----------------------------------------------------------------------------+
|i17 : J = I + ideal(x^2, y^2);                                               |
|                                                                             |
|o17 : Ideal of R                                                             |
+-----------------------------------------------------------------------------+
|i18 : g = inducedMap(module J, module I)                                     |
|                                                                             |
|o18 = {2} | 1 0 0 |                                                          |
|      {2} | 0 1 0 |                                                          |
|      {2} | 0 0 1 |                                                          |
|      {2} | 0 0 0 |                                                          |
|      {2} | 0 0 0 |                                                          |
|                                                                             |
|o18 : Matrix                                                                 |
+-----------------------------------------------------------------------------+
|i19 : assert isWellDefined g                                                 |
+-----------------------------------------------------------------------------+
|i20 : C = simplicialModule(complex {g}, 3, Degeneracy => true)               |
|                                                                             |
|o20 = image | xy xz yz x2 y2 | <-- image | xy xz yz x2 y2 0  0  0  | <--     |
|                                         | 0  0  0  0  0  xy xz yz |         |
|      0                                                                      |
|                                   1                                         |
|                                                                             |
|                                                                             |
|                                                                             |
|o20 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|image | xy xz yz x2 y2 0  0  0  0  0  0  | <-- image | xy xz yz x2 y2 0  0   |
|      | 0  0  0  0  0  xy xz yz 0  0  0  |           | 0  0  0  0  0  xy xz  |
|      | 0  0  0  0  0  0  0  0  xy xz yz |           | 0  0  0  0  0  0  0   |
|                                                     | 0  0  0  0  0  0  0   |
|2                                                                            |
|                                               3                             |
|-----------------------------------------------------------------------------|
|0  0  0  0  0  0  0  |<-- ...                                                |
|yz 0  0  0  0  0  0  |                                                       |
|0  xy xz yz 0  0  0  |                                                       |
|0  0  0  0  xy xz yz |                                                       |
+-----------------------------------------------------------------------------+
|i21 : D1 = phi C                                                             |
|                                                                             |
|o21 = cokernel {2} | -t  -t 0  s    0   -s-t | <-- cokernel {2} | -t  -t 0   |
|               {2} | s+t 0  s  0    0   0    |              {2} | s+t 0  s   |
|               {2} | 0   s  0  0    s+t 0    |              {2} | 0   s  0   |
|               {2} | 0   0  -t -s-t 0   0    |              {2} | 0   0  -t  |
|               {2} | 0   0  0  0    -t  s    |              {2} | 0   0  0   |
|                                                            {2} | 0   0  0   |
|      0                                                     {2} | 0   0  0   |
|                                                            {2} | 0   0  0   |
|                                                                             |
|                                                   1                         |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o21 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|s    0   -s-t 0   0  | <-- cokernel {2} | -t  -t 0  s    0   -s-t 0   0  0   |
|0    0   0    0   0  |              {2} | s+t 0  s  0    0   0    0   0  0   |
|0    s+t 0    0   0  |              {2} | 0   s  0  0    s+t 0    0   0  0   |
|-s-t 0   0    0   0  |              {2} | 0   0  -t -s-t 0   0    0   0  0   |
|0    -t  s    0   0  |              {2} | 0   0  0  0    -t  s    0   0  0   |
|0    0   0    -t  -t |              {2} | 0   0  0  0    0   0    -t  -t 0   |
|0    0   0    s+t 0  |              {2} | 0   0  0  0    0   0    s+t 0  0   |
|0    0   0    0   s  |              {2} | 0   0  0  0    0   0    0   s  0   |
|                                    {2} | 0   0  0  0    0   0    0   0  -t  |
|                                    {2} | 0   0  0  0    0   0    0   0  s+t |
|                                    {2} | 0   0  0  0    0   0    0   0  0   |
|                                                                             |
|                           2                                                 |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|0  | <-- cokernel {2} | -t  -t 0  s    0   -s-t 0   0  0   0  0   0  |<-- ...|
|0  |              {2} | s+t 0  s  0    0   0    0   0  0   0  0   0  |       |
|0  |              {2} | 0   s  0  0    s+t 0    0   0  0   0  0   0  |       |
|0  |              {2} | 0   0  -t -s-t 0   0    0   0  0   0  0   0  |       |
|0  |              {2} | 0   0  0  0    -t  s    0   0  0   0  0   0  |       |
|0  |              {2} | 0   0  0  0    0   0    -t  -t 0   0  0   0  |       |
|0  |              {2} | 0   0  0  0    0   0    s+t 0  0   0  0   0  |       |
|0  |              {2} | 0   0  0  0    0   0    0   s  0   0  0   0  |       |
|-t |              {2} | 0   0  0  0    0   0    0   0  -t  -t 0   0  |       |
|0  |              {2} | 0   0  0  0    0   0    0   0  s+t 0  0   0  |       |
|s  |              {2} | 0   0  0  0    0   0    0   0  0   s  0   0  |       |
|                  {2} | 0   0  0  0    0   0    0   0  0   0  -t  -t |       |
|                  {2} | 0   0  0  0    0   0    0   0  0   0  s+t 0  |       |
|                  {2} | 0   0  0  0    0   0    0   0  0   0  0   s  |       |
|                                                                             |
|         3                                                                   |
+-----------------------------------------------------------------------------+
|i22 : assert isWellDefined D1                                                |
+-----------------------------------------------------------------------------+
|i23 : D2 = phi ** C                                                          |
|                                                                             |
|o23 = cokernel {2} | -t  -t 0  s    0   -s-t | <-- cokernel {2} | -t  -t 0   |
|               {2} | s+t 0  s  0    0   0    |              {2} | s+t 0  s   |
|               {2} | 0   s  0  0    s+t 0    |              {2} | 0   s  0   |
|               {2} | 0   0  -t -s-t 0   0    |              {2} | 0   0  -t  |
|               {2} | 0   0  0  0    -t  s    |              {2} | 0   0  0   |
|                                                            {2} | 0   0  0   |
|      0                                                     {2} | 0   0  0   |
|                                                            {2} | 0   0  0   |
|                                                                             |
|                                                   1                         |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o23 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|s    0   -s-t 0   0  | <-- cokernel {2} | -t  -t 0  s    0   -s-t 0   0  0   |
|0    0   0    0   0  |              {2} | s+t 0  s  0    0   0    0   0  0   |
|0    s+t 0    0   0  |              {2} | 0   s  0  0    s+t 0    0   0  0   |
|-s-t 0   0    0   0  |              {2} | 0   0  -t -s-t 0   0    0   0  0   |
|0    -t  s    0   0  |              {2} | 0   0  0  0    -t  s    0   0  0   |
|0    0   0    -t  -t |              {2} | 0   0  0  0    0   0    -t  -t 0   |
|0    0   0    s+t 0  |              {2} | 0   0  0  0    0   0    s+t 0  0   |
|0    0   0    0   s  |              {2} | 0   0  0  0    0   0    0   s  0   |
|                                    {2} | 0   0  0  0    0   0    0   0  -t  |
|                                    {2} | 0   0  0  0    0   0    0   0  s+t |
|                                    {2} | 0   0  0  0    0   0    0   0  0   |
|                                                                             |
|                           2                                                 |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|0  | <-- cokernel {2} | -t  -t 0  s    0   -s-t 0   0  0   0  0   0  |<-- ...|
|0  |              {2} | s+t 0  s  0    0   0    0   0  0   0  0   0  |       |
|0  |              {2} | 0   s  0  0    s+t 0    0   0  0   0  0   0  |       |
|0  |              {2} | 0   0  -t -s-t 0   0    0   0  0   0  0   0  |       |
|0  |              {2} | 0   0  0  0    -t  s    0   0  0   0  0   0  |       |
|0  |              {2} | 0   0  0  0    0   0    -t  -t 0   0  0   0  |       |
|0  |              {2} | 0   0  0  0    0   0    s+t 0  0   0  0   0  |       |
|0  |              {2} | 0   0  0  0    0   0    0   s  0   0  0   0  |       |
|-t |              {2} | 0   0  0  0    0   0    0   0  -t  -t 0   0  |       |
|0  |              {2} | 0   0  0  0    0   0    0   0  s+t 0  0   0  |       |
|s  |              {2} | 0   0  0  0    0   0    0   0  0   s  0   0  |       |
|                  {2} | 0   0  0  0    0   0    0   0  0   0  -t  -t |       |
|                  {2} | 0   0  0  0    0   0    0   0  0   0  s+t 0  |       |
|                  {2} | 0   0  0  0    0   0    0   0  0   0  0   s  |       |
|                                                                             |
|         3                                                                   |
+-----------------------------------------------------------------------------+
|i24 : assert isWellDefined D2                                                |
+-----------------------------------------------------------------------------+
|i25 : prune D1                                                               |
|                                                                             |
|o25 = cokernel {2} | -t  -t -t -s+t -t -s-t | <-- cokernel {2} | -t  -t -t   |
|               {2} | s+t 0  t  -t   0  0    |              {2} | s+t 0  t    |
|               {2} | 0   s  0  0    -t 0    |              {2} | 0   s  0    |
|               {2} | 0   0  t  s    0  0    |              {2} | 0   0  t    |
|               {2} | 0   0  0  0    t  s    |              {2} | 0   0  0    |
|                                                           {2} | 0   0  0    |
|      0                                                    {2} | 0   0  0    |
|                                                           {2} | 0   0  0    |
|                                                                             |
|                                                  1                          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o25 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|-s+t -t -s-t 0   0  | <-- cokernel {2} | -t  -t -t -s+t -t -s-t 0   0  0   0 |
|-t   0  0    0   0  |              {2} | s+t 0  t  -t   0  0    0   0  0   0 |
|0    -t 0    0   0  |              {2} | 0   s  0  0    -t 0    0   0  0   0 |
|s    0  0    0   0  |              {2} | 0   0  t  s    0  0    0   0  0   0 |
|0    t  s    0   0  |              {2} | 0   0  0  0    t  s    0   0  0   0 |
|0    0  0    -t  -t |              {2} | 0   0  0  0    0  0    -t  -t 0   0 |
|0    0  0    s+t 0  |              {2} | 0   0  0  0    0  0    s+t 0  0   0 |
|0    0  0    0   s  |              {2} | 0   0  0  0    0  0    0   s  0   0 |
|                                   {2} | 0   0  0  0    0  0    0   0  -t  -t|
|                                   {2} | 0   0  0  0    0  0    0   0  s+t 0 |
|                                   {2} | 0   0  0  0    0  0    0   0  0   s |
|                                                                             |
|                          2                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|| <-- cokernel {2} | -t  -t -t -s+t -t -s-t 0   0  0   0  0   0  |<-- ...    |
||              {2} | s+t 0  t  -t   0  0    0   0  0   0  0   0  |           |
||              {2} | 0   s  0  0    -t 0    0   0  0   0  0   0  |           |
||              {2} | 0   0  t  s    0  0    0   0  0   0  0   0  |           |
||              {2} | 0   0  0  0    t  s    0   0  0   0  0   0  |           |
||              {2} | 0   0  0  0    0  0    -t  -t 0   0  0   0  |           |
||              {2} | 0   0  0  0    0  0    s+t 0  0   0  0   0  |           |
||              {2} | 0   0  0  0    0  0    0   s  0   0  0   0  |           |
||              {2} | 0   0  0  0    0  0    0   0  -t  -t 0   0  |           |
||              {2} | 0   0  0  0    0  0    0   0  s+t 0  0   0  |           |
||              {2} | 0   0  0  0    0  0    0   0  0   s  0   0  |           |
|               {2} | 0   0  0  0    0  0    0   0  0   0  -t  -t |           |
|               {2} | 0   0  0  0    0  0    0   0  0   0  s+t 0  |           |
|               {2} | 0   0  0  0    0  0    0   0  0   0  0   s  |           |
|                                                                             |
|      3                                                                      |
+-----------------------------------------------------------------------------+
|i26 : prune D2                                                               |
|                                                                             |
|o26 = cokernel {2} | -t  -t -t -s+t -t -s-t | <-- cokernel {2} | -t  -t -t   |
|               {2} | s+t 0  t  -t   0  0    |              {2} | s+t 0  t    |
|               {2} | 0   s  0  0    -t 0    |              {2} | 0   s  0    |
|               {2} | 0   0  t  s    0  0    |              {2} | 0   0  t    |
|               {2} | 0   0  0  0    t  s    |              {2} | 0   0  0    |
|                                                           {2} | 0   0  0    |
|      0                                                    {2} | 0   0  0    |
|                                                           {2} | 0   0  0    |
|                                                                             |
|                                                  1                          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o26 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|-s+t -t -s-t 0   0  | <-- cokernel {2} | -t  -t -t -s+t -t -s-t 0   0  0   0 |
|-t   0  0    0   0  |              {2} | s+t 0  t  -t   0  0    0   0  0   0 |
|0    -t 0    0   0  |              {2} | 0   s  0  0    -t 0    0   0  0   0 |
|s    0  0    0   0  |              {2} | 0   0  t  s    0  0    0   0  0   0 |
|0    t  s    0   0  |              {2} | 0   0  0  0    t  s    0   0  0   0 |
|0    0  0    -t  -t |              {2} | 0   0  0  0    0  0    -t  -t 0   0 |
|0    0  0    s+t 0  |              {2} | 0   0  0  0    0  0    s+t 0  0   0 |
|0    0  0    0   s  |              {2} | 0   0  0  0    0  0    0   s  0   0 |
|                                   {2} | 0   0  0  0    0  0    0   0  -t  -t|
|                                   {2} | 0   0  0  0    0  0    0   0  s+t 0 |
|                                   {2} | 0   0  0  0    0  0    0   0  0   s |
|                                                                             |
|                          2                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|| <-- cokernel {2} | -t  -t -t -s+t -t -s-t 0   0  0   0  0   0  |<-- ...    |
||              {2} | s+t 0  t  -t   0  0    0   0  0   0  0   0  |           |
||              {2} | 0   s  0  0    -t 0    0   0  0   0  0   0  |           |
||              {2} | 0   0  t  s    0  0    0   0  0   0  0   0  |           |
||              {2} | 0   0  0  0    t  s    0   0  0   0  0   0  |           |
||              {2} | 0   0  0  0    0  0    -t  -t 0   0  0   0  |           |
||              {2} | 0   0  0  0    0  0    s+t 0  0   0  0   0  |           |
||              {2} | 0   0  0  0    0  0    0   s  0   0  0   0  |           |
||              {2} | 0   0  0  0    0  0    0   0  -t  -t 0   0  |           |
||              {2} | 0   0  0  0    0  0    0   0  s+t 0  0   0  |           |
||              {2} | 0   0  0  0    0  0    0   0  0   s  0   0  |           |
|               {2} | 0   0  0  0    0  0    0   0  0   0  -t  -t |           |
|               {2} | 0   0  0  0    0  0    0   0  0   0  s+t 0  |           |
|               {2} | 0   0  0  0    0  0    0   0  0   0  0   s  |           |
|                                                                             |
|      3                                                                      |
+-----------------------------------------------------------------------------+

When the ring map doesn't preserve homogeneity, the DegreeMap (Macaulay2Doc),
option is needed to determine the degrees of the image
free modules in the complex.

+-----------------------------------------------------------------------------+
|i27 : R = ZZ/101[a..d];                                                      |
+-----------------------------------------------------------------------------+
|i28 : S = ZZ/101[s,t];                                                       |
+-----------------------------------------------------------------------------+
|i29 : f = map(S, R, {s^4, s^3*t, s*t^3, t^4}, DegreeMap => i -> 4*i)         |
|                                                                             |
|                   4   3      3   4                                          |
|o29 = map (S, R, {s , s t, s*t , t })                                        |
|                                                                             |
|o29 : RingMap S <-- R                                                        |
+-----------------------------------------------------------------------------+
|i30 : C = simplicialModule(freeResolution coker vars R, 3, Degeneracy =>     |
|                                                                             |
|       1      5      15      35                                              |
|o30 = R  <-- R  <-- R   <-- R  <-- ...                                       |
|                                                                             |
|      0      1      2       3                                                |
|                                                                             |
|o30 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|true)                                                                        |
+-----------------------------------------------------------------------------+
|i31 : D = f ** C                                                             |
|                                                                             |
|       1      5      15      35                                              |
|o31 = S  <-- S  <-- S   <-- S  <-- ...                                       |
|                                                                             |
|      0      1      2       3                                                |
|                                                                             |
|o31 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i32 : D == f C                                                               |
|                                                                             |
|o32 = true                                                                   |
+-----------------------------------------------------------------------------+
|i33 : assert isWellDefined D                                                 |
+-----------------------------------------------------------------------------+

See also
========

  * RingMap ** SimplicialModuleMap,
    -- tensor a map of simplicial modules along a ring map
  * RingMap SimplicialModule -- apply a ring
    map to a simplicial module

Ways to use this method:
========================

  * "Ring ** SimplicialModule"
  * RingMap ** SimplicialModule, --
    tensor a simplicial module along a ring map
  * "SimplicialModule ** Ring"
  * "SimplicialModule ** RingMap"
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1170:0.

RingMap ** SimplicialModuleMap -- tensor a map of simplicial modules along a ring map
*************************************************************************************

  * Operator: ** (Macaulay2Doc),
  * Usage:
        phi ** f
        tensor(phi, f)
        S ** f
        f ** S
  * Inputs:
      * phi, a ring map (Macaulay2Doc), whose source is a ring
        $R$, and whose target is a ring $S$
      * f, a Map of Simplicial Modules, over the
        ring $R$
  * Outputs:
      * a Map of Simplicial Modules, over the ring
        $S$

Description
===========

These methods implement the base change of rings.  As input, one can either
give a ring map $\phi$, or the ring $S$ (when there is a canonical map from $R$
to $S$).

We illustrate the tensor product of a map of simplicial modules along a ring
map.

+-----------------------------------------------------------------------------+
|i1 : R = QQ[a,b,c,d];                                                        |
+-----------------------------------------------------------------------------+
|i2 : S = QQ[s,t];                                                            |
+-----------------------------------------------------------------------------+
|i3 : phi = map(S, R, {s, s+t, t, s-t})                                       |
|                                                                             |
|o3 = map (S, R, {s, s + t, t, s - t})                                        |
|                                                                             |
|o3 : RingMap S <-- R                                                         |
+-----------------------------------------------------------------------------+
|i4 : I = ideal(a*b, b*c, c*d)                                                |
|                                                                             |
|o4 = ideal (a*b, b*c, c*d)                                                   |
|                                                                             |
|o4 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i5 : J = I + ideal(a^2, b^2, c^2, d^2)                                       |
|                                                                             |
|                            2   2   2   2                                    |
|o5 = ideal (a*b, b*c, c*d, a , b , c , d )                                   |
|                                                                             |
|o5 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i6 : CI = simplicialModule(freeResolution I, 4, Degeneracy => true)          |
|                                                                             |
|      1      4      9      16      25                                        |
|o6 = R  <-- R  <-- R  <-- R   <-- R  <-- ...                                 |
|                                                                             |
|     0      1      2      3       4                                          |
|                                                                             |
|o6 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i7 : CJ = simplicialModule(freeResolution J, Degeneracy => true)             |
|                                                                             |
|      1      8      28      71      150                                      |
|o7 = R  <-- R  <-- R   <-- R   <-- R   <-- ...                               |
|                                                                             |
|     0      1      2       3       4                                         |
|                                                                             |
|o7 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i8 : f = simplicialModule(extend(CJ.complex, CI.complex, map(CJ_0, CI_0, 1)),|
|                                                                             |
|          1             1                                                    |
|o8 = 0 : R  <--------- R  : 0                                                |
|               | 1 |                                                         |
|                                                                             |
|          8                       4                                          |
|     1 : R  <------------------- R  : 1                                      |
|               {0} | 1 0 0 0 |                                               |
|               {2} | 0 0 0 0 |                                               |
|               {2} | 0 1 0 0 |                                               |
|               {2} | 0 0 0 0 |                                               |
|               {2} | 0 0 1 0 |                                               |
|               {2} | 0 0 0 0 |                                               |
|               {2} | 0 0 0 1 |                                               |
|               {2} | 0 0 0 0 |                                               |
|                                                                             |
|          28                                 9                               |
|     2 : R   <----------------------------- R  : 2                           |
|                {0} | 1 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 1 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 1 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 1 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 1 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 1 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 1 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {3} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {3} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {3} | 0 0 0 0 0 0 0 1 0 |                                    |
|                {3} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {3} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {3} | 0 0 0 0 0 0 0 0 1 |                                    |
|                {3} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {3} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {4} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {4} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {4} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {4} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {4} | 0 0 0 0 0 0 0 0 0 |                                    |
|                                                                             |
|          71                                               16                |
|     3 : R   <------------------------------------------- R   : 3            |
|                {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                                                                             |
|          150                                                                |
|     4 : R    <------------------------------------------------------------- |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                                                                             |
|o8 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
| Degeneracy => true)                                                         |
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| 25                                                                          |
|R   : 4                                                                      |
+-----------------------------------------------------------------------------+
|i9 : assert isWellDefined f                                                  |
+-----------------------------------------------------------------------------+
|i10 : g = phi ** f                                                           |
|                                                                             |
|           1             1                                                   |
|o10 = 0 : S  <--------- S  : 0                                               |
|                | 1 |                                                        |
|                                                                             |
|           8                       4                                         |
|      1 : S  <------------------- S  : 1                                     |
|                {0} | 1 0 0 0 |                                              |
|                {2} | 0 0 0 0 |                                              |
|                {2} | 0 1 0 0 |                                              |
|                {2} | 0 0 0 0 |                                              |
|                {2} | 0 0 1 0 |                                              |
|                {2} | 0 0 0 0 |                                              |
|                {2} | 0 0 0 1 |                                              |
|                {2} | 0 0 0 0 |                                              |
|                                                                             |
|           28                                 9                              |
|      2 : S   <----------------------------- S  : 2                          |
|                 {0} | 1 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 1 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 1 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 1 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 1 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 1 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 1 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {3} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {3} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {3} | 0 0 0 0 0 0 0 1 0 |                                   |
|                 {3} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {3} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {3} | 0 0 0 0 0 0 0 0 1 |                                   |
|                 {3} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {3} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {4} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {4} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {4} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {4} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {4} | 0 0 0 0 0 0 0 0 0 |                                   |
|                                                                             |
|           71                                               16               |
|      3 : S   <------------------------------------------- S   : 3           |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                                                                             |
|           150                                                               |
|      4 : S    <-------------------------------------------------------------|
|                  {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                                                                             |
|o10 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
| 25                                                                          |
|S   : 4                                                                      |
+-----------------------------------------------------------------------------+
|i11 : assert isWellDefined g                                                 |
+-----------------------------------------------------------------------------+
|i12 : dd^(source g)                                                          |
|                                                                             |
|                1                               4                            |
|o12 = (0, 0) : S  <--------------------------- S  : (1, 0)                   |
|                     | 1 s2+st st+t2 st-t2 |                                 |
|                                                                             |
|                1                   4                                        |
|      (0, 1) : S  <--------------- S  : (1, 1)                               |
|                     | 1 0 0 0 |                                             |
|                                                                             |
|                4                                                 9          |
|      (1, 0) : S  <--------------------------------------------- S  : (2, 0) |
|                     {0} | 1 s2+st st+t2 st-t2 0 0 0 0  0    |               |
|                     {2} | 0 0     0     0     1 0 0 -t 0    |               |
|                     {2} | 0 0     0     0     0 1 0 s  -s+t |               |
|                     {2} | 0 0     0     0     0 0 1 0  s+t  |               |
|                                                                             |
|                4                                 9                          |
|      (1, 1) : S  <----------------------------- S  : (2, 1)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 1 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 1 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 1 0 0 |                               |
|                                                                             |
|                4                                 9                          |
|      (1, 2) : S  <----------------------------- S  : (2, 2)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                               |
|                                                                             |
|                9                                                            |
|      (2, 0) : S  <----------------------------------------------------------|
|                     {0} | 1 s2+st st+t2 st-t2 0 0 0 0 0 0 0  0    0  0    0 |
|                     {2} | 0 0     0     0     1 0 0 0 0 0 -t 0    0  0    0 |
|                     {2} | 0 0     0     0     0 1 0 0 0 0 s  -s+t 0  0    0 |
|                     {2} | 0 0     0     0     0 0 1 0 0 0 0  s+t  0  0    0 |
|                     {2} | 0 0     0     0     0 0 0 1 0 0 0  0    -t 0    0 |
|                     {2} | 0 0     0     0     0 0 0 0 1 0 0  0    s  -s+t 0 |
|                     {2} | 0 0     0     0     0 0 0 0 0 1 0  0    0  s+t  0 |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0  0    0  0    1 |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0  0    0  0    0 |
|                                                                             |
|                9                                               16           |
|      (2, 1) : S  <------------------------------------------- S   : (3, 1)  |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |                 |
|                                                                             |
|                9                                               16           |
|      (2, 2) : S  <------------------------------------------- S   : (3, 2)  |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |                 |
|                                                                             |
|                9                                               16           |
|      (2, 3) : S  <------------------------------------------- S   : (3, 3)  |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                 |
|                                                                             |
|                16                                                           |
|      (3, 0) : S   <---------------------------------------------------------|
|                      {0} | 1 s2+st st+t2 st-t2 0 0 0 0 0 0 0 0 0 0  0    0  |
|                      {2} | 0 0     0     0     1 0 0 0 0 0 0 0 0 -t 0    0  |
|                      {2} | 0 0     0     0     0 1 0 0 0 0 0 0 0 s  -s+t 0  |
|                      {2} | 0 0     0     0     0 0 1 0 0 0 0 0 0 0  s+t  0  |
|                      {2} | 0 0     0     0     0 0 0 1 0 0 0 0 0 0  0    -t |
|                      {2} | 0 0     0     0     0 0 0 0 1 0 0 0 0 0  0    s  |
|                      {2} | 0 0     0     0     0 0 0 0 0 1 0 0 0 0  0    0  |
|                      {2} | 0 0     0     0     0 0 0 0 0 0 1 0 0 0  0    0  |
|                      {2} | 0 0     0     0     0 0 0 0 0 0 0 1 0 0  0    0  |
|                      {2} | 0 0     0     0     0 0 0 0 0 0 0 0 1 0  0    0  |
|                      {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0    0  |
|                      {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0    0  |
|                      {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0    0  |
|                      {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0    0  |
|                      {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0    0  |
|                      {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0    0  |
|                                                                             |
|                16                                                           |
|      (3, 1) : S   <---------------------------------------------------------|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|                                                                             |
|                16                                                           |
|      (3, 2) : S   <---------------------------------------------------------|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1|
|                                                                             |
|                16                                                           |
|      (3, 3) : S   <---------------------------------------------------------|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0|
|                                                                             |
|                16                                                           |
|      (3, 4) : S   <---------------------------------------------------------|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|                                                                             |
|o12 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|       16                                                                    |
|----- S   : (3, 0)                                                           |
|0 |                                                                          |
|0 |                                                                          |
|0 |                                                                          |
|0 |                                                                          |
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|0 |                                                                          |
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|0 |                                                                          |
|1 |                                                                          |
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|                                                                             |
|                              25                                             |
|---------------------------- S   : (4, 0)                                    |
|0    0  0    0 0 0 0 0 0 |                                                   |
|0    0  0    0 0 0 0 0 0 |                                                   |
|0    0  0    0 0 0 0 0 0 |                                                   |
|0    0  0    0 0 0 0 0 0 |                                                   |
|0    0  0    0 0 0 0 0 0 |                                                   |
|-s+t 0  0    0 0 0 0 0 0 |                                                   |
|s+t  0  0    0 0 0 0 0 0 |                                                   |
|0    -t 0    0 0 0 0 0 0 |                                                   |
|0    s  -s+t 0 0 0 0 0 0 |                                                   |
|0    0  s+t  0 0 0 0 0 0 |                                                   |
|0    0  0    1 0 0 0 0 0 |                                                   |
|0    0  0    0 1 0 0 0 0 |                                                   |
|0    0  0    0 0 1 0 0 0 |                                                   |
|0    0  0    0 0 0 1 0 0 |                                                   |
|0    0  0    0 0 0 0 1 0 |                                                   |
|0    0  0    0 0 0 0 0 1 |                                                   |
|                                                                             |
|      25                                                                     |
|---- S   : (4, 1)                                                            |
| |                                                                           |
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|                                                                             |
|      25                                                                     |
|---- S   : (4, 2)                                                            |
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|                                                                             |
|      25                                                                     |
|---- S   : (4, 3)                                                            |
| |                                                                           |
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|                                                                             |
|      25                                                                     |
|---- S   : (4, 4)                                                            |
| |                                                                           |
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| |                                                                           |
+-----------------------------------------------------------------------------+
|i13 : ss^(source g)                                                          |
|                                                                             |
|                4                 1                                          |
|o13 = (1, 0) : S  <------------- S  : (0, 0)                                 |
|                     {0} | 1 |                                               |
|                     {2} | 0 |                                               |
|                     {2} | 0 |                                               |
|                     {2} | 0 |                                               |
|                                                                             |
|                9                       4                                    |
|      (2, 0) : S  <------------------- S  : (1, 0)                           |
|                     {0} | 1 0 0 0 |                                         |
|                     {2} | 0 0 0 0 |                                         |
|                     {2} | 0 0 0 0 |                                         |
|                     {2} | 0 0 0 0 |                                         |
|                     {2} | 0 1 0 0 |                                         |
|                     {2} | 0 0 1 0 |                                         |
|                     {2} | 0 0 0 1 |                                         |
|                     {3} | 0 0 0 0 |                                         |
|                     {3} | 0 0 0 0 |                                         |
|                                                                             |
|                9                       4                                    |
|      (2, 1) : S  <------------------- S  : (1, 1)                           |
|                     {0} | 1 0 0 0 |                                         |
|                     {2} | 0 1 0 0 |                                         |
|                     {2} | 0 0 1 0 |                                         |
|                     {2} | 0 0 0 1 |                                         |
|                     {2} | 0 0 0 0 |                                         |
|                     {2} | 0 0 0 0 |                                         |
|                     {2} | 0 0 0 0 |                                         |
|                     {3} | 0 0 0 0 |                                         |
|                     {3} | 0 0 0 0 |                                         |
|                                                                             |
|                16                                 9                         |
|      (3, 0) : S   <----------------------------- S  : (2, 0)                |
|                      {0} | 1 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 1 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 1 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 1 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 1 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 1 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 1 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 1 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 1 |                              |
|                                                                             |
|                16                                 9                         |
|      (3, 1) : S   <----------------------------- S  : (2, 1)                |
|                      {0} | 1 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 1 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 1 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 1 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 1 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 1 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 1 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 1 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 1 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                                                                             |
|                16                                 9                         |
|      (3, 2) : S   <----------------------------- S  : (2, 2)                |
|                      {0} | 1 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 1 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 1 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 1 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 1 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 1 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 1 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 1 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 1 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                                                                             |
|                25                                               16          |
|      (4, 0) : S   <------------------------------------------- S   : (3, 0) |
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                |
|                                                                             |
|                25                                               16          |
|      (4, 1) : S   <------------------------------------------- S   : (3, 1) |
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                |
|                                                                             |
|                25                                               16          |
|      (4, 2) : S   <------------------------------------------- S   : (3, 2) |
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                                                                             |
|                25                                               16          |
|      (4, 3) : S   <------------------------------------------- S   : (3, 3) |
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                                                                             |
|o13 : SimplicialModuleMap                                                    |
+-----------------------------------------------------------------------------+
|i14 : dd^(target g);                                                         |
+-----------------------------------------------------------------------------+
|i15 : simplicialModule prune HH normalize g                                  |
|                                                                             |
|o15 = 0 : cokernel | t2 st s2 | <--------- cokernel | t2 st s2 | : 0         |
|                                   | 1 |                                     |
|                                                                             |
|      1 : cokernel {0} | t2 st s2 0 0 0   0  0   0   0  0  | <-------------  |
|                   {2} | 0  0  0  t s 0   0  0   0   0  0  |    {0} | 1 |    |
|                   {2} | 0  0  0  0 0 s+t 2t -2t -4t t2 0  |    {2} | 0 |    |
|                   {2} | 0  0  0  0 0 0   t  s   0   0  0  |    {2} | 0 |    |
|                   {2} | 0  0  0  0 0 0   0  0   t   0  s2 |    {2} | 0 |    |
|                                                                {2} | 0 |    |
|                                                                             |
|      2 : cokernel {0} | t2 st s2 0 0 0   0  0   0   0  0  0 0 0   0  0   0  |
|                   {2} | 0  0  0  t s 0   0  0   0   0  0  0 0 0   0  0   0  |
|                   {2} | 0  0  0  0 0 s+t 2t -2t -4t t2 0  0 0 0   0  0   0  |
|                   {2} | 0  0  0  0 0 0   t  s   0   0  0  0 0 0   0  0   0  |
|                   {2} | 0  0  0  0 0 0   0  0   t   0  s2 0 0 0   0  0   0  |
|                   {2} | 0  0  0  0 0 0   0  0   0   0  0  t s 0   0  0   0  |
|                   {2} | 0  0  0  0 0 0   0  0   0   0  0  0 0 s+t 2t -2t -  |
|                   {2} | 0  0  0  0 0 0   0  0   0   0  0  0 0 0   t  s   0  |
|                   {2} | 0  0  0  0 0 0   0  0   0   0  0  0 0 0   0  0   t  |
|                   {4} | 0  0  0  0 0 0   0  0   0   0  0  0 0 0   0  0   0  |
|                   {4} | 0  0  0  0 0 0   0  0   0   0  0  0 0 0   0  0   0  |
|                   {4} | 0  0  0  0 0 0   0  0   0   0  0  0 0 0   0  0   0  |
|                                                                             |
|                                                                             |
|o15 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
| cokernel | t2 st s2 | : 1                                                   |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|   0  0  0 0 0 0 0 0 | <------------- cokernel | t2 st s2 | : 2              |
|   0  0  0 0 0 0 0 0 |    {0} | 1 |                                          |
|   0  0  0 0 0 0 0 0 |    {2} | 0 |                                          |
|   0  0  0 0 0 0 0 0 |    {2} | 0 |                                          |
|   0  0  0 0 0 0 0 0 |    {2} | 0 |                                          |
|   0  0  0 0 0 0 0 0 |    {2} | 0 |                                          |
|4t t2 0  0 0 0 0 0 0 |    {2} | 0 |                                          |
|   0  0  0 0 0 0 0 0 |    {2} | 0 |                                          |
|   0  s2 0 0 0 0 0 0 |    {2} | 0 |                                          |
|   0  0  t s 0 0 0 0 |    {2} | 0 |                                          |
|   0  0  0 0 t s 0 0 |    {4} | 0 |                                          |
|   0  0  0 0 0 0 t s |    {4} | 0 |                                          |
|                          {4} | 0 |                                          |
+-----------------------------------------------------------------------------+
|i16 : isSimplicialMorphism oo                                                |
|                                                                             |
|o16 = true                                                                   |
+-----------------------------------------------------------------------------+

See also
========

  * RingMap ** SimplicialModule, --
    tensor a simplicial module along a ring map
  * RingMap SimplicialModuleMap -- apply a
    ring map to a map of simplicial modules

Ways to use this method:
========================

  * "Ring ** SimplicialModuleMap"
  * RingMap ** SimplicialModuleMap,
    -- tensor a map of simplicial modules along a ring map
  * "SimplicialModuleMap ** Ring"
  * "SimplicialModuleMap ** RingMap"
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2455:0.

RingMap SimplicialModule -- apply a ring map to a simplicial module
*******************************************************************

  * Operator: SPACE (Macaulay2Doc),
  * Usage:
        phi C
  * Inputs:
      * phi, a ring map (Macaulay2Doc), whose source is a ring
        $R$, and whose target is a ring $S$
      * C, a Simplicial Module, over the ring $R$
  * Outputs:
      * a Simplicial Module, over the ring $S$

Description
===========

We illustrate the image of a simplicial module under a ring map.

+-----------------------------------------------------------------------------+
|i1 : R = QQ[x,y,z]                                                           |
|                                                                             |
|o1 = R                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : S = QQ[s,t]                                                             |
|                                                                             |
|o2 = S                                                                       |
|                                                                             |
|o2 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i3 : phi = map(S, R, {s, s+t, t})                                            |
|                                                                             |
|o3 = map (S, R, {s, s + t, t})                                               |
|                                                                             |
|o3 : RingMap S <-- R                                                         |
+-----------------------------------------------------------------------------+
|i4 : I = ideal(x^3, x^2*y, x*y^4, y*z^5)                                     |
|                                                                             |
|             3   2      4     5                                              |
|o4 = ideal (x , x y, x*y , y*z )                                             |
|                                                                             |
|o4 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i5 : C = simplicialModule(freeResolution I, 3, Degeneracy => true)           |
|                                                                             |
|      1      5      13      26                                               |
|o5 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o5 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i6 : D = phi C                                                               |
|                                                                             |
|      1      5      13      26                                               |
|o6 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o6 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i7 : isWellDefined D                                                         |
|                                                                             |
|o7 = true                                                                    |
+-----------------------------------------------------------------------------+
|i8 : dd^D                                                                    |
|                                                                             |
|               1                                                      5      |
|o8 = (0, 0) : S  <-------------------------------------------------- S  : (1,|
|                    | 1 s3 s3+s2t s5+4s4t+6s3t2+4s2t3+st4 st5+t6 |           |
|                                                                             |
|               1                     5                                       |
|     (0, 1) : S  <----------------- S  : (1, 1)                              |
|                    | 1 0 0 0 0 |                                            |
|                                                                             |
|               5                                                             |
|     (1, 0) : S  <-----------------------------------------------------------|
|                    {0} | 1 s3 s3+s2t s5+4s4t+6s3t2+4s2t3+st4 st5+t6 0 0 0 0 |
|                    {3} | 0 0  0      0                       0      1 0 0 0 |
|                    {3} | 0 0  0      0                       0      0 1 0 0 |
|                    {5} | 0 0  0      0                       0      0 0 1 0 |
|                    {6} | 0 0  0      0                       0      0 0 0 1 |
|                                                                             |
|               5                                         13                  |
|     (1, 1) : S  <------------------------------------- S   : (2, 1)         |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 |                        |
|                    {3} | 0 1 0 0 0 1 0 0 0 0 0 0 0 |                        |
|                    {3} | 0 0 1 0 0 0 1 0 0 0 0 0 0 |                        |
|                    {5} | 0 0 0 1 0 0 0 1 0 0 0 0 0 |                        |
|                    {6} | 0 0 0 0 1 0 0 0 1 0 0 0 0 |                        |
|                                                                             |
|               5                                         13                  |
|     (1, 2) : S  <------------------------------------- S   : (2, 2)         |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 |                        |
|                    {3} | 0 1 0 0 0 0 0 0 0 0 0 0 0 |                        |
|                    {3} | 0 0 1 0 0 0 0 0 0 0 0 0 0 |                        |
|                    {5} | 0 0 0 1 0 0 0 0 0 0 0 0 0 |                        |
|                    {6} | 0 0 0 0 1 0 0 0 0 0 0 0 0 |                        |
|                                                                             |
|               13                                                            |
|     (2, 0) : S   <----------------------------------------------------------|
|                     {0}  | 1 s3 s3+s2t s5+4s4t+6s3t2+4s2t3+st4 st5+t6 0 0 0 |
|                     {3}  | 0 0  0      0                       0      1 0 0 |
|                     {3}  | 0 0  0      0                       0      0 1 0 |
|                     {5}  | 0 0  0      0                       0      0 0 1 |
|                     {6}  | 0 0  0      0                       0      0 0 0 |
|                     {3}  | 0 0  0      0                       0      0 0 0 |
|                     {3}  | 0 0  0      0                       0      0 0 0 |
|                     {5}  | 0 0  0      0                       0      0 0 0 |
|                     {6}  | 0 0  0      0                       0      0 0 0 |
|                     {4}  | 0 0  0      0                       0      0 0 0 |
|                     {6}  | 0 0  0      0                       0      0 0 0 |
|                     {8}  | 0 0  0      0                       0      0 0 0 |
|                     {10} | 0 0  0      0                       0      0 0 0 |
|                                                                             |
|               13                                                            |
|     (2, 1) : S   <----------------------------------------------------------|
|                     {0}  | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {5}  | 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {5}  | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {4}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0|
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0|
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0|
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1|
|                                                                             |
|               13                                                            |
|     (2, 2) : S   <----------------------------------------------------------|
|                     {0}  | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {5}  | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {5}  | 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {4}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0|
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0|
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0|
|                                                                             |
|               13                                                            |
|     (2, 3) : S   <----------------------------------------------------------|
|                     {0}  | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {5}  | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {3}  | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {5}  | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                     {4}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|                     {6}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|                     {8}  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|                     {10} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|                                                                             |
|o8 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
| 0)                                                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                 13                          |
|----------------------------------------------- S   : (2, 0)                 |
|0    0                0   0                 |                                |
|-s-t 0                0   0                 |                                |
|s    -s3-3s2t-3st2-t3 -t5 0                 |                                |
|0    s                0   -t5               |                                |
|0    0                s2  s4+3s3t+3s2t2+st3 |                                |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0    0                0   0                 0    0                0|
|0 0 0 0 0 -s-t 0                0   0                 0    0                0|
|0 0 0 0 0 s    -s3-3s2t-3st2-t3 -t5 0                 0    0                0|
|0 0 0 0 0 0    s                0   -t5               0    0                0|
|1 0 0 0 0 0    0                s2  s4+3s3t+3s2t2+st3 0    0                0|
|0 1 0 0 0 0    0                0   0                 -s-t 0                0|
|0 0 1 0 0 0    0                0   0                 s    -s3-3s2t-3st2-t3 -|
|0 0 0 1 0 0    0                0   0                 0    s                0|
|0 0 0 0 1 0    0                0   0                 0    0                s|
|0 0 0 0 0 0    0                0   0                 0    0                0|
|0 0 0 0 0 0    0                0   0                 0    0                0|
|0 0 0 0 0 0    0                0   0                 0    0                0|
|0 0 0 0 0 0    0                0   0                 0    0                0|
|                                                                             |
|        26                                                                   |
|------ S   : (3, 1)                                                          |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
|                                                                             |
|        26                                                                   |
|------ S   : (3, 2)                                                          |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
|                                                                             |
|        26                                                                   |
|------ S   : (3, 3)                                                          |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
| 0 |                                                                         |
|-----------------------------------------------------------------------------|
|                                                   26                        |
|------------------------------------------------- S   : (3, 0)               |
|   0                 0 0 0 0 0                |                              |
|   0                 0 0 0 0 0                |                              |
|   0                 0 0 0 0 0                |                              |
|   0                 0 0 0 0 0                |                              |
|   0                 0 0 0 0 0                |                              |
|   0                 0 0 0 0 0                |                              |
|t5 0                 0 0 0 0 0                |                              |
|   -t5               0 0 0 0 0                |                              |
|2  s4+3s3t+3s2t2+st3 0 0 0 0 0                |                              |
|   0                 1 0 0 0 0                |                              |
|   0                 0 1 0 0 t5               |                              |
|   0                 0 0 1 0 -s3-3s2t-3st2-t3 |                              |
|   0                 0 0 0 1 s                |                              |
+-----------------------------------------------------------------------------+

When the ring map doesn't preserve homogeneity, the DegreeMap (Macaulay2Doc),
option is needed to determine the degrees of the image
free modules in the simplicial module.

+-----------------------------------------------------------------------------+
|i9 : R = ZZ/101[a..d]                                                        |
|                                                                             |
|o9 = R                                                                       |
|                                                                             |
|o9 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i10 : S = ZZ/101[s,t]                                                        |
|                                                                             |
|o10 = S                                                                      |
|                                                                             |
|o10 : PolynomialRing                                                         |
+-----------------------------------------------------------------------------+
|i11 : phi = map(S, R, {s^4, s^3*t, s*t^3, t^4}, DegreeMap => i -> 4*i)       |
|                                                                             |
|                   4   3      3   4                                          |
|o11 = map (S, R, {s , s t, s*t , t })                                        |
|                                                                             |
|o11 : RingMap S <-- R                                                        |
+-----------------------------------------------------------------------------+
|i12 : C = simplicialModule(freeResolution coker vars R, 4, Degeneracy =>     |
|                                                                             |
|       1      5      15      35      70                                      |
|o12 = R  <-- R  <-- R   <-- R   <-- R  <-- ...                               |
|                                                                             |
|      0      1      2       3       4                                        |
|                                                                             |
|o12 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|true)                                                                        |
+-----------------------------------------------------------------------------+
|i13 : D = phi C                                                              |
|                                                                             |
|       1      5      15      35      70                                      |
|o13 = S  <-- S  <-- S   <-- S   <-- S  <-- ...                               |
|                                                                             |
|      0      1      2       3       4                                        |
|                                                                             |
|o13 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i14 : assert isWellDefined D                                                 |
+-----------------------------------------------------------------------------+

Caveat
======

Every term in the simplicial module must be free or a submodule of a free
module. Otherwise, use tensor(RingMap,SimplicialModule).

See also
========

  * RingMap SimplicialModuleMap -- apply a
    ring map to a map of simplicial modules
  * RingMap ** SimplicialModule, --
    tensor a simplicial module along a ring map

Ways to use this method:
========================

  * RingMap SimplicialModule -- apply a ring
    map to a simplicial module
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1082:0.

RingMap SimplicialModuleMap -- apply a ring map to a map of simplicial modules
******************************************************************************

  * Operator: SPACE (Macaulay2Doc),
  * Usage:
        phi f
  * Inputs:
      * phi, a ring map (Macaulay2Doc), whose source is a ring
        $R$, and whose target is a ring $S$
      * f, a Map of Simplicial Modules, over the
        ring $R$
  * Outputs:
      * a Map of Simplicial Modules, over the ring
        $S$

Description
===========

We illustrate the image of a simplicial module map along a ring map.

+-----------------------------------------------------------------------------+
|i1 : R = QQ[a,b,c,d];                                                        |
+-----------------------------------------------------------------------------+
|i2 : S = QQ[s,t];                                                            |
+-----------------------------------------------------------------------------+
|i3 : phi = map(S, R, {s, s+t, t, s-t})                                       |
|                                                                             |
|o3 = map (S, R, {s, s + t, t, s - t})                                        |
|                                                                             |
|o3 : RingMap S <-- R                                                         |
+-----------------------------------------------------------------------------+
|i4 : I = ideal(a*b, b*c, c*d)                                                |
|                                                                             |
|o4 = ideal (a*b, b*c, c*d)                                                   |
|                                                                             |
|o4 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i5 : J = I + ideal(a^2, b^2, c^2, d^2)                                       |
|                                                                             |
|                            2   2   2   2                                    |
|o5 = ideal (a*b, b*c, c*d, a , b , c , d )                                   |
|                                                                             |
|o5 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i6 : CI = simplicialModule(freeResolution I, 4, Degeneracy => true)          |
|                                                                             |
|      1      4      9      16      25                                        |
|o6 = R  <-- R  <-- R  <-- R   <-- R  <-- ...                                 |
|                                                                             |
|     0      1      2      3       4                                          |
|                                                                             |
|o6 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i7 : CJ = simplicialModule(freeResolution J, Degeneracy => true)             |
|                                                                             |
|      1      8      28      71      150                                      |
|o7 = R  <-- R  <-- R   <-- R   <-- R   <-- ...                               |
|                                                                             |
|     0      1      2       3       4                                         |
|                                                                             |
|o7 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i8 : f = simplicialModule(extend(CJ.complex, CI.complex, map(CJ_0, CI_0, 1)),|
|                                                                             |
|          1             1                                                    |
|o8 = 0 : R  <--------- R  : 0                                                |
|               | 1 |                                                         |
|                                                                             |
|          8                       4                                          |
|     1 : R  <------------------- R  : 1                                      |
|               {0} | 1 0 0 0 |                                               |
|               {2} | 0 0 0 0 |                                               |
|               {2} | 0 1 0 0 |                                               |
|               {2} | 0 0 0 0 |                                               |
|               {2} | 0 0 1 0 |                                               |
|               {2} | 0 0 0 0 |                                               |
|               {2} | 0 0 0 1 |                                               |
|               {2} | 0 0 0 0 |                                               |
|                                                                             |
|          28                                 9                               |
|     2 : R   <----------------------------- R  : 2                           |
|                {0} | 1 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 1 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 1 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 1 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 1 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 1 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 1 0 0 |                                    |
|                {2} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {3} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {3} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {3} | 0 0 0 0 0 0 0 1 0 |                                    |
|                {3} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {3} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {3} | 0 0 0 0 0 0 0 0 1 |                                    |
|                {3} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {3} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {4} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {4} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {4} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {4} | 0 0 0 0 0 0 0 0 0 |                                    |
|                {4} | 0 0 0 0 0 0 0 0 0 |                                    |
|                                                                             |
|          71                                               16                |
|     3 : R   <------------------------------------------- R   : 3            |
|                {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                      |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                      |
|                                                                             |
|          150                                                                |
|     4 : R    <------------------------------------------------------------- |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                 {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|                                                                             |
|o8 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
| Degeneracy => true)                                                         |
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| 25                                                                          |
|R   : 4                                                                      |
+-----------------------------------------------------------------------------+
|i9 : assert isWellDefined f                                                  |
+-----------------------------------------------------------------------------+
|i10 : g = phi f                                                              |
|                                                                             |
|           1             1                                                   |
|o10 = 0 : S  <--------- S  : 0                                               |
|                | 1 |                                                        |
|                                                                             |
|           8                       4                                         |
|      1 : S  <------------------- S  : 1                                     |
|                {0} | 1 0 0 0 |                                              |
|                {2} | 0 0 0 0 |                                              |
|                {2} | 0 1 0 0 |                                              |
|                {2} | 0 0 0 0 |                                              |
|                {2} | 0 0 1 0 |                                              |
|                {2} | 0 0 0 0 |                                              |
|                {2} | 0 0 0 1 |                                              |
|                {2} | 0 0 0 0 |                                              |
|                                                                             |
|           28                                 9                              |
|      2 : S   <----------------------------- S  : 2                          |
|                 {0} | 1 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 1 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 1 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 1 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 1 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 1 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 1 0 0 |                                   |
|                 {2} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {3} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {3} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {3} | 0 0 0 0 0 0 0 1 0 |                                   |
|                 {3} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {3} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {3} | 0 0 0 0 0 0 0 0 1 |                                   |
|                 {3} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {3} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {4} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {4} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {4} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {4} | 0 0 0 0 0 0 0 0 0 |                                   |
|                 {4} | 0 0 0 0 0 0 0 0 0 |                                   |
|                                                                             |
|           71                                               16               |
|      3 : S   <------------------------------------------- S   : 3           |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                     |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                     |
|                                                                             |
|           150                                                               |
|      4 : S    <-------------------------------------------------------------|
|                  {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
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|                  {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
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|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
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|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
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|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                  {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  |
|                                                                             |
|o10 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
| 25                                                                          |
|S   : 4                                                                      |
+-----------------------------------------------------------------------------+
|i11 : assert isWellDefined g                                                 |
+-----------------------------------------------------------------------------+
|i12 : dd^(source g)                                                          |
|                                                                             |
|                1                               4                            |
|o12 = (0, 0) : S  <--------------------------- S  : (1, 0)                   |
|                     | 1 s2+st st+t2 st-t2 |                                 |
|                                                                             |
|                1                   4                                        |
|      (0, 1) : S  <--------------- S  : (1, 1)                               |
|                     | 1 0 0 0 |                                             |
|                                                                             |
|                4                                                 9          |
|      (1, 0) : S  <--------------------------------------------- S  : (2, 0) |
|                     {0} | 1 s2+st st+t2 st-t2 0 0 0 0  0    |               |
|                     {2} | 0 0     0     0     1 0 0 -t 0    |               |
|                     {2} | 0 0     0     0     0 1 0 s  -s+t |               |
|                     {2} | 0 0     0     0     0 0 1 0  s+t  |               |
|                                                                             |
|                4                                 9                          |
|      (1, 1) : S  <----------------------------- S  : (2, 1)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 1 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 1 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 1 0 0 |                               |
|                                                                             |
|                4                                 9                          |
|      (1, 2) : S  <----------------------------- S  : (2, 2)                 |
|                     {0} | 1 0 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 1 0 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 1 0 0 0 0 0 0 |                               |
|                     {2} | 0 0 0 1 0 0 0 0 0 |                               |
|                                                                             |
|                9                                                            |
|      (2, 0) : S  <----------------------------------------------------------|
|                     {0} | 1 s2+st st+t2 st-t2 0 0 0 0 0 0 0  0    0  0    0 |
|                     {2} | 0 0     0     0     1 0 0 0 0 0 -t 0    0  0    0 |
|                     {2} | 0 0     0     0     0 1 0 0 0 0 s  -s+t 0  0    0 |
|                     {2} | 0 0     0     0     0 0 1 0 0 0 0  s+t  0  0    0 |
|                     {2} | 0 0     0     0     0 0 0 1 0 0 0  0    -t 0    0 |
|                     {2} | 0 0     0     0     0 0 0 0 1 0 0  0    s  -s+t 0 |
|                     {2} | 0 0     0     0     0 0 0 0 0 1 0  0    0  s+t  0 |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0  0    0  0    1 |
|                     {3} | 0 0     0     0     0 0 0 0 0 0 0  0    0  0    0 |
|                                                                             |
|                9                                               16           |
|      (2, 1) : S  <------------------------------------------- S   : (3, 1)  |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |                 |
|                                                                             |
|                9                                               16           |
|      (2, 2) : S  <------------------------------------------- S   : (3, 2)  |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |                 |
|                                                                             |
|                9                                               16           |
|      (2, 3) : S  <------------------------------------------- S   : (3, 3)  |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                 |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                 |
|                                                                             |
|                16                                                           |
|      (3, 0) : S   <---------------------------------------------------------|
|                      {0} | 1 s2+st st+t2 st-t2 0 0 0 0 0 0 0 0 0 0  0    0  |
|                      {2} | 0 0     0     0     1 0 0 0 0 0 0 0 0 -t 0    0  |
|                      {2} | 0 0     0     0     0 1 0 0 0 0 0 0 0 s  -s+t 0  |
|                      {2} | 0 0     0     0     0 0 1 0 0 0 0 0 0 0  s+t  0  |
|                      {2} | 0 0     0     0     0 0 0 1 0 0 0 0 0 0  0    -t |
|                      {2} | 0 0     0     0     0 0 0 0 1 0 0 0 0 0  0    s  |
|                      {2} | 0 0     0     0     0 0 0 0 0 1 0 0 0 0  0    0  |
|                      {2} | 0 0     0     0     0 0 0 0 0 0 1 0 0 0  0    0  |
|                      {2} | 0 0     0     0     0 0 0 0 0 0 0 1 0 0  0    0  |
|                      {2} | 0 0     0     0     0 0 0 0 0 0 0 0 1 0  0    0  |
|                      {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0    0  |
|                      {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0    0  |
|                      {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0    0  |
|                      {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0    0  |
|                      {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0    0  |
|                      {3} | 0 0     0     0     0 0 0 0 0 0 0 0 0 0  0    0  |
|                                                                             |
|                16                                                           |
|      (3, 1) : S   <---------------------------------------------------------|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|                                                                             |
|                16                                                           |
|      (3, 2) : S   <---------------------------------------------------------|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1|
|                                                                             |
|                16                                                           |
|      (3, 3) : S   <---------------------------------------------------------|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0|
|                                                                             |
|                16                                                           |
|      (3, 4) : S   <---------------------------------------------------------|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|                                                                             |
|o12 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|       16                                                                    |
|----- S   : (3, 0)                                                           |
|0 |                                                                          |
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|1 |                                                                          |
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|                              25                                             |
|---------------------------- S   : (4, 0)                                    |
|0    0  0    0 0 0 0 0 0 |                                                   |
|0    0  0    0 0 0 0 0 0 |                                                   |
|0    0  0    0 0 0 0 0 0 |                                                   |
|0    0  0    0 0 0 0 0 0 |                                                   |
|0    0  0    0 0 0 0 0 0 |                                                   |
|-s+t 0  0    0 0 0 0 0 0 |                                                   |
|s+t  0  0    0 0 0 0 0 0 |                                                   |
|0    -t 0    0 0 0 0 0 0 |                                                   |
|0    s  -s+t 0 0 0 0 0 0 |                                                   |
|0    0  s+t  0 0 0 0 0 0 |                                                   |
|0    0  0    1 0 0 0 0 0 |                                                   |
|0    0  0    0 1 0 0 0 0 |                                                   |
|0    0  0    0 0 1 0 0 0 |                                                   |
|0    0  0    0 0 0 1 0 0 |                                                   |
|0    0  0    0 0 0 0 1 0 |                                                   |
|0    0  0    0 0 0 0 0 1 |                                                   |
|                                                                             |
|      25                                                                     |
|---- S   : (4, 1)                                                            |
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|      25                                                                     |
|---- S   : (4, 2)                                                            |
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|      25                                                                     |
|---- S   : (4, 3)                                                            |
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|      25                                                                     |
|---- S   : (4, 4)                                                            |
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+-----------------------------------------------------------------------------+
|i13 : ss^(source g)                                                          |
|                                                                             |
|                4                 1                                          |
|o13 = (1, 0) : S  <------------- S  : (0, 0)                                 |
|                     {0} | 1 |                                               |
|                     {2} | 0 |                                               |
|                     {2} | 0 |                                               |
|                     {2} | 0 |                                               |
|                                                                             |
|                9                       4                                    |
|      (2, 0) : S  <------------------- S  : (1, 0)                           |
|                     {0} | 1 0 0 0 |                                         |
|                     {2} | 0 0 0 0 |                                         |
|                     {2} | 0 0 0 0 |                                         |
|                     {2} | 0 0 0 0 |                                         |
|                     {2} | 0 1 0 0 |                                         |
|                     {2} | 0 0 1 0 |                                         |
|                     {2} | 0 0 0 1 |                                         |
|                     {3} | 0 0 0 0 |                                         |
|                     {3} | 0 0 0 0 |                                         |
|                                                                             |
|                9                       4                                    |
|      (2, 1) : S  <------------------- S  : (1, 1)                           |
|                     {0} | 1 0 0 0 |                                         |
|                     {2} | 0 1 0 0 |                                         |
|                     {2} | 0 0 1 0 |                                         |
|                     {2} | 0 0 0 1 |                                         |
|                     {2} | 0 0 0 0 |                                         |
|                     {2} | 0 0 0 0 |                                         |
|                     {2} | 0 0 0 0 |                                         |
|                     {3} | 0 0 0 0 |                                         |
|                     {3} | 0 0 0 0 |                                         |
|                                                                             |
|                16                                 9                         |
|      (3, 0) : S   <----------------------------- S  : (2, 0)                |
|                      {0} | 1 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 1 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 1 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 1 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 1 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 1 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 1 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 1 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 1 |                              |
|                                                                             |
|                16                                 9                         |
|      (3, 1) : S   <----------------------------- S  : (2, 1)                |
|                      {0} | 1 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 1 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 1 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 1 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 1 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 1 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 1 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 1 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 1 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                                                                             |
|                16                                 9                         |
|      (3, 2) : S   <----------------------------- S  : (2, 2)                |
|                      {0} | 1 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 1 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 1 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 1 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 1 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 1 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 1 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {2} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 1 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 1 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                      {3} | 0 0 0 0 0 0 0 0 0 |                              |
|                                                                             |
|                25                                               16          |
|      (4, 0) : S   <------------------------------------------- S   : (3, 0) |
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                |
|                                                                             |
|                25                                               16          |
|      (4, 1) : S   <------------------------------------------- S   : (3, 1) |
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                |
|                                                                             |
|                25                                               16          |
|      (4, 2) : S   <------------------------------------------- S   : (3, 2) |
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                                                                             |
|                25                                               16          |
|      (4, 3) : S   <------------------------------------------- S   : (3, 3) |
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                                                                             |
|o13 : SimplicialModuleMap                                                    |
+-----------------------------------------------------------------------------+
|i14 : dd^(target g);                                                         |
+-----------------------------------------------------------------------------+
|i15 : prune HH normalize g                                                   |
|                                                                             |
|o15 = 0 : cokernel | t2 st s2 | <--------- cokernel | t2 st s2 | : 0         |
|                                   | 1 |                                     |
|                                                                             |
|o15 : ComplexMap                                                             |
+-----------------------------------------------------------------------------+

See also
========

  * RingMap SimplicialModule -- apply a ring
    map to a simplicial module
  * RingMap ** SimplicialModuleMap,
    -- tensor a map of simplicial modules along a ring map

Ways to use this method:
========================

  * RingMap SimplicialModuleMap -- apply a
    ring map to a map of simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2403:0.

schurMap -- construct the Dold-Puppe extension of the Schur functor
*******************************************************************

  * Usage:
        schurMap(lambda, S, Options => {Degeneracy => false, TopDegree => null})
  * Inputs:
      * lambda, a list (Macaulay2Doc), A list representing a
        partition.
      * S, a Simplicial Module, or
        SimplicialModuleMap, or Complex (Complexes),
        or ComplexMap (Complexes), to
        apply the Schur functor extension.
  * Optional inputs (Macaulay2Doc),:
      * Degeneracy, => a
        Boolean value (Macaulay2Doc), default value false,
        Default value is false. Indicates whether to include degeneracy maps in
        the construction.
      * TopDegree, => an
        integer (Macaulay2Doc), default value null, Default is null.
        If provided, specifies the top degree for the construction of the
        simplicial module.
  * Outputs:
      * a Simplicial Module, The Dold-Puppe extension
        of the Schur functor applied to the simplicial module S.

Description
===========

This function constructs the Dold-Puppe extension of the Schur functor to the
category of simplicial modules, applied to a simplicial module S or a complex
(or maps thereof). The construction involves creating a simplicial module where
each component is a Schur functor applied to the corresponding component of S.

By default, degeneracy maps are not computed in this method since it adds
additional computation time that is not necessary for computing the
normalization or many of the invariants of interest. However, if the user is
interested in having degeneracy maps, use the option Degeneracy => true.

+-----------------------------------------------------+
|i1 : Q = ZZ/101[a..b]                                |
|                                                     |
|o1 = Q                                               |
|                                                     |
|o1 : PolynomialRing                                  |
+-----------------------------------------------------+
|i2 : K = koszulComplex vars Q;                       |
+-----------------------------------------------------+
|i3 : S = simplicialModule(K, 4, Degeneracy => true)  |
|                                                     |
|      1      3      6      10      15                |
|o3 = Q  <-- Q  <-- Q  <-- Q   <-- Q  <-- ...         |
|                                                     |
|     0      1      2      3       4                  |
|                                                     |
|o3 : SimplicialModule                                |
+-----------------------------------------------------+
|i4 : S2S = elapsedTime schurMap({2}, S)              |
| -- .165786s elapsed                                 |
|                                                     |
|      1      6      21      55      120              |
|o4 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...       |
|                                                     |
|     0      1      2       3       4                 |
|                                                     |
|o4 : SimplicialModule                                |
+-----------------------------------------------------+
|i5 : elapsedTime schurMap({2}, S, Degeneracy => true)|
| -- .346557s elapsed                                 |
|                                                     |
|      1      6      21      55      120              |
|o5 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...       |
|                                                     |
|     0      1      2       3       4                 |
|                                                     |
|o5 : SimplicialModule                                |
+-----------------------------------------------------+

In general, if a polynomial functor has degree $d$, the Dold-Puppe extension of
a functor applied to a chain complex of length $t$ will have length at most $d
\cdot t$. We can see this explicitly in the following example:

+--------------------------------------------------------+
|i6 : S = simplicialModule(K, 5)                         |
|                                                        |
|      1      3      6      10      15      21           |
|o6 = Q  <-- Q  <-- Q  <-- Q   <-- Q   <-- Q  <-- ...    |
|                                                        |
|     0      1      2      3       4       5             |
|                                                        |
|o6 : SimplicialModule                                   |
+--------------------------------------------------------+
|i7 : S2 = schurMap({2}, S)                              |
|                                                        |
|      1      6      21      55      120      231        |
|o7 = Q  <-- Q  <-- Q   <-- Q   <-- Q    <-- Q   <-- ... |
|                                                        |
|     0      1      2       3       4        5           |
|                                                        |
|o7 : SimplicialModule                                   |
+--------------------------------------------------------+
|i8 : prune normalize S2 --notice the output has length 4|
|                                                        |
|      1      5      10      9      3                    |
|o8 = Q  <-- Q  <-- Q   <-- Q  <-- Q                     |
|                                                        |
|     0      1      2       3      4                     |
|                                                        |
|o8 : Complex                                            |
+--------------------------------------------------------+
|i9 : minimize oo                                        |
|                                                        |
|      1      2      2      2      1                     |
|o9 = Q  <-- Q  <-- Q  <-- Q  <-- Q                      |
|                                                        |
|     0      1      2      3      4                      |
|                                                        |
|o9 : Complex                                            |
+--------------------------------------------------------+

If the input is a complex, then the default top degree is taken to be the
degree of the Schur functor multiplied by the length of the complex.
Computationally, this upper bound is often too big to be computed at the
moment, so the user may need to specify a top degree by using the TopDegree =>
d option.

+-----------------------------------------------------------------------------+
|i10 : Q = ZZ/101[a..c]                                                       |
|                                                                             |
|o10 = Q                                                                      |
|                                                                             |
|o10 : PolynomialRing                                                         |
+-----------------------------------------------------------------------------+
|i11 : K = koszulComplex vars Q;                                              |
+-----------------------------------------------------------------------------+
|i12 : S2K = elapsedTime prune schurMap({2}, K, TopDegree => 4)               |
| -- 1.09712s elapsed                                                         |
|                                                                             |
|       1      9      36      74      81                                      |
|o12 = Q  <-- Q  <-- Q   <-- Q   <-- Q                                        |
|                                                                             |
|      0      1      2       3       4                                        |
|                                                                             |
|o12 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i13 : (minimize S2K).dd --ignore the last differential                       |
|                                                                             |
|           1                 3                                               |
|o13 = 0 : Q  <------------- Q  : 1                                           |
|                | a b c |                                                    |
|                                                                             |
|           3                                 6                               |
|      1 : Q  <----------------------------- Q  : 2                           |
|                {1} | -b -c 0  b  c  0  |                                    |
|                {1} | a  0  -c -a 0  c  |                                    |
|                {1} | 0  a  b  0  -a -b |                                    |
|                                                                             |
|           6                                      10                         |
|      2 : Q  <---------------------------------- Q   : 3                     |
|                {2} | c  a 0 0  b 0 0 c  0 0 |                               |
|                {2} | -b 0 a 0  0 b 0 0  c 0 |                               |
|                {2} | a  0 0 a  0 0 b 0  0 c |                               |
|                {2} | 0  a 0 -c b c 0 0  0 0 |                               |
|                {2} | 0  0 a b  0 0 0 b  c 0 |                               |
|                {2} | 0  0 0 0  0 a b -a 0 c |                               |
|                                                                             |
|           10                                                                |
|      3 : Q   <--------------------------------------------------------------|
|                 {3} | -a -b -c 0  0  0  0  0  0  0 0 0 0 0 0 0   0 0   0 0  |
|                 {3} | c  0  0  -b -c 0  0  0  0  0 0 0 0 0 0 -bc 0 -c2 0 0  |
|                 {3} | -b 0  0  0  -b -c 0  0  0  0 0 0 0 0 0 b2  0 bc  0 0  |
|                 {3} | a  0  0  0  0  0  -b -c 0  0 0 0 0 0 0 -ab 0 -ac 0 0  |
|                 {3} | 0  c  0  a  0  0  -c 0  0  0 0 0 0 0 0 ac  0 0   0 -c2|
|                 {3} | 0  -b 0  0  a  0  0  -c 0  0 0 0 0 0 0 -ab 0 0   0 bc |
|                 {3} | 0  a  0  0  0  0  a  0  -c 0 0 0 0 0 0 a2  0 0   0 -ac|
|                 {3} | 0  0  c  0  a  0  b  0  0  0 0 0 0 0 0 0   0 ac  0 bc |
|                 {3} | 0  0  -b 0  0  a  0  b  0  0 0 0 0 0 0 0   0 -ab 0 -b2|
|                 {3} | 0  0  a  0  0  0  0  a  b  0 0 0 0 0 0 0   0 a2  0 ab |
|                                                                             |
|o13 : ComplexMap                                                             |
|-----------------------------------------------------------------------------|
|                                                            41               |
|---------------------------------------------------------- Q   : 4           |
| 0 0 0 0 0   0 0   0 0   0 0 0   0 0   0 0   0 0 0 0 0 |                     |
| 0 0 0 0 0   0 0   0 0   0 0 0   0 0   0 0   0 0 0 0 0 |                     |
| 0 0 0 0 b2  0 bc  0 0   0 0 -b2 0 -bc 0 0   0 0 0 0 0 |                     |
| 0 0 0 0 -ab 0 -ac 0 0   0 0 ab  0 ac  0 0   0 0 0 0 0 |                     |
| 0 0 0 0 0   0 0   0 0   0 0 0   0 0   0 0   0 0 0 0 0 |                     |
| 0 0 0 0 -ab 0 -ac 0 0   0 0 ab  0 ac  0 0   0 0 0 0 0 |                     |
| 0 0 0 0 a2  0 0   0 -ac 0 0 -a2 0 0   0 ac  0 0 0 0 0 |                     |
| 0 0 0 0 0   0 0   0 0   0 0 0   0 0   0 0   0 0 0 0 0 |                     |
| 0 0 0 0 0   0 0   0 0   0 0 0   0 0   0 0   0 0 0 0 0 |                     |
| 0 0 0 0 0   0 a2  0 ab  0 0 0   0 -a2 0 -ab 0 0 0 0 0 |                     |
+-----------------------------------------------------------------------------+

These functors are particularly interesting in the modular setting, i.e., when
the characteristic of the underlying field is small relative to the degree of
the Schur functor. In this case, the induced complexes will have different
homotopy classes as the characteristic varies.

+---------------------------------------------------------------------+
|i14 : needsPackage "ChainComplexOperations"                          |
|                                                                     |
|o14 = ChainComplexOperations                                         |
|                                                                     |
|o14 : Package                                                        |
+---------------------------------------------------------------------+
|i15 : Q = ZZ/2[a,b]                                                  |
|                                                                     |
|o15 = Q                                                              |
|                                                                     |
|o15 : PolynomialRing                                                 |
+---------------------------------------------------------------------+
|i16 : K = koszulComplex vars Q;                                      |
+---------------------------------------------------------------------+
|i17 : S2K = minimize prune schurMap({2}, K)                          |
|                                                                     |
|       1      2      2      2      1                                 |
|o17 = Q  <-- Q  <-- Q  <-- Q  <-- Q                                  |
|                                                                     |
|      0      1      2      3      4                                  |
|                                                                     |
|o17 : Complex                                                        |
+---------------------------------------------------------------------+
|i18 : S2K' = sym2 K --the "naive" sym2 functor                       |
|                                                                     |
|             2      2      2                                         |
|o18 = 0 <-- Q  <-- Q  <-- Q  <-- 0                                   |
|                                                                     |
|      0     1      2      3      4                                   |
|                                                                     |
|o18 : Complex                                                        |
+---------------------------------------------------------------------+
|i19 : S2K.dd                                                         |
|                                                                     |
|           1               2                                         |
|o19 = 0 : Q  <----------- Q  : 1                                     |
|                | a b |                                              |
|                                                                     |
|           2                   2                                     |
|      1 : Q  <--------------- Q  : 2                                 |
|                {1} | b b |                                          |
|                {1} | a a |                                          |
|                                                                     |
|           2                   2                                     |
|      2 : Q  <--------------- Q  : 3                                 |
|                {2} | a b |                                          |
|                {2} | a b |                                          |
|                                                                     |
|           2                 1                                       |
|      3 : Q  <------------- Q  : 4                                   |
|                {3} | b |                                            |
|                {3} | a |                                            |
|                                                                     |
|o19 : ComplexMap                                                     |
+---------------------------------------------------------------------+
|i20 : S2K'.dd                                                        |
|                                                                     |
|                    2                                                |
|o20 = 0 : 0 <----- Q  : 1                                            |
|               0                                                     |
|                                                                     |
|           2                   2                                     |
|      1 : Q  <--------------- Q  : 2                                 |
|                {1} | b b |                                          |
|                {1} | a a |                                          |
|                                                                     |
|           2                   2                                     |
|      2 : Q  <--------------- Q  : 3                                 |
|                {2} | a b |                                          |
|                {2} | a b |                                          |
|                                                                     |
|           2                                                         |
|      3 : Q  <----- 0 : 4                                            |
|                0                                                    |
|                                                                     |
|o20 : ComplexMap                                                     |
+---------------------------------------------------------------------+
|i21 : prune HH S2K                                                   |
|                                                                     |
|o21 = cokernel | b a | <-- 0 <-- cokernel {2} | b a |                |
|                                                                     |
|      0                    1     2                                   |
|                                                                     |
|o21 : Complex                                                        |
+---------------------------------------------------------------------+
|i22 : prune HH S2K' --not quasi-isomorphic!                          |
|                                                                     |
|                                                       1             |
|o22 = cokernel {1} | b | <-- cokernel {2} | b a | <-- Q              |
|               {1} | a |                                             |
|                             2                        3              |
|      1                                                              |
|                                                                     |
|o22 : Complex                                                        |
+---------------------------------------------------------------------+
|i23 : Q = ZZ/3[a,b];                                                 |
+---------------------------------------------------------------------+
|i24 : K = koszulComplex vars Q;                                      |
+---------------------------------------------------------------------+
|i25 : prune HH schurMap({2}, K)                                      |
|                                                                     |
|o25 = cokernel | b a | <-- 0 <-- cokernel {2} | b a |                |
|                                                                     |
|      0                    1     2                                   |
|                                                                     |
|o25 : Complex                                                        |
+---------------------------------------------------------------------+
|i26 : prune HH sym2 K --quasi-isomorphic in all other characteristics|
|                                                                     |
|o26 = cokernel | b a | <-- 0 <-- cokernel {2} | b a |                |
|                                                                     |
|      0                    1     2                                   |
|                                                                     |
|o26 : Complex                                                        |
+---------------------------------------------------------------------+

This method is also implemented in a functorial way.

+-----------------------------------------------------------------------------+
|i27 : Q = ZZ/2[a,b]                                                          |
|                                                                             |
|o27 = Q                                                                      |
|                                                                             |
|o27 : PolynomialRing                                                         |
+-----------------------------------------------------------------------------+
|i28 : K = koszulComplex vars Q;                                              |
+-----------------------------------------------------------------------------+
|i29 : F = freeResolution( (ideal vars Q)^2)                                  |
|                                                                             |
|       1      3      2                                                       |
|o29 = Q  <-- Q  <-- Q                                                        |
|                                                                             |
|      0      1      2                                                        |
|                                                                             |
|o29 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i30 : phi = extend(K, F, id_(K_0))                                           |
|                                                                             |
|           1             1                                                   |
|o30 = 0 : Q  <--------- Q  : 0                                               |
|                | 1 |                                                        |
|                                                                             |
|           2                     3                                           |
|      1 : Q  <----------------- Q  : 1                                       |
|                {1} | a b 0 |                                                |
|                {1} | 0 0 b |                                                |
|                                                                             |
|           1                   2                                             |
|      2 : Q  <--------------- Q  : 2                                         |
|                {2} | 0 b |                                                  |
|                                                                             |
|o30 : ComplexMap                                                             |
+-----------------------------------------------------------------------------+
|i31 : f = elapsedTime prune schurMap({2}, phi)                               |
| -- 1.11838s elapsed                                                         |
|                                                                             |
|           1             1                                                   |
|o31 = 0 : Q  <--------- Q  : 0                                               |
|                | 1 |                                                        |
|                                                                             |
|           5                                       9                         |
|      1 : Q  <----------------------------------- Q  : 1                     |
|                {1} | a b 0 0  0  0  0  0  0  |                              |
|                {1} | 0 0 b 0  0  0  0  0  0  |                              |
|                {2} | 0 0 0 a2 ab 0  b2 0  0  |                              |
|                {2} | 0 0 0 0  0  ab 0  b2 0  |                              |
|                {2} | 0 0 0 0  0  0  0  0  b2 |                              |
|                                                                             |
|           10                                                                |
|      2 : Q   <--------------------------------------------------------------|
|                 {2} | 0 b 0 0  0  0  0  0  0  0  0  0 0  0 0  0 0  0 0  0 0 |
|                 {2} | 0 0 0 ab b2 0  0  0  0  0  0  0 0  0 0  0 0  0 0  0 0 |
|                 {2} | 0 0 0 0  0  a2 ab 0  b2 0  0  0 0  0 0  0 0  0 0  0 0 |
|                 {2} | 0 0 0 0  0  0  0  ab 0  b2 0  0 0  0 0  0 0  0 0  0 0 |
|                 {2} | 0 0 0 0  0  0  0  0  0  0  b2 0 0  0 0  0 0  0 0  0 0 |
|                 {3} | 0 0 0 0  0  0  0  0  0  0  0  0 ab 0 b2 0 0  0 0  0 0 |
|                 {3} | 0 0 0 0  0  0  0  0  0  0  0  0 0  0 0  0 b2 0 0  0 0 |
|                 {3} | 0 0 0 0  0  0  0  0  0  0  0  0 0  0 0  0 0  0 ab 0 b2|
|                 {3} | 0 0 0 0  0  0  0  0  0  0  0  0 0  0 0  0 0  0 0  0 0 |
|                 {4} | 0 0 0 0  0  0  0  0  0  0  0  0 0  0 0  0 0  0 0  0 0 |
|                                                                             |
|           9                                                                 |
|      3 : Q  <---------------------------------------------------------------|
|                {3} | 0 ab 0 b2 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 |
|                {3} | 0 0  0 0  0 b2 0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 |
|                {3} | 0 0  0 0  0 0  0 ab 0 0  0 b2 0 0  0 0  0 0  0 0 0 0 0 |
|                {3} | 0 0  0 0  0 0  0 0  0 ab 0 0  0 b2 0 0  0 0  0 0 0 0 0 |
|                {3} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 b2 0 0  0 0 0 0 0 |
|                {3} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 b2 0 0 0 0 0 |
|                {4} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 b2|
|                {4} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 |
|                {4} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 |
|                                                                             |
|           3                                          12                     |
|      4 : Q  <-------------------------------------- Q   : 4                 |
|                {4} | 0 0 b2 0 0 0 0 0  0 0 0 0  |                           |
|                {4} | 0 0 0  0 0 0 0 b2 0 0 0 0  |                           |
|                {4} | 0 0 0  0 0 0 0 0  0 0 0 b2 |                           |
|                                                                             |
|o31 : ComplexMap                                                             |
|-----------------------------------------------------------------------------|
|                  26                                                         |
|---------------- Q   : 2                                                     |
| 0 0  0 0 0  |                                                               |
| 0 0  0 0 0  |                                                               |
| 0 0  0 0 0  |                                                               |
| 0 0  0 0 0  |                                                               |
| 0 0  0 0 0  |                                                               |
| 0 0  0 0 0  |                                                               |
| 0 0  0 0 0  |                                                               |
| 0 0  0 0 0  |                                                               |
| 0 b2 0 0 0  |                                                               |
| 0 0  0 0 b2 |                                                               |
|                                                                             |
|                      30                                                     |
|-------------------- Q   : 3                                                 |
| 0 0 0 0  0 0 0  |                                                           |
| 0 0 0 0  0 0 0  |                                                           |
| 0 0 0 0  0 0 0  |                                                           |
| 0 0 0 0  0 0 0  |                                                           |
| 0 0 0 0  0 0 0  |                                                           |
| 0 0 0 0  0 0 0  |                                                           |
| 0 0 0 0  0 0 0  |                                                           |
| 0 0 0 b2 0 0 0  |                                                           |
| 0 0 0 0  0 0 b2 |                                                           |
+-----------------------------------------------------------------------------+
|i32 : isCommutative f                                                        |
|                                                                             |
|o32 = true                                                                   |
+-----------------------------------------------------------------------------+
|i33 : isWellDefined f                                                        |
|                                                                             |
|o33 = true                                                                   |
+-----------------------------------------------------------------------------+
|i34 : prune HH source f                                                      |
|                                                                             |
|o34 = cokernel | b2 ab a2 | <-- 0 <-- cokernel {4} | b a 0 0 0 0 |           |
|                                               {4} | 0 0 b a 0 0 |           |
|      0                         1              {4} | 0 0 0 0 b a |           |
|                                                                             |
|                                      2                                      |
|                                                                             |
|o34 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i35 : prune HH target f                                                      |
|                                                                             |
|o35 = cokernel | b a | <-- 0 <-- cokernel {2} | b a |                        |
|                                                                             |
|      0                    1     2                                           |
|                                                                             |
|o35 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i36 : prune HH f                                                             |
|                                                                             |
|o36 = 0 : cokernel | b a | <--------- cokernel | b2 ab a2 | : 0              |
|                              | 1 |                                          |
|                                                                             |
|o36 : ComplexMap                                                             |
+-----------------------------------------------------------------------------+

Caveat
======

As many of the above examples show, this method can quickly become
computationally infeasible when the modules in the complex have too big of
rank. This seems to stem more from the efficiency of the schur (SchurFunctors),
method, since methods such as the tensor product still
run very quickly even with huge matrices.

See also
========

  * extPower -- compute the Dold-Puppe extension of the
    exterior power functor to simplicial modules
  * simplicialTensor -- compute the simplicial tensor
    product and cache direct sum indices for easy access
  * symmetricQuotient -- computes the image of the
    surjection from the simplicial tensor product onto the second symmetric
    power of a simplicial module

Ways to use schurMap:
=====================

  * "schurMap(List,Complex)"
  * "schurMap(List,ComplexMap)"
  * "schurMap(List,SimplicialModule)"
  * "schurMap(List,SimplicialModuleMap)"

For the programmer
==================

The object schurMap, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3545:0.

schurMap(...,Degeneracy=>...) -- whether to compute degeneracy maps
*******************************************************************

  * Usage:
        schurMap(..., Degeneracy => true)

Description
===========

If Degeneracy => true, the degeneracy maps are also computed alongside the face
maps. By default, degeneracy maps are not computed.

See also
========

  * schurMap -- construct the Dold-Puppe extension of the
    Schur functor

Functions with optional argument named Degeneracy:
==================================================

  * extPower(...,Degeneracy=>...)
    -- whether to compute
    degeneracy maps
  * schurMap(...,Degeneracy=>...)
    -- whether to compute
    degeneracy maps
  * simplicialModule(...,Degeneracy=>...)
    -- whether to
    compute degeneracy maps
  * simplicialTensor(...,Degeneracy=>...)
    -- whether to
    compute degeneracy maps

Further information
===================

  * Default value: false (Macaulay2Doc),
  * Function: schurMap, -- construct the Dold-Puppe extension
    of the Schur functor
  * Option key: Degeneracy, -- the class of all
    simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:4069:0.

schurMap(...,TopDegree=>...) -- top degree for the construction
***************************************************************

  * Usage:
        schurMap(..., TopDegree => n)

Description
===========

Specifies the top simplicial degree for the resulting simplicial module.

See also
========

  * schurMap -- construct the Dold-Puppe extension of the
    Schur functor

Functions with optional argument named TopDegree:
=================================================

  * extPower(...,TopDegree=>...)
    -- top degree for the
    construction
  * schurMap(...,TopDegree=>...)
    -- top degree for the
    construction
  * simplicialTensor(...,TopDegree=>...)
    -- top degree for
    the construction

Further information
===================

  * Default value: null
  * Function: schurMap, -- construct the Dold-Puppe extension
    of the Schur functor
  * Option key: TopDegree, -- construct the Dold-Puppe
    extension of the Schur functor
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:4083:0.

SimplicialModule -- the class of all simplicial modules
*******************************************************

Description
===========

A simplicial R-module is a presheaf on the so-called Simplex Category, with
values in the category of R-modules. Concretely, such objects can be viewed as
nonnegatively graded R-modules equipped with certain face and degeneracy
operators satisfying the simplicial identities. As an example, every R-module M
can be converted into a simplicial R-module whose degree n piece is equal to M
for all n, and with face/degeneracy operators simply given by the identity.

+-----------------------------------------------------------------------------+
|i1 : S = simplicialModule(ZZ^2,3,Degeneracy => true) --the integer 3         |
|                                                                             |
|       2       2       2       2                                             |
|o1 = ZZ  <-- ZZ  <-- ZZ  <-- ZZ <-- ...                                      |
|                                                                             |
|     0       1       2       3                                               |
|                                                                             |
|o1 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|specifies a top degree, the option Degeneracy specifies whether or not to    |
|-----------------------------------------------------------------------------|
|compute degeneracy maps                                                      |
+-----------------------------------------------------------------------------+

The output string for a simplicial module is meant to indicate that this object
is infinite in general, and the user can only compute a finite snapshot of the
object. The face/degeneracy maps can be accessed using the keys dd and ss,
respectively. In order to verify that the resulting face/degeneracy maps
satisfy the simplicial identities, one can use the isSimplicialModule,
command.

+---------------------------------------------------------------------+
|i2 : keys S                                                          |
|                                                                     |
|o2 = {complexLength, module, ss, dd, ring, complex, topDegree, cache}|
|                                                                     |
|o2 : List                                                            |
+---------------------------------------------------------------------+

See also
========

  * The Dold-Kan Correspondence in Macaulay2
    -- compute the image of a non-negatively graded complex under
    the Dold-Kan correspondence
  * Making simplicial modules -- information
    about the basic constructors

Functions and methods returning a Simplicial Module:
====================================================

  * basis(List,SimplicialModule) (missing documentation)
  * basis(ZZ,SimplicialModule) (missing documentation)
  * coimage(SimplicialModuleMap), --
    make the coimage of a map of simplicial modules
  * cokernel(SimplicialModuleMap), --
    make the cokernel of a map of simplicial modules
  * "SimplicialModule ++ SimplicialModule" -- see
    directSum(SimplicialModule), -- direct sum
    of simplicial modules
  * "extPower(ZZ,SimplicialModule)" -- see extPower, -- compute
    the Dold-Puppe extension of the exterior power functor to simplicial
    modules
  * forgetComplex(SimplicialModule),
    -- forget the underlying complex data of a simplicial module obtained as a
    Dold-Kan image
  * HH SimplicialModule (missing documentation)
  * image(SimplicialModuleMap) -- make
    the image of a map of simplicial modules
  * kernel(SimplicialModuleMap) -- make
    the kernel of a map of simplicial modules
  * minimalPresentation(SimplicialModule)
    -- minimal presentation of all
    terms in a simplicial module
  * "prune(SimplicialModule)" -- see
    minimalPresentation(SimplicialModule),
    -- minimal presentation of all
    terms in a simplicial module
  * "ring(SimplicialModuleMap)" -- see ring(SimplicialModule),
    -- access the ring of a simplicial module or a
    simplicial module map
  * "Ring ** SimplicialModule" -- see RingMap ** SimplicialModule,
    -- tensor a simplicial module along a ring
    map
  * RingMap ** SimplicialModule, --
    tensor a simplicial module along a ring map
  * "SimplicialModule ** Ring" -- see RingMap ** SimplicialModule,
    -- tensor a simplicial module along a ring
    map
  * "SimplicialModule ** RingMap" -- see RingMap ** SimplicialModule,
    -- tensor a simplicial module along a ring
    map
  * "tensor(RingMap,SimplicialModule)" -- see RingMap ** SimplicialModule,
    -- tensor a simplicial
    module along a ring map
  * "tensor(SimplicialModule,RingMap)" -- see RingMap ** SimplicialModule,
    -- tensor a simplicial
    module along a ring map
  * RingMap SimplicialModule -- apply a ring
    map to a simplicial module
  * "schurMap(List,SimplicialModule)" -- see schurMap, --
    construct the Dold-Puppe extension of the Schur functor
  * "simplicialModule(Complex,HashTable,HashTable,ZZ)" -- see
    simplicialModule, -- make a simplicial module
  * "simplicialModule(Complex,HashTable,ZZ)" -- see simplicialModule,
    -- make a simplicial module
  * "simplicialModule(HashTable,HashTable,HashTable,ZZ)" -- see
    simplicialModule, -- make a simplicial module
  * "simplicialModule(HashTable,HashTable,ZZ)" -- see simplicialModule,
    -- make a simplicial module
  * "Complex ** SimplicialModule" -- see SimplicialModule ** SimplicialModule,
    -- tensor
    product of simplicial modules
  * "Module ** SimplicialModule" -- see SimplicialModule ** SimplicialModule,
    -- tensor
    product of simplicial modules
  * "SimplicialModule ** Complex" -- see SimplicialModule ** SimplicialModule,
    -- tensor
    product of simplicial modules
  * "SimplicialModule ** Module" -- see SimplicialModule ** SimplicialModule,
    -- tensor
    product of simplicial modules
  * SimplicialModule ** SimplicialModule
    -- tensor product of simplicial modules
  * "simplicialModule(Ideal,ZZ)" -- see simplicialModule(Module,ZZ),
    -- make a simplicial module associated
    to a complex concentrated in degree 0
  * simplicialModule(Module,ZZ), --
    make a simplicial module associated to a complex concentrated in degree 0
  * "simplicialModule(Ring,ZZ)" -- see simplicialModule(Module,ZZ),
    -- make a simplicial module associated
    to a complex concentrated in degree 0
  * "simplicialTensor(List)" -- see simplicialTensor,
    -- compute the simplicial tensor product and cache direct sum indices for
    easy access
  * "simplicialTensor(ZZ,SimplicialModule)" -- see simplicialTensor,
    -- compute the simplicial tensor product and cache direct
    sum indices for easy access
  * simplicialTensor(SimplicialModule,SimplicialModule) (missing documentation)
  * source(SimplicialModuleMap) -- get
    the source of a map of simplicial modules
  * target(SimplicialModuleMap) -- get
    the target of a map of simplicial modules
  * "simplicialModule(Complex)" -- see The Dold-Kan Correspondence in Macaulay2,
    -- compute the image
    of a non-negatively graded complex under the Dold-Kan correspondence
  * "simplicialModule(Complex,ZZ)" -- see The Dold-Kan Correspondence in Macaulay2,
    -- compute the image
    of a non-negatively graded complex under the Dold-Kan correspondence
  * truncate(List,SimplicialModule)
    -- truncation of a simplicial module
    at a specified degree or set of degrees
  * "truncate(ZZ,SimplicialModule)" -- see
    truncate(List,SimplicialModule), --
    truncation of a simplicial module at a specified degree or set of degrees

Methods that use a Simplicial Module:
=====================================

  * components(SimplicialModule), --
    list the components of a direct sum
  * directSum(SimplicialModule), --
    direct sum of simplicial modules
  * "exteriorInclusion(SimplicialModule)" -- see exteriorInclusion,
    -- computes the image of the 2nd exterior power into the
    tensor product
  * "Symbol ^ SimplicialModule" -- see face/degeneracy maps of a simplicial module,
    -- access
    the face and degeneracy maps of a simplicial module
  * forgetDegeneracy(SimplicialModule)
    -- forget the data of degeneracy
    maps of a simplicial object
  * inducedMap(SimplicialModule,SimplicialModule)
    -- make the map of
    simplicial modules induced at each term by the identity map
  * isSimplicialModule(SimplicialModule)
    -- check if the simplicial
    identities hold for a simplicial object with degeneracy map keys
  * isWellDefined(SimplicialModule),
    -- whether a simplicial module is well-defined
  * map(SimplicialModule,SimplicialModule,HashTable)
    -- make a map of
    simplicial modules
  * map(SimplicialModule,SimplicialModule,SimplicialModuleMap)
    -- make
    a new map of simplicial modules, induced from an existing one
  * map(SimplicialModule,SimplicialModule,ZZ)
    -- make the zero map or
    identity between simplicial modules
  * "naiveNorm(SimplicialModule)" -- see naiveNorm(SimplicialModule,ZZ),
    -- compute the naive normalization of
    a simplicial object
  * naiveNorm(SimplicialModule,ZZ),
    -- compute the naive normalization of a simplicial object
  * net(SimplicialModule) (missing documentation)
  * "normalize(SimplicialModule)" -- see normalize(SimplicialModule,ZZ),
    -- normalization functor from
    simplicial modules to nonnegatively-graded chain complexes
  * normalize(SimplicialModule,ZZ),
    -- normalization functor from simplicial modules to nonnegatively-graded
    chain complexes
  * randomSimplicialMap(SimplicialModule,SimplicialModule)
    -- a random
    map of simplicial modules
  * ring(SimplicialModule) -- access the
    ring of a simplicial module or a simplicial module map
  * "Matrix ** SimplicialModule" -- see SimplicialModule ** Matrix,
    -- create the tensor product of a
    simplicial module and a map of modules
  * SimplicialModule ** Matrix -- create
    the tensor product of a simplicial module and a map of modules
  * SimplicialModule == SimplicialModule
    -- whether two simplicial modules are equal
  * "SimplicialModule == ZZ" -- see SimplicialModule == SimplicialModule,
    -- whether two simplicial modules are
    equal
  * "ZZ == SimplicialModule" -- see SimplicialModule == SimplicialModule,
    -- whether two simplicial modules are
    equal
  * "SimplicialModule ^ Array" -- see SimplicialModule _ Array,
    -- the canonical inclusion or projection map of
    a direct sum
  * SimplicialModule _ Array -- the
    canonical inclusion or projection map of a direct sum
  * "SimplicialModule _ Sequence" -- see SimplicialModule _ ZZ,
    -- access individual objects of a simplicial
    module
  * SimplicialModule _ ZZ -- access individual
    objects of a simplicial module
  * SimplicialModule Array -- shift a simplicial
    module or simplicial module map
  * "SimplicialModule ** SimplicialModuleMap" -- see SimplicialModuleMap ** SimplicialModuleMap,
    --
    the map of simplicial modules between tensor simplicial modules
  * "SimplicialModuleMap ** SimplicialModule" -- see SimplicialModuleMap ** SimplicialModuleMap,
    --
    the map of simplicial modules between tensor simplicial modules
  * "symmetricQuotient(SimplicialModule)" -- see symmetricQuotient,
    -- computes the image of the surjection from the
    simplicial tensor product onto the second symmetric power of a simplicial
    module
  * "topDegree(SimplicialModule)" -- see topDegree, -- top
    degree of a simplicial module

For the programmer
==================

The object SimplicialModule, is a type (Macaulay2Doc),
with ancestor classes MutableHashTable (Macaulay2Doc),
< HashTable (Macaulay2Doc) <
Thing (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:161:0.

simplicialModule -- make a simplicial module
********************************************

  * Usage:
        simplicialModule(H1, H2, H3, d)
  * Inputs:
      * H1, a hash table (Macaulay2Doc), or Complex (Complexes),
        used to determine the modules in each degree
      * H2, a hash table (Macaulay2Doc), used to specify the
        face maps
      * H3, a hash table (Macaulay2Doc), optional, used to
        specify degeneracy maps if needed
      * d, an integer (Macaulay2Doc), an integer, specifying the top
        degree to compute the simplicial module up to
  * Optional inputs (Macaulay2Doc),:
      * Degeneracy,
        => a Boolean value (Macaulay2Doc), default value
        false, option specifying whether or not to also compute the degeneracy
        maps
      * Base (missing documentation) => ..., default value 0,
  * Outputs:
      * a Simplicial Module,

Description
===========

A simplicial module is a sequence of objects (e.g. modules), connected by maps
called face/degeneracy maps denoted by $d$ and $s$, respectively. These maps
satisfy the simplicial identities: 1. For face maps: \[ d_j d_i = d_i d_{j-1}
\text{ for } 0 \leq i < j \leq n \] 2. For face and degeneracy maps: \[ d_i s_j
= s_{j-1} d_i \text{ for } i < j \] \[ d_j s_j = \text{id} \] \[ d_{j+1} s_j =
\text{id} \] \[ d_k s_j = s_j d_{k-1} \text{ for } k > j+1 \] 3. For degeneracy
maps: \[ s_j s_i = s_i s_{j+1} \text{ for } i \leq j \]

This constructor is the most basic constructor for building a simplicial
module, and is called by all of the more user friendly constructors. It is
highly recommended that the user sees simplicialModule(Complex),
to quickly build simplicial modules.

+-----------------------------------------------------------------+
|i1 : S = ZZ/101[a..d]                                            |
|                                                                 |
|o1 = S                                                           |
|                                                                 |
|o1 : PolynomialRing                                              |
+-----------------------------------------------------------------+
|i2 : moduleHash = hashTable { 0 => S^1,                          |
|                              1 => S^1++S^2,                     |
|                              2 => S^1++S^2++S^2++S^1}           |
|                                                                 |
|                     1                                           |
|o2 = HashTable{0 => S }                                          |
|                     3                                           |
|               1 => S                                            |
|                     6                                           |
|               2 => S                                            |
|                                                                 |
|o2 : HashTable                                                   |
+-----------------------------------------------------------------+
|i3 : faceHash = hashTable {(1,0) => matrix {{1, a, b}},          |
|                           (1,1) => matrix {{1_S, 0, 0}},        |
|                           (2,0) => matrix {{1, a, b, 0, 0, 0},  |
|                                           {0, 0, 0, 1, 0, -b},  |
|                                           {0, 0, 0, 0, 1, a}},  |
|                           (2,1) => matrix {{1_S, 0, 0, 0, 0, 0},|
|                                            {0, 1, 0, 1, 0,0},   |
|                                            {0, 0, 1, 0, 1, 0}}, |
|                           (2,2) => matrix {{1_S, 0, 0, 0, 0, 0},|
|                                            {0, 1, 0, 0, 0, 0},  |
|                                            {0, 0, 1, 0, 0, 0}}} |
|                                                                 |
|o3 = HashTable{(1, 0) => | 1 a b |       }                       |
|               (1, 1) => | 1 0 0 |                               |
|               (2, 0) => | 1 a b 0 0 0  |                        |
|                         | 0 0 0 1 0 -b |                        |
|                         | 0 0 0 0 1 a  |                        |
|               (2, 1) => | 1 0 0 0 0 0 |                         |
|                         | 0 1 0 1 0 0 |                         |
|                         | 0 0 1 0 1 0 |                         |
|               (2, 2) => | 1 0 0 0 0 0 |                         |
|                         | 0 1 0 0 0 0 |                         |
|                         | 0 0 1 0 0 0 |                         |
|                                                                 |
|o3 : HashTable                                                   |
+-----------------------------------------------------------------+
|i4 : degenHash = hashTable {(0,0) => matrix {{1_S}, {0}, {0}},   |
|                            (1,0) => matrix {{1_S, 0, 0},        |
|                                             {0, 0, 0},          |
|                                             {0, 0, 0},          |
|                                             {0, 1, 0},          |
|                                             {0, 0, 1},          |
|                                             {0, 0, 0}},         |
|                            (1,1) => matrix {{1_S, 0, 0},        |
|                                             {0, 1, 0},          |
|                                             {0, 0, 1},          |
|                                             {0, 0, 0},          |
|                                             {0, 0, 0},          |
|                                             {0, 0, 0}}}         |
|                                                                 |
|o4 = HashTable{(0, 0) => | 1 |    }                              |
|                         | 0 |                                   |
|                         | 0 |                                   |
|               (1, 0) => | 1 0 0 |                               |
|                         | 0 0 0 |                               |
|                         | 0 0 0 |                               |
|                         | 0 1 0 |                               |
|                         | 0 0 1 |                               |
|                         | 0 0 0 |                               |
|               (1, 1) => | 1 0 0 |                               |
|                         | 0 1 0 |                               |
|                         | 0 0 1 |                               |
|                         | 0 0 0 |                               |
|                         | 0 0 0 |                               |
|                         | 0 0 0 |                               |
|                                                                 |
|o4 : HashTable                                                   |
+-----------------------------------------------------------------+
|i5 : T = simplicialModule(moduleHash, faceHash, degenHash, 2)    |
|                                                                 |
|      1      3      6                                            |
|o5 = S  <-- S  <-- S <-- ...                                     |
|                                                                 |
|     0      1      2                                             |
|                                                                 |
|o5 : SimplicialModule                                            |
+-----------------------------------------------------------------+
|i6 : T.module                                                    |
|                                                                 |
|                     1                                           |
|o6 = HashTable{0 => S }                                          |
|                     3                                           |
|               1 => S                                            |
|                     6                                           |
|               2 => S                                            |
|                                                                 |
|o6 : HashTable                                                   |
+-----------------------------------------------------------------+
|i7 : T.dd                                                        |
|                                                                 |
|               1                 3                               |
|o7 = (0, 0) : S  <------------- S  : (1, 0)                      |
|                    | 1 a b |                                    |
|                                                                 |
|               1                 3                               |
|     (0, 1) : S  <------------- S  : (1, 1)                      |
|                    | 1 0 0 |                                    |
|                                                                 |
|               3                        6                        |
|     (1, 0) : S  <-------------------- S  : (2, 0)               |
|                    | 1 a b 0 0 0  |                             |
|                    | 0 0 0 1 0 -b |                             |
|                    | 0 0 0 0 1 a  |                             |
|                                                                 |
|               3                       6                         |
|     (1, 1) : S  <------------------- S  : (2, 1)                |
|                    | 1 0 0 0 0 0 |                              |
|                    | 0 1 0 1 0 0 |                              |
|                    | 0 0 1 0 1 0 |                              |
|                                                                 |
|               3                       6                         |
|     (1, 2) : S  <------------------- S  : (2, 2)                |
|                    | 1 0 0 0 0 0 |                              |
|                    | 0 1 0 0 0 0 |                              |
|                    | 0 0 1 0 0 0 |                              |
|                                                                 |
|o7 : SimplicialModuleMap                                         |
+-----------------------------------------------------------------+
|i8 : T.ss                                                        |
|                                                                 |
|               3             1                                   |
|o8 = (1, 0) : S  <--------- S  : (0, 0)                          |
|                    | 1 |                                        |
|                    | 0 |                                        |
|                    | 0 |                                        |
|                                                                 |
|               6                 3                               |
|     (2, 0) : S  <------------- S  : (1, 0)                      |
|                    | 1 0 0 |                                    |
|                    | 0 0 0 |                                    |
|                    | 0 0 0 |                                    |
|                    | 0 1 0 |                                    |
|                    | 0 0 1 |                                    |
|                    | 0 0 0 |                                    |
|                                                                 |
|               6                 3                               |
|     (2, 1) : S  <------------- S  : (1, 1)                      |
|                    | 1 0 0 |                                    |
|                    | 0 1 0 |                                    |
|                    | 0 0 1 |                                    |
|                    | 0 0 0 |                                    |
|                    | 0 0 0 |                                    |
|                    | 0 0 0 |                                    |
|                                                                 |
|o8 : SimplicialModuleMap                                         |
+-----------------------------------------------------------------+
|i9 : T' = simplicialModule(moduleHash, faceHash, 2)              |
|                                                                 |
|      1      3      6                                            |
|o9 = S  <-- S  <-- S <-- ...                                     |
|                                                                 |
|     0      1      2                                             |
|                                                                 |
|o9 : SimplicialModule                                            |
+-----------------------------------------------------------------+
|i10 : T'.?ss                                                     |
|                                                                 |
|o10 = false                                                      |
+-----------------------------------------------------------------+

In the above, notice that if the user does not provide a hash table specifying
the degeneracy maps, then the simplicial module is still constructed using only
the data of the face maps. This feature is intentional since for the purposes
of efficiency, storing the data of the degeneracy maps may slow down
computations (and one can compute normalizations using only the face maps). In
general, the method forgetDegeneracy,
allows the user to ignore the data of
the degeneracy maps if needed.

In the following example, we see that one can construct simplicial modules that
are not well-defined. The user should use
isWellDefined(SimplicialModule), in order
to check that a simplicial module is indeed well-defined.

+---------------------------------------------------+
|i11 : H1 = hashTable {0 => S^1, 1 => S^1, 2 => S^1}|
|                                                   |
|                      1                            |
|o11 = HashTable{0 => S }                           |
|                      1                            |
|                1 => S                             |
|                      1                            |
|                2 => S                             |
|                                                   |
|o11 : HashTable                                    |
+---------------------------------------------------+
|i12 : H2 = hashTable {(1,0) => map(S^1, S^1, 0),   |
|                      (1,1) => map(S^1, S^1, 0),   |
|                      (2,0) => map(S^1, S^1, 0),   |
|                      (2,1) => map(S^1, S^1, 0),   |
|                      (2,2) => map(S^1, S^1, 0)}   |
|                                                   |
|o12 = HashTable{(1, 0) => 0}                       |
|                (1, 1) => 0                        |
|                (2, 0) => 0                        |
|                (2, 1) => 0                        |
|                (2, 2) => 0                        |
|                                                   |
|o12 : HashTable                                    |
+---------------------------------------------------+
|i13 : H3 = hashTable {(0,0) => map(S^1, S^1, 0),   |
|                      (1,0) => map(S^1, S^1, 0),   |
|                      (1,1) => map(S^1, S^1, 0)}   |
|                                                   |
|o13 = HashTable{(0, 0) => 0}                       |
|                (1, 0) => 0                        |
|                (1, 1) => 0                        |
|                                                   |
|o13 : HashTable                                    |
+---------------------------------------------------+
|i14 : U = simplicialModule(H1,H2,H3,2)             |
|                                                   |
|       1      1      1                             |
|o14 = S  <-- S  <-- S <-- ...                      |
|                                                   |
|      0      1      2                              |
|                                                   |
|o14 : SimplicialModule                             |
+---------------------------------------------------+
|i15 : U.dd                                         |
|                                                   |
|                1         1                        |
|o15 = (0, 0) : S  <----- S  : (1, 0)               |
|                     0                             |
|                                                   |
|                1         1                        |
|      (0, 1) : S  <----- S  : (1, 1)               |
|                     0                             |
|                                                   |
|                1         1                        |
|      (1, 0) : S  <----- S  : (2, 0)               |
|                     0                             |
|                                                   |
|                1         1                        |
|      (1, 1) : S  <----- S  : (2, 1)               |
|                     0                             |
|                                                   |
|                1         1                        |
|      (1, 2) : S  <----- S  : (2, 2)               |
|                     0                             |
|                                                   |
|o15 : SimplicialModuleMap                          |
+---------------------------------------------------+
|i16 : U.ss                                         |
|                                                   |
|                1         1                        |
|o16 = (1, 0) : S  <----- S  : (0, 0)               |
|                     0                             |
|                                                   |
|                1         1                        |
|      (2, 0) : S  <----- S  : (1, 0)               |
|                     0                             |
|                                                   |
|                1         1                        |
|      (2, 1) : S  <----- S  : (1, 1)               |
|                     0                             |
|                                                   |
|o16 : SimplicialModuleMap                          |
+---------------------------------------------------+
|i17 : isWellDefined U                              |
|                                                   |
|o17 = false                                        |
+---------------------------------------------------+

Caveat
======

This constructor minimizes computation and does very little error checking. To
verify that a complex is well constructed, use
isWellDefined(SimplicialModule).

See also
========

  * Making simplicial modules -- information
    about the basic constructors
  * isWellDefined(SimplicialModule),
    -- whether a simplicial module is well-defined
  * simplicialModule(HashTable,HashTable,HashTable,ZZ),
    -- make a simplicial module
  * simplicialModule(Module,ZZ), --
    make a simplicial module associated to a complex concentrated in degree 0
  * SimplicialModule Array -- shift a simplicial
    module or simplicial module map

Ways to use simplicialModule:
=============================

  * "simplicialModule(Complex,HashTable,HashTable,ZZ)"
  * "simplicialModule(Complex,HashTable,ZZ)"
  * "simplicialModule(HashTable,HashTable,HashTable,ZZ)"
  * "simplicialModule(HashTable,HashTable,ZZ)"
  * simplicialModule(ComplexMap) (missing documentation)
  * simplicialModule(ComplexMap,ZZ) (missing documentation)
  * "simplicialModule(Ideal,ZZ)" -- see simplicialModule(Module,ZZ),
    -- make a simplicial module associated
    to a complex concentrated in degree 0
  * simplicialModule(Module,ZZ), --
    make a simplicial module associated to a complex concentrated in degree 0
  * "simplicialModule(Ring,ZZ)" -- see simplicialModule(Module,ZZ),
    -- make a simplicial module associated
    to a complex concentrated in degree 0
  * "simplicialModule(Complex)" -- see The Dold-Kan Correspondence in Macaulay2,
    -- compute the image
    of a non-negatively graded complex under the Dold-Kan correspondence
  * "simplicialModule(Complex,ZZ)" -- see The Dold-Kan Correspondence in Macaulay2,
    -- compute the image
    of a non-negatively graded complex under the Dold-Kan correspondence

For the programmer
==================

The object simplicialModule, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:522:0.

SimplicialModule ** Matrix -- create the tensor product of a simplicial module and a map of modules
***************************************************************************************************

  * Operator: ** (Macaulay2Doc),
  * Usage:
        h = C ** f
        h = f ** C
  * Inputs:
      * C, a Simplicial Module, over a ring $R$
      * f, a matrix (Macaulay2Doc), defining a homomorphism from
        the $R$-module $M$ to the $R$-module $N$
  * Outputs:
      * h, a Map of Simplicial Modules, from $C
        \otimes M$ to $C \otimes N$

Description
===========

For any simplicial module $C$, a map $f \colon M \to N$ of $R$-modules induces
a morphism $C \otimes f$ of simplicial modules from $C \otimes M$ to $C \otimes
N$.  This method returns this map of simplicial modules.

+-------------------------------------------------------------------+
|i1 : R = ZZ/101[a..d];                                             |
+-------------------------------------------------------------------+
|i2 : I = ideal(c^2-b*d, b*c-a*d, b^2-a*c)                          |
|                                                                   |
|             2                    2                                |
|o2 = ideal (c  - b*d, b*c - a*d, b  - a*c)                         |
|                                                                   |
|o2 : Ideal of R                                                    |
+-------------------------------------------------------------------+
|i3 : J = ideal(I_0, I_1)                                           |
|                                                                   |
|             2                                                     |
|o3 = ideal (c  - b*d, b*c - a*d)                                   |
|                                                                   |
|o3 : Ideal of R                                                    |
+-------------------------------------------------------------------+
|i4 : C = simplicialModule(koszulComplex vars R, Degeneracy => true)|
|                                                                   |
|      1      5      15      35      70                             |
|o4 = R  <-- R  <-- R   <-- R   <-- R  <-- ...                      |
|                                                                   |
|     0      1      2       3       4                               |
|                                                                   |
|o4 : SimplicialModule                                              |
+-------------------------------------------------------------------+
|i5 : f = map(R^1/I, R^1/J, 1)                                      |
|                                                                   |
|o5 = | 1 |                                                         |
|                                                                   |
|o5 : Matrix                                                        |
+-------------------------------------------------------------------+
|i6 : C ** f;                                                       |
+-------------------------------------------------------------------+
|i7 : isSimplicialMorphism oo                                       |
|                                                                   |
|o7 = true                                                          |
+-------------------------------------------------------------------+
|i8 : f ** C;                                                       |
+-------------------------------------------------------------------+
|i9 : isSimplicialMorphism oo                                       |
|                                                                   |
|o9 = true                                                          |
+-------------------------------------------------------------------+
|i10 : f' = random(R^2, R^{-1, -1, -1})                             |
|                                                                   |
|o10 = | 24a-36b-30c-29d -8a-22b-29c-24d  34a+19b-47c-39d  |        |
|      | 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |        |
|                                                                   |
|              2      3                                             |
|o10 : Matrix R  <-- R                                              |
+-------------------------------------------------------------------+
|i11 : C ** f';                                                     |
+-------------------------------------------------------------------+
|i12 : f' ** C;                                                     |
+-------------------------------------------------------------------+
|i13 : assert isWellDefined(C ** f')                                |
+-------------------------------------------------------------------+
|i14 : assert isWellDefined(f' ** C)                                |
+-------------------------------------------------------------------+

Tensoring with a simplicial module defines a functor from the category of
$R$-modules to the category of simplicial modules over $R$.

+--------------------------------------------------------------+
|i15 : f'' = random(source f', R^{-2,-2})                      |
|                                                              |
|o15 = {1} | -28a-47b+38c+2d  -16a+7b+15c-23d |                |
|      {1} | 16a+22b+45c-34d  39a+43b-17c-11d |                |
|      {1} | -48a-47b+47c+19d 48a+36b+35c+11d |                |
|                                                              |
|              3      2                                        |
|o15 : Matrix R  <-- R                                         |
+--------------------------------------------------------------+
|i16 : assert((C ** f') * (C ** f'') == C ** (f' * f''))       |
+--------------------------------------------------------------+
|i17 : assert(C ** id_(R^{-1,-2,-3}) == id_(C ** R^{-1,-2,-3}))|
+--------------------------------------------------------------+

See also
========

  * Making maps between simplicial modules
    -- information about the basic constructors
  * SimplicialModule ** SimplicialModule
    -- tensor product of simplicial modules

Ways to use this method:
========================

  * "Matrix ** SimplicialModule"
  * SimplicialModule ** Matrix -- create
    the tensor product of a simplicial module and a map of modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2229:0.

SimplicialModule ** SimplicialModule -- tensor product of simplicial modules
****************************************************************************

  * Operator: ** (Macaulay2Doc),
  * Usage:
        D = C1 ** C2
  * Inputs:
      * C1, a Simplicial Module, or a module (Macaulay2Doc),
      * C2, a Simplicial Module, or a module (Macaulay2Doc),
  * Outputs:
      * D, a Simplicial Module, tensor product of C1
        and C2

Description
===========

The tensor product is a simplicial module $D$ whose $i$th component is the
tensor product of the degree i components of $C1$ and $C2$. The face/degeneracy
maps are given by the tensor products of the face and degeneracy maps of the
original objects.

As the next example illustrates, the simplicial tensor product in general does
not normalize to give an object that is isomorphic to the classically defined
tensor product of complexes.

+---------------------------------------------------------------------------+
|i1 : S = ZZ/101[a..c]                                                      |
|                                                                           |
|o1 = S                                                                     |
|                                                                           |
|o1 : PolynomialRing                                                        |
+---------------------------------------------------------------------------+
|i2 : Ca = simplicialModule(complex {matrix{{a}}}, 3)                       |
|                                                                           |
|      1      2      3      4                                               |
|o2 = S  <-- S  <-- S  <-- S <-- ...                                        |
|                                                                           |
|     0      1      2      3                                                |
|                                                                           |
|o2 : SimplicialModule                                                      |
+---------------------------------------------------------------------------+
|i3 : Cb = simplicialModule(complex {matrix{{b}}}, 3)                       |
|                                                                           |
|      1      2      3      4                                               |
|o3 = S  <-- S  <-- S  <-- S <-- ...                                        |
|                                                                           |
|     0      1      2      3                                                |
|                                                                           |
|o3 : SimplicialModule                                                      |
+---------------------------------------------------------------------------+
|i4 : Cc = simplicialModule(complex {matrix{{c}}}, 3)                       |
|                                                                           |
|      1      2      3      4                                               |
|o4 = S  <-- S  <-- S  <-- S <-- ...                                        |
|                                                                           |
|     0      1      2      3                                                |
|                                                                           |
|o4 : SimplicialModule                                                      |
+---------------------------------------------------------------------------+
|i5 : Cab = Cb ** Ca                                                        |
|                                                                           |
|      1      4      9      16                                              |
|o5 = S  <-- S  <-- S  <-- S  <-- ...                                       |
|                                                                           |
|     0      1      2      3                                                |
|                                                                           |
|o5 : SimplicialModule                                                      |
+---------------------------------------------------------------------------+
|i6 : dd^Cab                                                                |
|                                                                           |
|               1                    4                                      |
|o6 = (0, 0) : S  <---------------- S  : (1, 0)                             |
|                    | 1 a b ab |                                           |
|                                                                           |
|               1                   4                                       |
|     (0, 1) : S  <--------------- S  : (1, 1)                              |
|                    | 1 0 0 0 |                                            |
|                                                                           |
|               4                                  9                        |
|     (1, 0) : S  <------------------------------ S  : (2, 0)               |
|                    {0} | 1 a 0 b ab 0 0 0 0 |                             |
|                    {1} | 0 0 1 0 0  b 0 0 0 |                             |
|                    {1} | 0 0 0 0 0  0 1 a 0 |                             |
|                    {2} | 0 0 0 0 0  0 0 0 1 |                             |
|                                                                           |
|               4                                 9                         |
|     (1, 1) : S  <----------------------------- S  : (2, 1)                |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                              |
|                    {1} | 0 1 1 0 0 0 0 0 0 |                              |
|                    {1} | 0 0 0 1 0 0 1 0 0 |                              |
|                    {2} | 0 0 0 0 1 1 0 1 1 |                              |
|                                                                           |
|               4                                 9                         |
|     (1, 2) : S  <----------------------------- S  : (2, 2)                |
|                    {0} | 1 0 0 0 0 0 0 0 0 |                              |
|                    {1} | 0 1 0 0 0 0 0 0 0 |                              |
|                    {1} | 0 0 0 1 0 0 0 0 0 |                              |
|                    {2} | 0 0 0 0 1 0 0 0 0 |                              |
|                                                                           |
|               9                                                16         |
|     (2, 0) : S  <-------------------------------------------- S   : (3, 0)|
|                    {0} | 1 a 0 0 b ab 0 0 0 0 0 0 0 0 0 0 |               |
|                    {1} | 0 0 1 0 0 0  b 0 0 0 0 0 0 0 0 0 |               |
|                    {1} | 0 0 0 1 0 0  0 b 0 0 0 0 0 0 0 0 |               |
|                    {1} | 0 0 0 0 0 0  0 0 1 a 0 0 0 0 0 0 |               |
|                    {2} | 0 0 0 0 0 0  0 0 0 0 1 0 0 0 0 0 |               |
|                    {2} | 0 0 0 0 0 0  0 0 0 0 0 1 0 0 0 0 |               |
|                    {1} | 0 0 0 0 0 0  0 0 0 0 0 0 1 a 0 0 |               |
|                    {2} | 0 0 0 0 0 0  0 0 0 0 0 0 0 0 1 0 |               |
|                    {2} | 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 1 |               |
|                                                                           |
|               9                                               16          |
|     (2, 1) : S  <------------------------------------------- S   : (3, 1) |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                    {1} | 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                    {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                    {1} | 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 |                |
|                    {2} | 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 |                |
|                    {2} | 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 |                |
|                    {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                |
|                    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 |                |
|                    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                |
|                                                                           |
|               9                                               16          |
|     (2, 2) : S  <------------------------------------------- S   : (3, 2) |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                    {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                    {1} | 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                    {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                |
|                    {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                |
|                    {2} | 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 |                |
|                    {1} | 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 |                |
|                    {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 |                |
|                    {2} | 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 |                |
|                                                                           |
|               9                                               16          |
|     (2, 3) : S  <------------------------------------------- S   : (3, 3) |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                    {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                    {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                    {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                |
|                    {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                |
|                    {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                |
|                    {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                |
|                    {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                |
|                    {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                |
|                                                                           |
|o6 : SimplicialModuleMap                                                   |
+---------------------------------------------------------------------------+
|i7 : (prune normalize Cab).dd                                              |
|                                                                           |
|          1                  3                                             |
|o7 = 0 : S  <-------------- S  : 1                                         |
|               | a b ab |                                                  |
|                                                                           |
|          3                     2                                          |
|     1 : S  <----------------- S  : 2                                      |
|               {1} | -b -b |                                               |
|               {1} | a  0  |                                               |
|               {2} | 0  1  |                                               |
|                                                                           |
|o7 : ComplexMap                                                            |
+---------------------------------------------------------------------------+
|i8 : assert isWellDefined Cab                                              |
+---------------------------------------------------------------------------+
|i9 : Cabc = Cc ** Cab                                                      |
|                                                                           |
|      1      8      27      64                                             |
|o9 = S  <-- S  <-- S   <-- S  <-- ...                                      |
|                                                                           |
|     0      1      2       3                                               |
|                                                                           |
|o9 : SimplicialModule                                                      |
+---------------------------------------------------------------------------+
|i10 : Cc ** Cb ** Ca                                                       |
|                                                                           |
|       1      8      27      64                                            |
|o10 = S  <-- S  <-- S   <-- S  <-- ...                                     |
|                                                                           |
|      0      1      2       3                                              |
|                                                                           |
|o10 : SimplicialModule                                                     |
+---------------------------------------------------------------------------+
|i11 : dd^(nC = prune normalize Cabc)                                       |
|                                                                           |
|           1                              7                                |
|o11 = 0 : S  <-------------------------- S  : 1                            |
|                | a b c ab ac bc abc |                                     |
|                                                                           |
|           7                                                         12    |
|      1 : S  <----------------------------------------------------- S   : 2|
|                {1} | -b -b -c -c 0  0  -bc -bc -bc -bc -bc -bc |          |
|                {1} | a  0  0  0  -c -c ac  0   0   0   0   0   |          |
|                {1} | 0  0  a  0  b  0  0   0   ab  0   0   0   |          |
|                {2} | 0  1  0  0  0  0  0   c   0   0   0   0   |          |
|                {2} | 0  0  0  1  0  0  0   0   0   b   0   0   |          |
|                {2} | 0  0  0  0  0  1  0   0   0   0   a   0   |          |
|                {3} | 0  0  0  0  0  0  0   0   0   0   0   1   |          |
|                                                                           |
|           12                                 6                            |
|      2 : S   <----------------------------- S  : 3                        |
|                 {2} | -c 0  c  0  0  -c |                                 |
|                 {2} | 0  -c 0  0  0  0  |                                 |
|                 {2} | 0  0  -b -b 0  0  |                                 |
|                 {2} | 0  0  0  0  -b 0  |                                 |
|                 {2} | 0  0  a  0  0  0  |                                 |
|                 {2} | 0  0  0  0  0  -a |                                 |
|                 {3} | 1  0  0  0  0  0  |                                 |
|                 {3} | 0  1  0  0  0  0  |                                 |
|                 {3} | 0  0  0  1  0  0  |                                 |
|                 {3} | 0  0  0  0  1  0  |                                 |
|                 {3} | 0  0  0  0  0  1  |                                 |
|                 {3} | 0  0  0  0  0  0  |                                 |
|                                                                           |
|o11 : ComplexMap                                                           |
+---------------------------------------------------------------------------+
|i12 : assert isWellDefined nC                                              |
+---------------------------------------------------------------------------+

If one of the arguments is a module, it is considered as a complex concentrated
in homological degree 0.

+----------------------------------------+
|i13 : Cabc ** (S^1/(a,b,c));            |
+----------------------------------------+
|i14 : S^2 ** Cabc                       |
|                                        |
|       2      16      54      128       |
|o14 = S  <-- S   <-- S   <-- S   <-- ...|
|                                        |
|      0      1       2       3          |
|                                        |
|o14 : SimplicialModule                  |
+----------------------------------------+

Because the tensor product can be regarded as the total complex of a double
complex, each term of the tensor product comes with pairs of indices, labelling
the summands.

+-----------------------------------------------------------------------------+
|i15 : indices Cabc_1                                                         |
|                                                                             |
|o15 = {{0, 0}, {0, 1}, {0, 2}, {0, 3}, {1, 0}, {1, 1}, {1, 2}, {1, 3}}       |
|                                                                             |
|o15 : List                                                                   |
+-----------------------------------------------------------------------------+
|i16 : components Cabc_1                                                      |
|                                                                             |
|        1   1   1   1   1   1   1   1                                        |
|o16 = {S , S , S , S , S , S , S , S }                                       |
|                                                                             |
|o16 : List                                                                   |
+-----------------------------------------------------------------------------+
|i17 : Cabc_1_[{1,0}]                                                         |
|                                                                             |
|o17 = {0} | 0 |                                                              |
|      {1} | 0 |                                                              |
|      {1} | 0 |                                                              |
|      {2} | 0 |                                                              |
|      {1} | 1 |                                                              |
|      {2} | 0 |                                                              |
|      {2} | 0 |                                                              |
|      {3} | 0 |                                                              |
|                                                                             |
|              8      1                                                       |
|o17 : Matrix S  <-- S                                                        |
+-----------------------------------------------------------------------------+
|i18 : indices Cabc_2                                                         |
|                                                                             |
|o18 = {{0, 0}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {0, 5}, {0, 6}, {0, 7}, {0,   |
|      -----------------------------------------------------------------------|
|      8}, {1, 0}, {1, 1}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {1,|
|      -----------------------------------------------------------------------|
|      8}, {2, 0}, {2, 1}, {2, 2}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {2,|
|      -----------------------------------------------------------------------|
|      8}}                                                                    |
|                                                                             |
|o18 : List                                                                   |
+-----------------------------------------------------------------------------+
|i19 : components Cabc_2                                                      |
|                                                                             |
|        1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1    |
|o19 = {S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S ,   |
|      -----------------------------------------------------------------------|
|       1   1   1   1   1   1   1   1   1   1                                 |
|      S , S , S , S , S , S , S , S , S , S }                                |
|                                                                             |
|o19 : List                                                                   |
+-----------------------------------------------------------------------------+
|i20 : Cabc_2_[{0,2}]                                                         |
|                                                                             |
|o20 = {0} | 0 |                                                              |
|      {1} | 0 |                                                              |
|      {1} | 1 |                                                              |
|      {1} | 0 |                                                              |
|      {2} | 0 |                                                              |
|      {2} | 0 |                                                              |
|      {1} | 0 |                                                              |
|      {2} | 0 |                                                              |
|      {2} | 0 |                                                              |
|      {1} | 0 |                                                              |
|      {2} | 0 |                                                              |
|      {2} | 0 |                                                              |
|      {2} | 0 |                                                              |
|      {3} | 0 |                                                              |
|      {3} | 0 |                                                              |
|      {2} | 0 |                                                              |
|      {3} | 0 |                                                              |
|      {3} | 0 |                                                              |
|      {1} | 0 |                                                              |
|      {2} | 0 |                                                              |
|      {2} | 0 |                                                              |
|      {2} | 0 |                                                              |
|      {3} | 0 |                                                              |
|      {3} | 0 |                                                              |
|      {2} | 0 |                                                              |
|      {3} | 0 |                                                              |
|      {3} | 0 |                                                              |
|                                                                             |
|              27      1                                                      |
|o20 : Matrix S   <-- S                                                       |
+-----------------------------------------------------------------------------+

See also
========

  * indices (Macaulay2Doc) -- indices of a polynomial; also
    components for a direct sum
  * components (Macaulay2Doc) -- list the components of a
    direct sum
  * directSum (Macaulay2Doc) -- direct sum of modules or maps

Ways to use this method:
========================

  * "Complex ** SimplicialModule"
  * "Module ** SimplicialModule"
  * "SimplicialModule ** Complex"
  * "SimplicialModule ** Module"
  * SimplicialModule ** SimplicialModule
    -- tensor product of simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:976:0.

SimplicialModule == SimplicialModule -- whether two simplicial modules are equal
********************************************************************************

  * Operator: == (Macaulay2Doc),
  * Usage:
        C == D
        C == 0
  * Inputs:
      * C, a Simplicial Module,
      * D, a Simplicial Module,
  * Outputs:
      * a Boolean value (Macaulay2Doc), that is true when C and
        D are equal

Description
===========

Two simplicial modules are equal if the corresponding objects and corresponding
maps at each index are equal.

+-------------------------------------------------------------+
|i1 : S = ZZ/101[a..c]                                        |
|                                                             |
|o1 = S                                                       |
|                                                             |
|o1 : PolynomialRing                                          |
+-------------------------------------------------------------+
|i2 : C = simplicialModule(K = freeResolution coker vars S, 4)|
|                                                             |
|      1      4      10      20      35                       |
|o2 = S  <-- S  <-- S   <-- S   <-- S  <-- ...                |
|                                                             |
|     0      1      2       3       4                         |
|                                                             |
|o2 : SimplicialModule                                        |
+-------------------------------------------------------------+
|i3 : D = image id_C;                                         |
+-------------------------------------------------------------+
|i4 : C === D                                                 |
|                                                             |
|o4 = false                                                   |
+-------------------------------------------------------------+
|i5 : C == D                                                  |
|                                                             |
|o5 = true                                                    |
+-------------------------------------------------------------+

Both the maps and the objects must be equal.

+-----------------------------------------------------------------------------+
|i6 : E = simplicialModule(complex for i from 1 to 3 list 0*dd^K_i, 4)        |
|                                                                             |
|      1      4      10      20      35                                       |
|o6 = S  <-- S  <-- S   <-- S   <-- S  <-- ...                                |
|                                                                             |
|     0      1      2       3       4                                         |
|                                                                             |
|o6 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i7 : dd^E                                                                    |
|                                                                             |
|               1                   4                                         |
|o7 = (0, 0) : S  <--------------- S  : (1, 0)                                |
|                    | 1 0 0 0 |                                              |
|                                                                             |
|               1                   4                                         |
|     (0, 1) : S  <--------------- S  : (1, 1)                                |
|                    | 1 0 0 0 |                                              |
|                                                                             |
|               4                                   10                        |
|     (1, 0) : S  <------------------------------- S   : (2, 0)               |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 |                              |
|                    {1} | 0 0 0 0 1 0 0 0 0 0 |                              |
|                    {1} | 0 0 0 0 0 1 0 0 0 0 |                              |
|                    {1} | 0 0 0 0 0 0 1 0 0 0 |                              |
|                                                                             |
|               4                                   10                        |
|     (1, 1) : S  <------------------------------- S   : (2, 1)               |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 |                              |
|                    {1} | 0 1 0 0 1 0 0 0 0 0 |                              |
|                    {1} | 0 0 1 0 0 1 0 0 0 0 |                              |
|                    {1} | 0 0 0 1 0 0 1 0 0 0 |                              |
|                                                                             |
|               4                                   10                        |
|     (1, 2) : S  <------------------------------- S   : (2, 2)               |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 |                              |
|                    {1} | 0 1 0 0 0 0 0 0 0 0 |                              |
|                    {1} | 0 0 1 0 0 0 0 0 0 0 |                              |
|                    {1} | 0 0 0 1 0 0 0 0 0 0 |                              |
|                                                                             |
|               10                                                       20   |
|     (2, 0) : S   <--------------------------------------------------- S   : |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |         |
|                                                                             |
|               10                                                       20   |
|     (2, 1) : S   <--------------------------------------------------- S   : |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 |         |
|                                                                             |
|               10                                                       20   |
|     (2, 2) : S   <--------------------------------------------------- S   : |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 |         |
|                                                                             |
|               10                                                       20   |
|     (2, 3) : S   <--------------------------------------------------- S   : |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |         |
|                                                                             |
|               20                                                            |
|     (3, 0) : S   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|               20                                                            |
|     (3, 1) : S   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|               20                                                            |
|     (3, 2) : S   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|               20                                                            |
|     (3, 3) : S   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|               20                                                            |
|     (3, 4) : S   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|(3, 0)                                                                       |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|(3, 1)                                                                       |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|(3, 2)                                                                       |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|(3, 3)                                                                       |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                         35                                                  |
|----------------------- S   : (4, 0)                                         |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|1 0 0 0 0 0 0 0 0 0 |                                                        |
|0 1 0 0 0 0 0 0 0 0 |                                                        |
|0 0 1 0 0 0 0 0 0 0 |                                                        |
|0 0 0 1 0 0 0 0 0 0 |                                                        |
|0 0 0 0 1 0 0 0 0 0 |                                                        |
|0 0 0 0 0 1 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 1 |                                                        |
|                                                                             |
|                         35                                                  |
|----------------------- S   : (4, 1)                                         |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|1 0 0 0 0 0 0 0 0 0 |                                                        |
|0 1 0 0 0 0 0 0 0 0 |                                                        |
|0 0 1 0 0 0 0 0 0 0 |                                                        |
|0 0 0 1 0 0 0 0 0 0 |                                                        |
|0 0 0 0 1 0 0 0 0 0 |                                                        |
|0 0 0 0 0 1 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 1 1 |                                                        |
|                                                                             |
|                         35                                                  |
|----------------------- S   : (4, 2)                                         |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|1 0 0 1 0 0 0 0 0 0 |                                                        |
|0 1 0 0 1 0 0 0 0 0 |                                                        |
|0 0 1 0 0 1 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 1 1 0 |                                                        |
|                                                                             |
|                         35                                                  |
|----------------------- S   : (4, 3)                                         |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|1 0 0 0 0 0 0 0 0 0 |                                                        |
|0 1 0 0 0 0 0 0 0 0 |                                                        |
|0 0 1 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 1 1 0 0 |                                                        |
|                                                                             |
|                         35                                                  |
|----------------------- S   : (4, 4)                                         |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 1 0 0 0 |                                                        |
+-----------------------------------------------------------------------------+
|i8 : C == E                                                                  |
|                                                                             |
|o8 = false                                                                   |
+-----------------------------------------------------------------------------+
|i9 : E == 0                                                                  |
|                                                                             |
|o9 = false                                                                   |
+-----------------------------------------------------------------------------+

A simplicial module is equal to zero if all the objects and maps are zero. This
could require computation to determine if something that is superficially not
zero is in fact zero.

+-----------------------------------------------------------------------------+
|i10 : f = id_C                                                               |
|                                                                             |
|           1             1                                                   |
|o10 = 0 : S  <--------- S  : 0                                               |
|                | 1 |                                                        |
|                                                                             |
|           4                       4                                         |
|      1 : S  <------------------- S  : 1                                     |
|                {0} | 1 0 0 0 |                                              |
|                {1} | 0 1 0 0 |                                              |
|                {1} | 0 0 1 0 |                                              |
|                {1} | 0 0 0 1 |                                              |
|                                                                             |
|           10                                   10                           |
|      2 : S   <------------------------------- S   : 2                       |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 |                                 |
|                 {1} | 0 1 0 0 0 0 0 0 0 0 |                                 |
|                 {1} | 0 0 1 0 0 0 0 0 0 0 |                                 |
|                 {1} | 0 0 0 1 0 0 0 0 0 0 |                                 |
|                 {1} | 0 0 0 0 1 0 0 0 0 0 |                                 |
|                 {1} | 0 0 0 0 0 1 0 0 0 0 |                                 |
|                 {1} | 0 0 0 0 0 0 1 0 0 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 1 0 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 0 1 0 |                                 |
|                 {2} | 0 0 0 0 0 0 0 0 0 1 |                                 |
|                                                                             |
|           20                                                       20       |
|      3 : S   <--------------------------------------------------- S   : 3   |
|                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |             |
|                 {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |             |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |             |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |             |
|                                                                             |
|           35                                                                |
|      4 : S   <--------------------------------------------------------------|
|                 {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|o10 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                     35                                                      |
|------------------- S   : 4                                                  |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|0 0 0 0 0 0 0 0 |                                                            |
|1 0 0 0 0 0 0 0 |                                                            |
|0 1 0 0 0 0 0 0 |                                                            |
|0 0 1 0 0 0 0 0 |                                                            |
|0 0 0 1 0 0 0 0 |                                                            |
|0 0 0 0 1 0 0 0 |                                                            |
|0 0 0 0 0 1 0 0 |                                                            |
|0 0 0 0 0 0 1 0 |                                                            |
|0 0 0 0 0 0 0 1 |                                                            |
+-----------------------------------------------------------------------------+
|i11 : D = coker f                                                            |
|                                                                             |
|o11 = cokernel | 1 | <-- cokernel {0} | 1 0 0 0 | <-- cokernel {0} | 1 0 0 0 |
|                                  {1} | 0 1 0 0 |              {1} | 0 1 0 0 |
|      0                           {1} | 0 0 1 0 |              {1} | 0 0 1 0 |
|                                  {1} | 0 0 0 1 |              {1} | 0 0 0 1 |
|                                                               {1} | 0 0 0 0 |
|                         1                                     {1} | 0 0 0 0 |
|                                                               {1} | 0 0 0 0 |
|                                                               {2} | 0 0 0 0 |
|                                                               {2} | 0 0 0 0 |
|                                                               {2} | 0 0 0 0 |
|                                                                             |
|                                                      2                      |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o11 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 | <-- cokernel {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|0 0 0 0 0 0 |              {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|0 0 0 0 0 0 |              {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|0 0 0 0 0 0 |              {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|1 0 0 0 0 0 |              {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|0 1 0 0 0 0 |              {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|0 0 1 0 0 0 |              {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|0 0 0 1 0 0 |              {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |   |
|0 0 0 0 1 0 |              {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |   |
|0 0 0 0 0 1 |              {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |   |
|                           {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |   |
|                           {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |   |
|                           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |   |
|                           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |   |
|                           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |   |
|                           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |   |
|                           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |   |
|                           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |   |
|                           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |   |
|                           {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |   |
|                                                                             |
|                  3                                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|<-- cokernel {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|             {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|    4                                                                        |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 |<-- ...                                                         |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|1 0 0 0 0 0 |                                                                |
|0 1 0 0 0 0 |                                                                |
|0 0 1 0 0 0 |                                                                |
|0 0 0 1 0 0 |                                                                |
|0 0 0 0 1 0 |                                                                |
|0 0 0 0 0 1 |                                                                |
+-----------------------------------------------------------------------------+
|i12 : D == 0                                                                 |
|                                                                             |
|o12 = true                                                                   |
+-----------------------------------------------------------------------------+
+---------------------------------------------------------------------------+
|i13 : C0 = simplicialModule( S^0, 5, Degeneracy => true)                   |
|                                                                           |
|o13 = 0 <-- 0 <-- 0 <-- 0 <-- 0 <-- 0<-- ...                               |
|                                                                           |
|      0     1     2     3     4     5                                      |
|                                                                           |
|o13 : SimplicialModule                                                     |
+---------------------------------------------------------------------------+
|i14 : C1 = simplicialModule(complex(S^0, Base => 2), 5, Degeneracy => true)|
|                                                                           |
|o14 = 0 <-- 0 <-- 0 <-- 0 <-- 0 <-- 0<-- ...                               |
|                                                                           |
|      0     1     2     3     4     5                                      |
|                                                                           |
|o14 : SimplicialModule                                                     |
+---------------------------------------------------------------------------+
|i15 : topDegree C0 == topDegree C1                                         |
|                                                                           |
|o15 = true                                                                 |
+---------------------------------------------------------------------------+
|i16 : C0 == C1                                                             |
|                                                                           |
|o16 = true                                                                 |
+---------------------------------------------------------------------------+
|i17 : C0 == 0                                                              |
|                                                                           |
|o17 = true                                                                 |
+---------------------------------------------------------------------------+
|i18 : C1 == 0                                                              |
|                                                                           |
|o18 = true                                                                 |
+---------------------------------------------------------------------------+

Caveat
======

Testing for equality is not the same as testing for isomorphism.

Ways to use this method:
========================

  * SimplicialModule == SimplicialModule
    -- whether two simplicial modules are equal
  * "SimplicialModule == ZZ"
  * "ZZ == SimplicialModule"
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:735:0.

SimplicialModule _ Array -- the canonical inclusion or projection map of a direct sum
*************************************************************************************

  * Operator: _ (Macaulay2Doc),
  * Usage:
        i = C_[name]
        p = C^[name]
  * Inputs:
      * C, a Simplicial Module,
      * name, an array (Macaulay2Doc),
  * Outputs:
      * a Map of Simplicial Modules, i is the
        canonical inclusion and p is the canonical projection

Description
===========

The direct sum is an n-ary operator with projection and inclusion maps from
each component satisfying appropriate identities.

One can access these maps as follows.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a,b,c];                                                      |
+-----------------------------------------------------------------------------+
|i2 : C1 = simplicialModule(freeResolution coker vars S, 5, Degeneracy =>     |
|                                                                             |
|      1      4      10      20      35      56                               |
|o2 = S  <-- S  <-- S   <-- S   <-- S   <-- S  <-- ...                        |
|                                                                             |
|     0      1      2       3       4       5                                 |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|true)                                                                        |
+-----------------------------------------------------------------------------+
|i3 : C2 = simplicialModule(complex (ideal(a,b,c)) , 5, Degeneracy => true)   |
|                                                                             |
|o3 = image | a b c | <-- image | a b c | <-- image | a b c | <-- image | a b |
|                                                                             |
|     0                   1                   2                   3           |
|                                                                             |
|o3 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|c | <-- image | a b c | <-- image | a b c |<-- ...                           |
|                                                                             |
|        4                   5                                                |
+-----------------------------------------------------------------------------+
|i4 : D = C1 ++ C2                                                            |
|                                                                             |
|o4 = image | 1 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 | <-- image {0} | 1 0 0 |
|           | 0 a b c |           {1} | 0 1 0 0 0 0 0 |           {1} | 0 1 0 |
|                                 {1} | 0 0 1 0 0 0 0 |           {1} | 0 0 1 |
|     0                           {1} | 0 0 0 1 0 0 0 |           {1} | 0 0 0 |
|                                 {0} | 0 0 0 0 a b c |           {1} | 0 0 0 |
|                                                                 {1} | 0 0 0 |
|                           1                                     {1} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {2} | 0 0 0 |
|                                                                 {0} | 0 0 0 |
|                                                                             |
|                                                           2                 |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
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|i5 : D_[0]                                                                   |
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|o5 = 0 : image | 1 0 0 0 |                                                   |
|               | 0 a b c |                                                   |
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|     1 : image {0} | 1 0 0                                                   |
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|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {0} | 0 0 0                                                   |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|     3 : image {0} | 1 0 0                                                   |
|               {1} | 0 1 0                                                   |
|               {1} | 0 0 1                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {0} | 0 0 0                                                   |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|     4 : image {0} | 1 0 0                                                   |
|               {1} | 0 1 0                                                   |
|               {1} | 0 0 1                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {0} | 0 0 0                                                   |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|     5 : image {0} | 1 0 0                                                   |
|               {1} | 0 1 0                                                   |
|               {1} | 0 0 1                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {0} | 0 0 0                                                   |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o5 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                1                                                            |
|<------------- S  : 0                                                        |
|   {0} | 1 |                                                                 |
|   {1} | 0 |                                                                 |
|   {1} | 0 |                                                                 |
|   {1} | 0 |                                                                 |
|                                                                             |
|                                4                                            |
|0 0 0 0 | <------------------- S  : 1                                        |
|0 0 0 0 |    {0} | 1 0 0 0 |                                                 |
|0 0 0 0 |    {1} | 0 1 0 0 |                                                 |
|1 0 0 0 |    {1} | 0 0 1 0 |                                                 |
|0 a b c |    {1} | 0 0 0 1 |                                                 |
|             {1} | 0 0 0 0 |                                                 |
|             {1} | 0 0 0 0 |                                                 |
|             {1} | 0 0 0 0 |                                                 |
|                                                                             |
|                                                        10                   |
|0 0 0 0 0 0 0 0 0 0 | <------------------------------- S   : 2               |
|0 0 0 0 0 0 0 0 0 0 |    {0} | 1 0 0 0 0 0 0 0 0 0 |                         |
|0 0 0 0 0 0 0 0 0 0 |    {1} | 0 1 0 0 0 0 0 0 0 0 |                         |
|1 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 1 0 0 0 0 0 0 0 |                         |
|0 1 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 1 0 0 0 0 0 0 |                         |
|0 0 1 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 1 0 0 0 0 0 |                         |
|0 0 0 1 0 0 0 0 0 0 |    {1} | 0 0 0 0 0 1 0 0 0 0 |                         |
|0 0 0 0 1 0 0 0 0 0 |    {1} | 0 0 0 0 0 0 1 0 0 0 |                         |
|0 0 0 0 0 1 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 1 0 0 |                         |
|0 0 0 0 0 0 1 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 1 0 |                         |
|0 0 0 0 0 0 0 a b c |    {2} | 0 0 0 0 0 0 0 0 0 1 |                         |
|                         {1} | 0 0 0 0 0 0 0 0 0 0 |                         |
|                         {1} | 0 0 0 0 0 0 0 0 0 0 |                         |
|                         {1} | 0 0 0 0 0 0 0 0 0 0 |                         |
|                                                                             |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <----------------------------------|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 |
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 |
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 |
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 |
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 |
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 |
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 |
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 |
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 |
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 |
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                             {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                             {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                             {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <----|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |    {3|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |    {3|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |    {3|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |    {3|
|                                                                           {1|
|                                                                           {1|
|                                                                           {1|
|                                                                             |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                   20                                                        |
|----------------- S   : 3                                                    |
|0 0 0 0 0 0 0 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|1 0 0 0 0 0 0 |                                                              |
|0 1 0 0 0 0 0 |                                                              |
|0 0 1 0 0 0 0 |                                                              |
|0 0 0 1 0 0 0 |                                                              |
|0 0 0 0 1 0 0 |                                                              |
|0 0 0 0 0 1 0 |                                                              |
|0 0 0 0 0 0 1 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|0 0 0 0 0 0 0 |                                                              |
|                                                                             |
|                                                                             |
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|i6 : isCommutative D_[0]                                                     |
|                                                                             |
|o6 = true                                                                    |
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|i7 : D_[1]                                                                   |
|                                                                             |
|o7 = 0 : image | 1 0 0 0 |                                                   |
|               | 0 a b c |                                                   |
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|     1 : image {0} | 1 0 0                                                   |
|               {1} | 0 1 0                                                   |
|               {1} | 0 0 1                                                   |
|               {1} | 0 0 0                                                   |
|               {0} | 0 0 0                                                   |
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|     2 : image {0} | 1 0 0                                                   |
|               {1} | 0 1 0                                                   |
|               {1} | 0 0 1                                                   |
|               {1} | 0 0 0                                                   |
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|               {0} | 0 0 0                                                   |
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|     3 : image {0} | 1 0 0                                                   |
|               {1} | 0 1 0                                                   |
|               {1} | 0 0 1                                                   |
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|               {1} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {0} | 0 0 0                                                   |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|     4 : image {0} | 1 0 0                                                   |
|               {1} | 0 1 0                                                   |
|               {1} | 0 0 1                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {0} | 0 0 0                                                   |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|     5 : image {0} | 1 0 0                                                   |
|               {1} | 0 1 0                                                   |
|               {1} | 0 0 1                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {1} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {2} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {3} | 0 0 0                                                   |
|               {0} | 0 0 0                                                   |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|<----------------- image | a b c | : 0                                       |
|   {0} | 0 0 0 |                                                             |
|   {1} | 1 0 0 |                                                             |
|   {1} | 0 1 0 |                                                             |
|   {1} | 0 0 1 |                                                             |
|                                                                             |
|0 0 0 0 | <----------------- image | a b c | : 1                             |
|0 0 0 0 |    {0} | 0 0 0 |                                                   |
|0 0 0 0 |    {1} | 0 0 0 |                                                   |
|1 0 0 0 |    {1} | 0 0 0 |                                                   |
|0 a b c |    {1} | 0 0 0 |                                                   |
|             {1} | 1 0 0 |                                                   |
|             {1} | 0 1 0 |                                                   |
|             {1} | 0 0 1 |                                                   |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 | <----------------- image | a b c | : 2                 |
|0 0 0 0 0 0 0 0 0 0 |    {0} | 0 0 0 |                                       |
|0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|1 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 1 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 0 1 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 0 0 1 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 0 0 0 1 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 0 0 0 0 1 0 0 0 0 |    {2} | 0 0 0 |                                       |
|0 0 0 0 0 0 1 0 0 0 |    {2} | 0 0 0 |                                       |
|0 0 0 0 0 0 0 a b c |    {2} | 0 0 0 |                                       |
|                         {1} | 1 0 0 |                                       |
|                         {1} | 0 1 0 |                                       |
|                         {1} | 0 0 1 |                                       |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <----------------- image | a b     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {0} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |    {3} | 0 0 0 |                   |
|                                             {1} | 1 0 0 |                   |
|                                             {1} | 0 1 0 |                   |
|                                             {1} | 0 0 1 |                   |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |      |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
| c | : 3                                                                     |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|<----------------- image | a b c | : 4                                       |
|   {0} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {3} | 0 0 0 |                                                             |
|   {3} | 0 0 0 |                                                             |
|   {3} | 0 0 0 |                                                             |
|   {3} | 0 0 0 |                                                             |
|   {1} | 1 0 0 |                                                             |
|   {1} | 0 1 0 |                                                             |
|   {1} | 0 0 1 |                                                             |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <----------------- image | a b c | |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {0} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |    {3} | 0 0 0 |                   |
|                                             {1} | 1 0 0 |                   |
|                                             {1} | 0 1 0 |                   |
|                                             {1} | 0 0 1 |                   |
|-----------------------------------------------------------------------------|
|: 5                                                                          |
+-----------------------------------------------------------------------------+
|i8 : D^[0] * D_[0] == 1                                                      |
|                                                                             |
|o8 = true                                                                    |
+-----------------------------------------------------------------------------+
|i9 : D^[1] * D_[1] == 1                                                      |
|                                                                             |
|o9 = true                                                                    |
+-----------------------------------------------------------------------------+
|i10 : D^[0] * D_[1] == 0                                                     |
|                                                                             |
|o10 = true                                                                   |
+-----------------------------------------------------------------------------+
|i11 : D^[1] * D_[0] == 0                                                     |
|                                                                             |
|o11 = true                                                                   |
+-----------------------------------------------------------------------------+
|i12 : D_[0] * D^[0] + D_[1] * D^[1] == 1                                     |
|                                                                             |
|o12 = true                                                                   |
+-----------------------------------------------------------------------------+

The default names for the components are the non-negative integers.  However,
one can choose any name.

+-----------------------------------------------------------------------------+
|i13 : E = (chicken => C1) ++ (nuggets => C2)                                 |
|                                                                             |
|o13 = image | 1 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 | <-- image {0} | 1 0 0|
|            | 0 a b c |           {1} | 0 1 0 0 0 0 0 |           {1} | 0 1 0|
|                                  {1} | 0 0 1 0 0 0 0 |           {1} | 0 0 1|
|      0                           {1} | 0 0 0 1 0 0 0 |           {1} | 0 0 0|
|                                  {0} | 0 0 0 0 a b c |           {1} | 0 0 0|
|                                                                  {1} | 0 0 0|
|                            1                                     {1} | 0 0 0|
|                                                                  {2} | 0 0 0|
|                                                                  {2} | 0 0 0|
|                                                                  {2} | 0 0 0|
|                                                                  {0} | 0 0 0|
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|o13 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 |           {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|1 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 1 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 1 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 1 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 1 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 1 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 1 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 a b c |           {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0|
|                                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|                                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|                                {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|                          3                                                  |
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|-----------------------------------------------------------------------------|
|0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
|a b c |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0|
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                  {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|            4                                                                |
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|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|1 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 1 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 1 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 1 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 1 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 1 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 1 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 1 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 a b c |           {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                  {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|                            5                                                |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
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|-----------------------------------------------------------------------------|
|0 |<-- ...                                                                   |
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|c |                                                                          |
+-----------------------------------------------------------------------------+
|i14 : E_[chicken]                                                            |
|                                                                             |
|                                                                             |
|o14 = 0 : image | 1 0 0 0 |                                                  |
|                | 0 a b c |                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      1 : image {0} | 1 0 0                                                  |
|                {1} | 0 1 0                                                  |
|                {1} | 0 0 1                                                  |
|                {1} | 0 0 0                                                  |
|                {0} | 0 0 0                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      2 : image {0} | 1 0 0                                                  |
|                {1} | 0 1 0                                                  |
|                {1} | 0 0 1                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {0} | 0 0 0                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      3 : image {0} | 1 0 0                                                  |
|                {1} | 0 1 0                                                  |
|                {1} | 0 0 1                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {0} | 0 0 0                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      4 : image {0} | 1 0 0                                                  |
|                {1} | 0 1 0                                                  |
|                {1} | 0 0 1                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {0} | 0 0 0                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      5 : image {0} | 1 0 0                                                  |
|                {1} | 0 1 0                                                  |
|                {1} | 0 0 1                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {0} | 0 0 0                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o14 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                1                                                            |
|<------------- S  : 0                                                        |
|   {0} | 1 |                                                                 |
|   {1} | 0 |                                                                 |
|   {1} | 0 |                                                                 |
|   {1} | 0 |                                                                 |
|                                                                             |
|                                4                                            |
|0 0 0 0 | <------------------- S  : 1                                        |
|0 0 0 0 |    {0} | 1 0 0 0 |                                                 |
|0 0 0 0 |    {1} | 0 1 0 0 |                                                 |
|1 0 0 0 |    {1} | 0 0 1 0 |                                                 |
|0 a b c |    {1} | 0 0 0 1 |                                                 |
|             {1} | 0 0 0 0 |                                                 |
|             {1} | 0 0 0 0 |                                                 |
|             {1} | 0 0 0 0 |                                                 |
|                                                                             |
|                                                        10                   |
|0 0 0 0 0 0 0 0 0 0 | <------------------------------- S   : 2               |
|0 0 0 0 0 0 0 0 0 0 |    {0} | 1 0 0 0 0 0 0 0 0 0 |                         |
|0 0 0 0 0 0 0 0 0 0 |    {1} | 0 1 0 0 0 0 0 0 0 0 |                         |
|1 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 1 0 0 0 0 0 0 0 |                         |
|0 1 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 1 0 0 0 0 0 0 |                         |
|0 0 1 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 1 0 0 0 0 0 |                         |
|0 0 0 1 0 0 0 0 0 0 |    {1} | 0 0 0 0 0 1 0 0 0 0 |                         |
|0 0 0 0 1 0 0 0 0 0 |    {1} | 0 0 0 0 0 0 1 0 0 0 |                         |
|0 0 0 0 0 1 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 1 0 0 |                         |
|0 0 0 0 0 0 1 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 1 0 |                         |
|0 0 0 0 0 0 0 a b c |    {2} | 0 0 0 0 0 0 0 0 0 1 |                         |
|                         {1} | 0 0 0 0 0 0 0 0 0 0 |                         |
|                         {1} | 0 0 0 0 0 0 0 0 0 0 |                         |
|                         {1} | 0 0 0 0 0 0 0 0 0 0 |                         |
|                                                                             |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <----------------------------------|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 |
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 |
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 |
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 |
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 |
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 |
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 |
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 |
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 |
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 |
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |    {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                             {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                             {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                             {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <----|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1|
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |    {2|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |    {3|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |    {3|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |    {3|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |    {3|
|                                                                           {1|
|                                                                           {1|
|                                                                           {1|
|                                                                             |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
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|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
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|-----------------------------------------------------------------------------|
|         56                                                                  |
|------- S   : 5                                                              |
|0 0 |                                                                        |
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|1 0 |                                                                        |
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+-----------------------------------------------------------------------------+
|i15 : E_[nuggets]                                                            |
|                                                                             |
|o15 = 0 : image | 1 0 0 0 |                                                  |
|                | 0 a b c |                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      1 : image {0} | 1 0 0                                                  |
|                {1} | 0 1 0                                                  |
|                {1} | 0 0 1                                                  |
|                {1} | 0 0 0                                                  |
|                {0} | 0 0 0                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      2 : image {0} | 1 0 0                                                  |
|                {1} | 0 1 0                                                  |
|                {1} | 0 0 1                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {0} | 0 0 0                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      3 : image {0} | 1 0 0                                                  |
|                {1} | 0 1 0                                                  |
|                {1} | 0 0 1                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {0} | 0 0 0                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      4 : image {0} | 1 0 0                                                  |
|                {1} | 0 1 0                                                  |
|                {1} | 0 0 1                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {0} | 0 0 0                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      5 : image {0} | 1 0 0                                                  |
|                {1} | 0 1 0                                                  |
|                {1} | 0 0 1                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {1} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
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|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
|                {2} | 0 0 0                                                  |
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|                {2} | 0 0 0                                                  |
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|                {2} | 0 0 0                                                  |
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|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
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|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
|                {3} | 0 0 0                                                  |
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|                {0} | 0 0 0                                                  |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o15 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|<----------------- image | a b c | : 0                                       |
|   {0} | 0 0 0 |                                                             |
|   {1} | 1 0 0 |                                                             |
|   {1} | 0 1 0 |                                                             |
|   {1} | 0 0 1 |                                                             |
|                                                                             |
|0 0 0 0 | <----------------- image | a b c | : 1                             |
|0 0 0 0 |    {0} | 0 0 0 |                                                   |
|0 0 0 0 |    {1} | 0 0 0 |                                                   |
|1 0 0 0 |    {1} | 0 0 0 |                                                   |
|0 a b c |    {1} | 0 0 0 |                                                   |
|             {1} | 1 0 0 |                                                   |
|             {1} | 0 1 0 |                                                   |
|             {1} | 0 0 1 |                                                   |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 | <----------------- image | a b c | : 2                 |
|0 0 0 0 0 0 0 0 0 0 |    {0} | 0 0 0 |                                       |
|0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|1 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 1 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 0 1 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 0 0 1 0 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 0 0 0 1 0 0 0 0 0 |    {1} | 0 0 0 |                                       |
|0 0 0 0 0 1 0 0 0 0 |    {2} | 0 0 0 |                                       |
|0 0 0 0 0 0 1 0 0 0 |    {2} | 0 0 0 |                                       |
|0 0 0 0 0 0 0 a b c |    {2} | 0 0 0 |                                       |
|                         {1} | 1 0 0 |                                       |
|                         {1} | 0 1 0 |                                       |
|                         {1} | 0 0 1 |                                       |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <----------------- image | a b     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {0} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
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|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
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|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |    {3} | 0 0 0 |                   |
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|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      |
|                                                                             |
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|                                                                             |
|-----------------------------------------------------------------------------|
| c | : 3                                                                     |
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|<----------------- image | a b c | : 4                                       |
|   {0} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {1} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {2} | 0 0 0 |                                                             |
|   {3} | 0 0 0 |                                                             |
|   {3} | 0 0 0 |                                                             |
|   {3} | 0 0 0 |                                                             |
|   {3} | 0 0 0 |                                                             |
|   {1} | 1 0 0 |                                                             |
|   {1} | 0 1 0 |                                                             |
|   {1} | 0 0 1 |                                                             |
|                                                                             |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <----------------- image | a b c | |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {0} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {1} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |    {2} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |    {3} | 0 0 0 |                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c |    {3} | 0 0 0 |                   |
|                                             {1} | 1 0 0 |                   |
|                                             {1} | 0 1 0 |                   |
|                                             {1} | 0 0 1 |                   |
|-----------------------------------------------------------------------------|
|: 5                                                                          |
+-----------------------------------------------------------------------------+
|i16 : E^[chicken] * E_[chicken] == 1                                         |
|                                                                             |
|o16 = true                                                                   |
+-----------------------------------------------------------------------------+
|i17 : E^[nuggets] * E_[nuggets] == 1                                         |
|                                                                             |
|o17 = true                                                                   |
+-----------------------------------------------------------------------------+
|i18 : E^[chicken] * E_[nuggets] == 0                                         |
|                                                                             |
|o18 = true                                                                   |
+-----------------------------------------------------------------------------+
|i19 : E^[nuggets] * E_[chicken] == 0                                         |
|                                                                             |
|o19 = true                                                                   |
+-----------------------------------------------------------------------------+
|i20 : E_[chicken] * E^[chicken] + E_[nuggets] * E^[nuggets] == 1             |
|                                                                             |
|o20 = true                                                                   |
+-----------------------------------------------------------------------------+

One can also access inclusion and projection maps of sub-direct sums.

+-----------------------------------------------------------------------------+
|i21 : F = directSum(C1, C2, simplicialModule(complex(S^2, Base => 1),        |
|                                                                             |
|o21 = image | 1 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 | <-- image        |
|            | 0 a b c |           {1} | 0 1 0 0 0 0 0 0 0 |                  |
|                                  {1} | 0 0 1 0 0 0 0 0 0 |                  |
|      0                           {1} | 0 0 0 1 0 0 0 0 0 |                  |
|                                  {0} | 0 0 0 0 a b c 0 0 |                  |
|                                  {0} | 0 0 0 0 0 0 0 1 0 |                  |
|                                  {0} | 0 0 0 0 0 0 0 0 1 |                  |
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|o21 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|5, Degeneracy => true))                                                      |
|                                                                             |
|{0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0|
|{1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 1 0 0 0 0 0 0 0 0|
|{1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0|
|{1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0|
|{1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 1 0 0 0 0 0|
|{1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 1 0 0 0 0|
|{1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 1 0 0 0|
|{2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 1 0 0|
|{2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 0|
|{2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 1|
|{0} | 0 0 0 0 0 0 0 0 0 0 a b c 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0|
|{0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0|
|{0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0|
|{0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |           {2} | 0 0 0 0 0 0 0 0 0 0|
|{0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |           {2} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {2} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {2} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {2} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {2} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {3} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {0} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {0} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {0} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {0} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {0} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {0} | 0 0 0 0 0 0 0 0 0 0|
|                                                    {0} | 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|                                              3                              |
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|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 1 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 1 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 1 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 1 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 1 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 1 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 1 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 1 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 1 0|
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 0 1|
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 a b c 0 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |           {2} | 0 0 0 0 0 0 0 0 0 0 0|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |           {2} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {2} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {2} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {2} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {2} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {3} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {3} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {3} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {3} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {0} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {0} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {0} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {0} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {0} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {0} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {0} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {0} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                  {0} | 0 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|                                            4                                |
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|i22 : prune (F^[0,1])                                                        |
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|      2 : cokernel {0} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {1} | -b 0  -c |                                          |
|                   {1} | a  -c 0  |                                          |
|                   {1} | 0  b  a  |                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      3 : cokernel {0} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {1} | -b 0  -c |                                          |
|                   {1} | a  -c 0  |                                          |
|                   {1} | 0  b  a  |                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      4 : cokernel {0} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {1} | -b 0  -c |                                          |
|                   {1} | a  -c 0  |                                          |
|                   {1} | 0  b  a  |                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|      5 : cokernel {0} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {1} | -b 0  -c |                                          |
|                   {1} | a  -c 0  |                                          |
|                   {1} | 0  b  a  |                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o22 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|<------------------- cokernel {0} | 0  0  0  | : 0                           |
|   {0} | 1 0 0 0 |            {1} | -b 0  -c |                               |
|   {1} | 0 1 0 0 |            {1} | a  -c 0  |                               |
|   {1} | 0 0 1 0 |            {1} | 0  b  a  |                               |
|   {1} | 0 0 0 1 |                                                           |
|                                                                             |
|<----------------------------- cokernel {0} | 0  0  0  | : 1                 |
|   {0} | 1 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |                     |
|   {1} | 0 1 0 0 0 0 0 0 0 |            {1} | 0  0  0  |                     |
|   {1} | 0 0 1 0 0 0 0 0 0 |            {1} | 0  0  0  |                     |
|   {1} | 0 0 0 1 0 0 0 0 0 |            {1} | -b 0  -c |                     |
|   {1} | 0 0 0 0 1 0 0 0 0 |            {1} | a  -c 0  |                     |
|   {1} | 0 0 0 0 0 1 0 0 0 |            {1} | 0  b  a  |                     |
|   {1} | 0 0 0 0 0 0 1 0 0 |            {0} | 0  0  0  |                     |
|                                        {0} | 0  0  0  |                     |
|                                                                             |
|<--------------------------------------------- cokernel {0} | 0  0  0  | : 2 |
|   {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |     |
|   {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |     |
|   {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |     |
|   {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |     |
|   {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |     |
|   {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |     |
|   {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |     |
|   {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |     |
|   {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |     |
|   {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |            {1} | -b 0  -c |     |
|   {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |            {1} | a  -c 0  |     |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |            {1} | 0  b  a  |     |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |            {0} | 0  0  0  |     |
|                                                        {0} | 0  0  0  |     |
|                                                        {0} | 0  0  0  |     |
|                                                        {0} | 0  0  0  |     |
|                                                                             |
|<--------------------------------------------------------------------- cokern|
|   {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|   {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|   {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|   {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|   {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|   {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|   {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|   {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|   {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|   {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|   {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
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|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |         |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |         |
|   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |         |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |         |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |         |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |         |
|                                                                             |
|                                                                             |
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|                                                                             |
|                                                                             |
|<----------------------------------------------------------------------------|
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|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
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|                                                                             |
|                                                                             |
|                                                                             |
|<----------------------------------------------------------------------------|
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|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|el {0} | 0  0  0  | : 3                                                      |
|   {1} | 0  0  0  |                                                          |
|   {1} | 0  0  0  |                                                          |
|   {1} | 0  0  0  |                                                          |
|   {1} | 0  0  0  |                                                          |
|   {1} | 0  0  0  |                                                          |
|   {1} | 0  0  0  |                                                          |
|   {1} | 0  0  0  |                                                          |
|   {1} | 0  0  0  |                                                          |
|   {1} | 0  0  0  |                                                          |
|   {2} | 0  0  0  |                                                          |
|   {2} | 0  0  0  |                                                          |
|   {2} | 0  0  0  |                                                          |
|   {2} | 0  0  0  |                                                          |
|   {2} | 0  0  0  |                                                          |
|   {2} | 0  0  0  |                                                          |
|   {2} | 0  0  0  |                                                          |
|   {2} | 0  0  0  |                                                          |
|   {2} | 0  0  0  |                                                          |
|   {3} | 0  0  0  |                                                          |
|   {1} | -b 0  -c |                                                          |
|   {1} | a  -c 0  |                                                          |
|   {1} | 0  b  a  |                                                          |
|   {0} | 0  0  0  |                                                          |
|   {0} | 0  0  0  |                                                          |
|   {0} | 0  0  0  |                                                          |
|   {0} | 0  0  0  |                                                          |
|   {0} | 0  0  0  |                                                          |
|   {0} | 0  0  0  |                                                          |
|                                                                             |
|--------------------------- cokernel {0} | 0  0  0  | : 4                    |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {1} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {2} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {3} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {3} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {3} | 0  0  0  |                        |
|0 0 0 0 0 0 0 0 0 0 0 0 |            {3} | 0  0  0  |                        |
|1 0 0 0 0 0 0 0 0 0 0 0 |            {1} | -b 0  -c |                        |
|0 1 0 0 0 0 0 0 0 0 0 0 |            {1} | a  -c 0  |                        |
|0 0 1 0 0 0 0 0 0 0 0 0 |            {1} | 0  b  a  |                        |
|0 0 0 1 0 0 0 0 0 0 0 0 |            {0} | 0  0  0  |                        |
|                                     {0} | 0  0  0  |                        |
|                                     {0} | 0  0  0  |                        |
|                                     {0} | 0  0  0  |                        |
|                                     {0} | 0  0  0  |                        |
|                                     {0} | 0  0  0  |                        |
|                                     {0} | 0  0  0  |                        |
|                                     {0} | 0  0  0  |                        |
|                                                                             |
|-------------------------------------------------------------------------    |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |      |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |      |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|cokernel {0} | 0  0  0  | : 5                                                |
|         {1} | 0  0  0  |                                                    |
|         {1} | 0  0  0  |                                                    |
|         {1} | 0  0  0  |                                                    |
|         {1} | 0  0  0  |                                                    |
|         {1} | 0  0  0  |                                                    |
|         {1} | 0  0  0  |                                                    |
|         {1} | 0  0  0  |                                                    |
|         {1} | 0  0  0  |                                                    |
|         {1} | 0  0  0  |                                                    |
|         {1} | 0  0  0  |                                                    |
|         {1} | 0  0  0  |                                                    |
|         {1} | 0  0  0  |                                                    |
|         {1} | 0  0  0  |                                                    |
|         {1} | 0  0  0  |                                                    |
|         {1} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {2} | 0  0  0  |                                                    |
|         {3} | 0  0  0  |                                                    |
|         {3} | 0  0  0  |                                                    |
|         {3} | 0  0  0  |                                                    |
|         {3} | 0  0  0  |                                                    |
|         {3} | 0  0  0  |                                                    |
|         {3} | 0  0  0  |                                                    |
|         {3} | 0  0  0  |                                                    |
|         {3} | 0  0  0  |                                                    |
|         {3} | 0  0  0  |                                                    |
|         {3} | 0  0  0  |                                                    |
|         {1} | -b 0  -c |                                                    |
|         {1} | a  -c 0  |                                                    |
|         {1} | 0  b  a  |                                                    |
|         {0} | 0  0  0  |                                                    |
|         {0} | 0  0  0  |                                                    |
|         {0} | 0  0  0  |                                                    |
|         {0} | 0  0  0  |                                                    |
|         {0} | 0  0  0  |                                                    |
|         {0} | 0  0  0  |                                                    |
|         {0} | 0  0  0  |                                                    |
|         {0} | 0  0  0  |                                                    |
|         {0} | 0  0  0  |                                                    |
|         {0} | 0  0  0  |                                                    |
+-----------------------------------------------------------------------------+
|i23 : isSimplicialMorphism oo                                                |
|                                                                             |
|o23 = true                                                                   |
+-----------------------------------------------------------------------------+
|i24 : prune (F_[0,2])                                                        |
|                                                                             |
|                                                                             |
|o24 = 0 : cokernel {0} | 0  0  0  |                                          |
|                   {1} | -b 0  -c |                                          |
|                   {1} | a  -c 0  |                                          |
|                   {1} | 0  b  a  |                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|      1 : cokernel {0} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | -b 0  -c |                                          |
|                   {1} | a  -c 0  |                                          |
|                   {1} | 0  b  a  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|      2 : cokernel {0} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {1} | -b 0  -c |                                          |
|                   {1} | a  -c 0  |                                          |
|                   {1} | 0  b  a  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|      3 : cokernel {0} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {1} | -b 0  -c |                                          |
|                   {1} | a  -c 0  |                                          |
|                   {1} | 0  b  a  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|      4 : cokernel {0} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {1} | -b 0  -c |                                          |
|                   {1} | a  -c 0  |                                          |
|                   {1} | 0  b  a  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|      5 : cokernel {0} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {1} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {2} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {3} | 0  0  0  |                                          |
|                   {1} | -b 0  -c |                                          |
|                   {1} | a  -c 0  |                                          |
|                   {1} | 0  b  a  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                   {0} | 0  0  0  |                                          |
|                                                                             |
|                                                                             |
|o24 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                1                                                            |
|<------------- S  : 0                                                        |
|   {0} | 1 |                                                                 |
|   {1} | 0 |                                                                 |
|   {1} | 0 |                                                                 |
|   {1} | 0 |                                                                 |
|                                                                             |
|                          6                                                  |
|<----------------------- S  : 1                                              |
|   {0} | 1 0 0 0 0 0 |                                                       |
|   {1} | 0 1 0 0 0 0 |                                                       |
|   {1} | 0 0 1 0 0 0 |                                                       |
|   {1} | 0 0 0 1 0 0 |                                                       |
|   {1} | 0 0 0 0 0 0 |                                                       |
|   {1} | 0 0 0 0 0 0 |                                                       |
|   {1} | 0 0 0 0 0 0 |                                                       |
|   {0} | 0 0 0 0 1 0 |                                                       |
|   {0} | 0 0 0 0 0 1 |                                                       |
|                                                                             |
|                                          14                                 |
|<--------------------------------------- S   : 2                             |
|   {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                       |
|   {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                                       |
|   {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                                       |
|   {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                                       |
|   {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                                       |
|   {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                                       |
|   {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                                       |
|   {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                                       |
|   {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                                       |
|   {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                                       |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                       |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                       |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                       |
|   {0} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                                       |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                                       |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                                       |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                                       |
|                                                                             |
|                                                                  26         |
|<--------------------------------------------------------------- S   : 3     |
|   {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |               |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |               |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |               |
|   {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |               |
|   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |               |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |               |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |               |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |               |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |               |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |               |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |               |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |               |
|                                                                             |
|                                                                             |
|<----------------------------------------------------------------------------|
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|<----------------------------------------------------------------------------|
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|   {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|   {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|-----------------------------------------------------------------------------|
|                       43                                                    |
|--------------------- S   : 4                                                |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|1 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 0 0 0 0 0 0 0 0 |                                                          |
|0 1 0 0 0 0 0 0 0 |                                                          |
|0 0 1 0 0 0 0 0 0 |                                                          |
|0 0 0 1 0 0 0 0 0 |                                                          |
|0 0 0 0 1 0 0 0 0 |                                                          |
|0 0 0 0 0 1 0 0 0 |                                                          |
|0 0 0 0 0 0 1 0 0 |                                                          |
|0 0 0 0 0 0 0 1 0 |                                                          |
|0 0 0 0 0 0 0 0 1 |                                                          |
|                                                                             |
|                                                                     66      |
|------------------------------------------------------------------- S   : 5  |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |            |
+-----------------------------------------------------------------------------+
|i25 : isSimplicialMorphism oo                                                |
|                                                                             |
|o25 = true                                                                   |
+-----------------------------------------------------------------------------+

See also
========

  * directSum(SimplicialModule), --
    direct sum of simplicial modules
  * components(SimplicialModule), --
    list the components of a direct sum
  * indices (Macaulay2Doc) -- indices of a polynomial; also
    components for a direct sum

Ways to use this method:
========================

  * SimplicialModule _ Array -- the
    canonical inclusion or projection map of a direct sum
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:872:0.

SimplicialModule _ ZZ -- access individual objects of a simplicial module
*************************************************************************

  * Operator: _ (Macaulay2Doc),
  * Usage:
        C_i
  * Inputs:
      * C, a Simplicial Module,
      * i, an integer (Macaulay2Doc), or Sequence (Macaulay2Doc),
        indicating a term in some degree
  * Outputs:
      * a module (Macaulay2Doc), the i-th object

Description
===========

The data of the modules of a simplicial module is stored in two different ways,
depending on whether the simplicial module $S$ is obtained as the Dold-Kan
image of some complex. If the simplicial module is obtained as a Dold-Kan image
of some complex $C$, then for each $i$ there is a decomposition $$S_i = C_0
\oplus C_1^{\binom{i}{1}} \oplus \cdots \oplus C_j^{\binom{i}{j}} \oplus
\cdots.$$ Thus the individual terms $C_j^{\binom{i}{j}}$ may be accessed as the
$(i,j)$-component of the simplicial module. If $S$ is not obtained as a
Dold-Kan image, then each term is considered singly indexed with no additional
components.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a..c]                                                        |
|                                                                             |
|o1 = S                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution coker vars S, 4, Degeneracy => true)|
|                                                                             |
|      1      4      10      20      35                                       |
|o2 = S  <-- S  <-- S   <-- S   <-- S  <-- ...                                |
|                                                                             |
|     0      1      2       3       4                                         |
|                                                                             |
|o2 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i3 : C.?complex                                                              |
|                                                                             |
|o3 = true                                                                    |
+-----------------------------------------------------------------------------+
|i4 : C.complex                                                               |
|                                                                             |
|      1      3      3      1                                                 |
|o4 = S  <-- S  <-- S  <-- S                                                  |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o4 : Complex                                                                 |
+-----------------------------------------------------------------------------+
|i5 : C_(1,0) == C.complex_0                                                  |
|                                                                             |
|o5 = true                                                                    |
+-----------------------------------------------------------------------------+
|i6 : C_(1,1) == C.complex_1                                                  |
|                                                                             |
|o6 = true                                                                    |
+-----------------------------------------------------------------------------+
|i7 : C_(2,1) == C.complex_1 ++ C.complex_1                                   |
|                                                                             |
|o7 = true                                                                    |
+-----------------------------------------------------------------------------+
|i8 : tC = C ** C                                                             |
|                                                                             |
|      1      16      100      400      1225                                  |
|o8 = S  <-- S   <-- S    <-- S    <-- S    <-- ...                           |
|                                                                             |
|     0      1       2        3        4                                      |
|                                                                             |
|o8 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i9 : tC.?complex                                                             |
|                                                                             |
|o9 = false                                                                   |
+-----------------------------------------------------------------------------+
|i10 : tC_2                                                                   |
|                                                                             |
|       100                                                                   |
|o10 = S                                                                      |
|                                                                             |
|o10 : S-module, free, degrees {0..1, 5:1, 3:2, 1..2, 5:2, 3:3, 1..2, 5:2,    |
|-----------------------------------------------------------------------------|
|3:3, 1..2, 5:2, 3:3, 1..2, 5:2, 3:3, 1..2, 5:2, 3:3, 1..2, 5:2, 3:3, 2..3,   |
|-----------------------------------------------------------------------------|
|5:3, 3:4, 2..3, 5:3, 3:4, 2..3, 5:3, 3:4}                                    |
+-----------------------------------------------------------------------------+

If the simplicial object is obtained as a Dold-Kan image, then using an integer
subscript will automatically take the direct sum over all the terms of the
complex appearing in that degree (and cache the components of that direct sum
accordingly).

+--------------------------------------------------------------+
|i11 : C_3                                                     |
|                                                              |
|       20                                                     |
|o11 = S                                                       |
|                                                              |
|o11 : S-module, free, degrees {0..1, 8:1, 9:2, 3}             |
+--------------------------------------------------------------+
|i12 : components C_3                                          |
|                                                              |
|        1   3   3   3   3   3   3   1                         |
|o12 = {S , S , S , S , S , S , S , S }                        |
|                                                              |
|o12 : List                                                    |
+--------------------------------------------------------------+
|i13 : (C_3)_[1,2,3] --extract the inclusion of (C.complex_1)^3|
|                                                              |
|o13 = {0} | 0 0 0 0 0 0 0 0 0 |                               |
|      {1} | 1 0 0 0 0 0 0 0 0 |                               |
|      {1} | 0 1 0 0 0 0 0 0 0 |                               |
|      {1} | 0 0 1 0 0 0 0 0 0 |                               |
|      {1} | 0 0 0 1 0 0 0 0 0 |                               |
|      {1} | 0 0 0 0 1 0 0 0 0 |                               |
|      {1} | 0 0 0 0 0 1 0 0 0 |                               |
|      {1} | 0 0 0 0 0 0 1 0 0 |                               |
|      {1} | 0 0 0 0 0 0 0 1 0 |                               |
|      {1} | 0 0 0 0 0 0 0 0 1 |                               |
|      {2} | 0 0 0 0 0 0 0 0 0 |                               |
|      {2} | 0 0 0 0 0 0 0 0 0 |                               |
|      {2} | 0 0 0 0 0 0 0 0 0 |                               |
|      {2} | 0 0 0 0 0 0 0 0 0 |                               |
|      {2} | 0 0 0 0 0 0 0 0 0 |                               |
|      {2} | 0 0 0 0 0 0 0 0 0 |                               |
|      {2} | 0 0 0 0 0 0 0 0 0 |                               |
|      {2} | 0 0 0 0 0 0 0 0 0 |                               |
|      {2} | 0 0 0 0 0 0 0 0 0 |                               |
|      {3} | 0 0 0 0 0 0 0 0 0 |                               |
|                                                              |
|              20      9                                       |
|o13 : Matrix S   <-- S                                        |
+--------------------------------------------------------------+

Indices that are outside of the top degree automatically return the zero
object.

+--------------+
|i14 : C_(-1,3)|
|              |
|o14 = 0       |
|              |
|o14 : S-module|
+--------------+
|i15 : C_-7    |
|              |
|o15 = 0       |
|              |
|o15 : S-module|
+--------------+

See also
========

  * SimplicialModuleMap _ ZZ -- access
    individual matrices in a simplicial module map

Ways to use this method:
========================

  * "SimplicialModule _ Sequence"
  * SimplicialModule _ ZZ -- access individual
    objects of a simplicial module
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:681:0.

SimplicialModule Array -- shift a simplicial module or simplicial module map
****************************************************************************

  * Operator: SPACE (Macaulay2Doc),
  * Usage:
        D = C[i]
        g = f[i]
  * Inputs:
      * C, a Simplicial Module, or f, a
        SimplicialModuleMap,
      * an array (Macaulay2Doc), [i], where i is an integer
  * Outputs:
      * D, a Simplicial Module, or g, a
        SimplicialModuleMap.

Description
===========

The shifted simplicial module $D$ is not as simple to define as the shift in
the category of chain complexes. This method naively normalizes the given
simplicial module/map, applies the shift in the category of chain complexes,
then applies the Dold-Kan functor to the result.

As the following example shows, this is not the same thing as simply shifting
all terms of the simplicial module.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a..d]                                                        |
|                                                                             |
|o1 = S                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : D = simplicialModule(S^1, 5)                                            |
|                                                                             |
|      1      1      1      1      1      1                                   |
|o2 = S  <-- S  <-- S  <-- S  <-- S  <-- S <-- ...                            |
|                                                                             |
|     0      1      2      3      4      5                                    |
|                                                                             |
|o2 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i3 : D[-1]                                                                   |
|                                                                             |
|            1      2      3      4      5                                    |
|o3 = 0 <-- S  <-- S  <-- S  <-- S  <-- S <-- ...                             |
|                                                                             |
|     0     1      2      3      4      5                                     |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : oo.dd                                                                   |
|                                                                             |
|                        1                                                    |
|o4 = (0, 0) : 0 <----- S  : (1, 0)                                           |
|                   0                                                         |
|                                                                             |
|                        1                                                    |
|     (0, 1) : 0 <----- S  : (1, 1)                                           |
|                   0                                                         |
|                                                                             |
|               1               2                                             |
|     (1, 0) : S  <----------- S  : (2, 0)                                    |
|                    | 0 1 |                                                  |
|                                                                             |
|               1               2                                             |
|     (1, 1) : S  <----------- S  : (2, 1)                                    |
|                    | 1 1 |                                                  |
|                                                                             |
|               1               2                                             |
|     (1, 2) : S  <----------- S  : (2, 2)                                    |
|                    | 1 0 |                                                  |
|                                                                             |
|               2                 3                                           |
|     (2, 0) : S  <------------- S  : (3, 0)                                  |
|                    | 0 1 0 |                                                |
|                    | 0 0 1 |                                                |
|                                                                             |
|               2                 3                                           |
|     (2, 1) : S  <------------- S  : (3, 1)                                  |
|                    | 1 1 0 |                                                |
|                    | 0 0 1 |                                                |
|                                                                             |
|               2                 3                                           |
|     (2, 2) : S  <------------- S  : (3, 2)                                  |
|                    | 1 0 0 |                                                |
|                    | 0 1 1 |                                                |
|                                                                             |
|               2                 3                                           |
|     (2, 3) : S  <------------- S  : (3, 3)                                  |
|                    | 1 0 0 |                                                |
|                    | 0 1 0 |                                                |
|                                                                             |
|               3                   4                                         |
|     (3, 0) : S  <--------------- S  : (4, 0)                                |
|                    | 0 1 0 0 |                                              |
|                    | 0 0 1 0 |                                              |
|                    | 0 0 0 1 |                                              |
|                                                                             |
|               3                   4                                         |
|     (3, 1) : S  <--------------- S  : (4, 1)                                |
|                    | 1 1 0 0 |                                              |
|                    | 0 0 1 0 |                                              |
|                    | 0 0 0 1 |                                              |
|                                                                             |
|               3                   4                                         |
|     (3, 2) : S  <--------------- S  : (4, 2)                                |
|                    | 1 0 0 0 |                                              |
|                    | 0 1 1 0 |                                              |
|                    | 0 0 0 1 |                                              |
|                                                                             |
|               3                   4                                         |
|     (3, 3) : S  <--------------- S  : (4, 3)                                |
|                    | 1 0 0 0 |                                              |
|                    | 0 1 0 0 |                                              |
|                    | 0 0 1 1 |                                              |
|                                                                             |
|               3                   4                                         |
|     (3, 4) : S  <--------------- S  : (4, 4)                                |
|                    | 1 0 0 0 |                                              |
|                    | 0 1 0 0 |                                              |
|                    | 0 0 1 0 |                                              |
|                                                                             |
|               4                     5                                       |
|     (4, 0) : S  <----------------- S  : (5, 0)                              |
|                    | 0 1 0 0 0 |                                            |
|                    | 0 0 1 0 0 |                                            |
|                    | 0 0 0 1 0 |                                            |
|                    | 0 0 0 0 1 |                                            |
|                                                                             |
|               4                     5                                       |
|     (4, 1) : S  <----------------- S  : (5, 1)                              |
|                    | 1 1 0 0 0 |                                            |
|                    | 0 0 1 0 0 |                                            |
|                    | 0 0 0 1 0 |                                            |
|                    | 0 0 0 0 1 |                                            |
|                                                                             |
|               4                     5                                       |
|     (4, 2) : S  <----------------- S  : (5, 2)                              |
|                    | 1 0 0 0 0 |                                            |
|                    | 0 1 1 0 0 |                                            |
|                    | 0 0 0 1 0 |                                            |
|                    | 0 0 0 0 1 |                                            |
|                                                                             |
|               4                     5                                       |
|     (4, 3) : S  <----------------- S  : (5, 3)                              |
|                    | 1 0 0 0 0 |                                            |
|                    | 0 1 0 0 0 |                                            |
|                    | 0 0 1 1 0 |                                            |
|                    | 0 0 0 0 1 |                                            |
|                                                                             |
|               4                     5                                       |
|     (4, 4) : S  <----------------- S  : (5, 4)                              |
|                    | 1 0 0 0 0 |                                            |
|                    | 0 1 0 0 0 |                                            |
|                    | 0 0 1 0 0 |                                            |
|                    | 0 0 0 1 1 |                                            |
|                                                                             |
|               4                     5                                       |
|     (4, 5) : S  <----------------- S  : (5, 5)                              |
|                    | 1 0 0 0 0 |                                            |
|                    | 0 1 0 0 0 |                                            |
|                    | 0 0 1 0 0 |                                            |
|                    | 0 0 0 1 0 |                                            |
|                                                                             |
|o4 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+
|i5 : D[-2]                                                                   |
|                                                                             |
|                  1      3      6      10                                    |
|o5 = 0 <-- 0 <-- S  <-- S  <-- S  <-- S  <-- ...                             |
|                                                                             |
|     0     1     2      3      4      5                                      |
|                                                                             |
|o5 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i6 : C = simplicialModule(freeResolution coker vars S, Degeneracy => true)   |
|                                                                             |
|      1      5      15      35      70                                       |
|o6 = S  <-- S  <-- S   <-- S   <-- S  <-- ...                                |
|                                                                             |
|     0      1      2       3       4                                         |
|                                                                             |
|o6 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i7 : dd^C_(3,0)                                                              |
|                                                                             |
|o7 = {0} | 1 a b c d 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0 0 |
|     {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 -b -c 0  -d 0  0  0  0  0  0  0  0  0 0 |
|     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 a  0  -c 0  -d 0  0  0  0  0  0  0  0 0 |
|     {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0  a  b  0  0  -d 0  0  0  0  0  0  0 0 |
|     {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0  0  0  a  b  c  0  0  0  0  0  0  0 0 |
|     {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0  0  0  0  0  0  -b -c 0  -d 0  0  0 0 |
|     {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0  0  0  0  0  0  a  0  -c 0  -d 0  0 0 |
|     {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0  0  0  0  0  0  0  a  b  0  0  -d 0 0 |
|     {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0  0  0  0  0  0  0  0  0  a  b  c  0 0 |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  1 0 |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0 1 |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0 0 |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0 0 |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0 0 |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0 0 |
|     ------------------------------------------------------------------------|
|     0 0 0 0 0  0  0  0  |                                                   |
|     0 0 0 0 0  0  0  0  |                                                   |
|     0 0 0 0 0  0  0  0  |                                                   |
|     0 0 0 0 0  0  0  0  |                                                   |
|     0 0 0 0 0  0  0  0  |                                                   |
|     0 0 0 0 0  0  0  0  |                                                   |
|     0 0 0 0 0  0  0  0  |                                                   |
|     0 0 0 0 0  0  0  0  |                                                   |
|     0 0 0 0 0  0  0  0  |                                                   |
|     0 0 0 0 c  d  0  0  |                                                   |
|     0 0 0 0 -b 0  d  0  |                                                   |
|     1 0 0 0 a  0  0  d  |                                                   |
|     0 1 0 0 0  -b -c 0  |                                                   |
|     0 0 1 0 0  a  0  -c |                                                   |
|     0 0 0 1 0  0  a  b  |                                                   |
|                                                                             |
|             15      35                                                      |
|o7 : Matrix S   <-- S                                                        |
+-----------------------------------------------------------------------------+
|i8 : ss^C_(2,0)                                                              |
|                                                                             |
|o8 = {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                                   |
|     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                                   |
|     {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                                   |
|     {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                                   |
|     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                                   |
|     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                   |
|                                                                             |
|             35      15                                                      |
|o8 : Matrix S   <-- S                                                        |
+-----------------------------------------------------------------------------+
|i9 : D = C[1]                                                                |
|                                                                             |
|      4      10      20      35      56                                      |
|o9 = S  <-- S   <-- S   <-- S   <-- S  <-- ...                               |
|                                                                             |
|     0      1       2       3       4                                        |
|                                                                             |
|o9 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i10 : assert isWellDefined D                                                 |
+-----------------------------------------------------------------------------+
|i11 : dd^D_(2,0)                                                             |
|                                                                             |
|o11 = {1} | 1 0 0 0 b  c  0  d  0  0  0 0 0 0 0 0 0  0  0  0  |              |
|      {1} | 0 1 0 0 -a 0  c  0  d  0  0 0 0 0 0 0 0  0  0  0  |              |
|      {1} | 0 0 1 0 0  -a -b 0  0  d  0 0 0 0 0 0 0  0  0  0  |              |
|      {1} | 0 0 0 1 0  0  0  -a -b -c 0 0 0 0 0 0 0  0  0  0  |              |
|      {2} | 0 0 0 0 0  0  0  0  0  0  1 0 0 0 0 0 -c -d 0  0  |              |
|      {2} | 0 0 0 0 0  0  0  0  0  0  0 1 0 0 0 0 b  0  -d 0  |              |
|      {2} | 0 0 0 0 0  0  0  0  0  0  0 0 1 0 0 0 -a 0  0  -d |              |
|      {2} | 0 0 0 0 0  0  0  0  0  0  0 0 0 1 0 0 0  b  c  0  |              |
|      {2} | 0 0 0 0 0  0  0  0  0  0  0 0 0 0 1 0 0  -a 0  c  |              |
|      {2} | 0 0 0 0 0  0  0  0  0  0  0 0 0 0 0 1 0  0  -a -b |              |
|                                                                             |
|              10      20                                                     |
|o11 : Matrix S   <-- S                                                       |
+-----------------------------------------------------------------------------+
|i12 : ss^D_(1,0)                                                             |
|                                                                             |
|o12 = {1} | 1 0 0 0 0 0 0 0 0 0 |                                            |
|      {1} | 0 1 0 0 0 0 0 0 0 0 |                                            |
|      {1} | 0 0 1 0 0 0 0 0 0 0 |                                            |
|      {1} | 0 0 0 1 0 0 0 0 0 0 |                                            |
|      {2} | 0 0 0 0 0 0 0 0 0 0 |                                            |
|      {2} | 0 0 0 0 0 0 0 0 0 0 |                                            |
|      {2} | 0 0 0 0 0 0 0 0 0 0 |                                            |
|      {2} | 0 0 0 0 0 0 0 0 0 0 |                                            |
|      {2} | 0 0 0 0 0 0 0 0 0 0 |                                            |
|      {2} | 0 0 0 0 0 0 0 0 0 0 |                                            |
|      {2} | 0 0 0 0 1 0 0 0 0 0 |                                            |
|      {2} | 0 0 0 0 0 1 0 0 0 0 |                                            |
|      {2} | 0 0 0 0 0 0 1 0 0 0 |                                            |
|      {2} | 0 0 0 0 0 0 0 1 0 0 |                                            |
|      {2} | 0 0 0 0 0 0 0 0 1 0 |                                            |
|      {2} | 0 0 0 0 0 0 0 0 0 1 |                                            |
|      {3} | 0 0 0 0 0 0 0 0 0 0 |                                            |
|      {3} | 0 0 0 0 0 0 0 0 0 0 |                                            |
|      {3} | 0 0 0 0 0 0 0 0 0 0 |                                            |
|      {3} | 0 0 0 0 0 0 0 0 0 0 |                                            |
|                                                                             |
|              20      10                                                     |
|o12 : Matrix S   <-- S                                                       |
+-----------------------------------------------------------------------------+

Notice that the face maps of the shifted simplicial module are not simply the
negation of the face maps of the original. The shift operator is functorial, as
illustrated below.

+-----------------------------------------------------------------------------+
|i13 : C2 = simplicialModule(freeResolution (S^1/(a^2, b^2, c^2, d^2)),       |
|                                                                             |
|       1      5      15      35      70                                      |
|o13 = S  <-- S  <-- S   <-- S   <-- S  <-- ...                               |
|                                                                             |
|      0      1      2       3       4                                        |
|                                                                             |
|o13 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i14 : C3 = simplicialModule(freeResolution (S^1/(a^2, b^3, c^4, d^5)),       |
|                                                                             |
|       1      5      15      35      70                                      |
|o14 = S  <-- S  <-- S   <-- S   <-- S  <-- ...                               |
|                                                                             |
|      0      1      2       3       4                                        |
|                                                                             |
|o14 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i15 : f2 = simplicialModule(extend(C.complex, C2.complex, map(C_0, C2_0,     |
|-----------------------------------------------------------------------------|
|1)));                                                                        |
+-----------------------------------------------------------------------------+
|i16 : f3 = simplicialModule(extend(C2.complex, C3.complex, map(C2_0, C3_0,   |
|-----------------------------------------------------------------------------|
|1)));                                                                        |
+-----------------------------------------------------------------------------+
|i17 : isWellDefined (f2[-1])                                                 |
|                                                                             |
|o17 = true                                                                   |
+-----------------------------------------------------------------------------+
|i18 : isSimplicialMorphism (f2[-1])                                          |
|                                                                             |
|o18 = true                                                                   |
+-----------------------------------------------------------------------------+
|i19 : isSimplicialMorphism (f3[-2])                                          |
|                                                                             |
|o19 = true                                                                   |
+-----------------------------------------------------------------------------+

See also
========

  * Making simplicial modules -- information
    about the basic constructors

Ways to use this method:
========================

  * SimplicialModule Array -- shift a simplicial
    module or simplicial module map
  * "SimplicialModuleMap Array"
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3149:0.

simplicialModule(...,Degeneracy=>...) -- whether to compute degeneracy maps
***************************************************************************

  * Usage:
        simplicialModule(..., Degeneracy => true)

Description
===========

If Degeneracy => true, the degeneracy maps are also computed alongside the face
maps. By default, degeneracy maps are not computed.

See also
========

  * simplicialModule -- make a simplicial module

Functions with optional argument named Degeneracy:
==================================================

  * extPower(...,Degeneracy=>...)
    -- whether to compute
    degeneracy maps
  * schurMap(...,Degeneracy=>...)
    -- whether to compute
    degeneracy maps
  * simplicialModule(...,Degeneracy=>...)
    -- whether to
    compute degeneracy maps
  * simplicialTensor(...,Degeneracy=>...)
    -- whether to
    compute degeneracy maps

Further information
===================

  * Default value: false (Macaulay2Doc),
  * Function: simplicialModule, -- make a simplicial
    module
  * Option key: Degeneracy, -- the class of all
    simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:4010:0.

simplicialModule(Module,ZZ) -- make a simplicial module associated to a complex concentrated in degree 0
********************************************************************************************************

  * Function: simplicialModule,
  * Usage:
        simplicialModule(M, d)
  * Inputs:
      * M, a module (Macaulay2Doc), or Ideal (Macaulay2Doc),
        or Ring (Macaulay2Doc).
      * d, an integer (Macaulay2Doc), top degree to compute the
        simplicial module up to
  * Optional inputs (Macaulay2Doc),:
      * Base (missing documentation) => ..., default value 0,
      * Degeneracy,
        => ..., default value false, whether to compute degeneracy maps
  * Outputs:
      * a Simplicial Module, returns the complex whose
        0-th component is M.

Description
===========

In contrast to simplicialModule(HashTable,HashTable,HashTable,ZZ),
or simplicialModule(Complex,ZZ),
this constructor provides a convenient method to
construct a simplicial module from a single module/ring/ideal.

We illustrate this with a free module.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a..d]                                                        |
|                                                                             |
|o1 = S                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : C0 = simplicialModule( S^2, 6, Degeneracy => true)                      |
|                                                                             |
|      2      2      2      2      2      2      2                            |
|o2 = S  <-- S  <-- S  <-- S  <-- S  <-- S  <-- S <-- ...                     |
|                                                                             |
|     0      1      2      3      4      5      6                             |
|                                                                             |
|o2 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i3 : f = dd^C0                                                               |
|                                                                             |
|               2               2                                             |
|o3 = (0, 0) : S  <----------- S  : (1, 0)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (0, 1) : S  <----------- S  : (1, 1)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (1, 0) : S  <----------- S  : (2, 0)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (1, 1) : S  <----------- S  : (2, 1)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (1, 2) : S  <----------- S  : (2, 2)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (2, 0) : S  <----------- S  : (3, 0)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (2, 1) : S  <----------- S  : (3, 1)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (2, 2) : S  <----------- S  : (3, 2)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (2, 3) : S  <----------- S  : (3, 3)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (3, 0) : S  <----------- S  : (4, 0)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (3, 1) : S  <----------- S  : (4, 1)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (3, 2) : S  <----------- S  : (4, 2)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (3, 3) : S  <----------- S  : (4, 3)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (3, 4) : S  <----------- S  : (4, 4)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (4, 0) : S  <----------- S  : (5, 0)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (4, 1) : S  <----------- S  : (5, 1)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (4, 2) : S  <----------- S  : (5, 2)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (4, 3) : S  <----------- S  : (5, 3)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (4, 4) : S  <----------- S  : (5, 4)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (4, 5) : S  <----------- S  : (5, 5)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (5, 0) : S  <----------- S  : (6, 0)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (5, 1) : S  <----------- S  : (6, 1)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (5, 2) : S  <----------- S  : (6, 2)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (5, 3) : S  <----------- S  : (6, 3)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (5, 4) : S  <----------- S  : (6, 4)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (5, 5) : S  <----------- S  : (6, 5)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|               2               2                                             |
|     (5, 6) : S  <----------- S  : (6, 6)                                    |
|                    | 1 0 |                                                  |
|                    | 0 1 |                                                  |
|                                                                             |
|o3 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+
|i4 : source f, target f                                                      |
|                                                                             |
|       2      2      2      2      2      2      2          2      2      2  |
|o4 = (S  <-- S  <-- S  <-- S  <-- S  <-- S  <-- S <-- ..., S  <-- S  <-- S   |
|                                                                             |
|      0      1      2      3      4      5      6          0      1      2   |
|     ------------------------------------------------------------------------|
|          2      2      2      2                                             |
|     <-- S  <-- S  <-- S  <-- S <-- ...)                                     |
|                                                                             |
|         3      4      5      6                                              |
|                                                                             |
|o4 : Sequence                                                                |
+-----------------------------------------------------------------------------+
|i5 : f == 0                                                                  |
|                                                                             |
|o5 = false                                                                   |
+-----------------------------------------------------------------------------+
|i6 : isWellDefined C0                                                        |
|                                                                             |
|o6 = true                                                                    |
+-----------------------------------------------------------------------------+
|i7 : C0 == 0                                                                 |
|                                                                             |
|o7 = false                                                                   |
+-----------------------------------------------------------------------------+
|i8 : topDegree C0                                                            |
|                                                                             |
|o8 = 6                                                                       |
+-----------------------------------------------------------------------------+
+------------------------------------------------------------------------+
|i9 : C1 = simplicialModule(complex(S^2, Base=>3), 6, Degeneracy => true)|
|                                                                        |
|                        2      8      20      40                        |
|o9 = 0 <-- 0 <-- 0 <-- S  <-- S  <-- S   <-- S  <-- ...                 |
|                                                                        |
|     0     1     2     3      4      5       6                          |
|                                                                        |
|o9 : SimplicialModule                                                   |
+------------------------------------------------------------------------+
|i10 : C1_3                                                              |
|                                                                        |
|       2                                                                |
|o10 = S                                                                 |
|                                                                        |
|o10 : S-module, free                                                    |
+------------------------------------------------------------------------+
|i11 : C1_0                                                              |
|                                                                        |
|o11 = 0                                                                 |
|                                                                        |
|o11 : S-module                                                          |
+------------------------------------------------------------------------+

A ring or an ideal will be converted to a module first.

+-----------------------------------------------------------------------------+
|i12 : C2 = simplicialModule( S, 5, Degeneracy => true)                       |
|                                                                             |
|       1      1      1      1      1      1                                  |
|o12 = S  <-- S  <-- S  <-- S  <-- S  <-- S <-- ...                           |
|                                                                             |
|      0      1      2      3      4      5                                   |
|                                                                             |
|o12 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i13 : I = ideal(a^2-b, c^3)                                                  |
|                                                                             |
|              2       3                                                      |
|o13 = ideal (a  - b, c )                                                     |
|                                                                             |
|o13 : Ideal of S                                                             |
+-----------------------------------------------------------------------------+
|i14 : C3 = simplicialModule( I, 7, Degeneracy => true)                       |
|                                                                             |
|o14 = image | a2-b c3 | <-- image | a2-b c3 | <-- image | a2-b c3 | <-- image|
|                                                                             |
|      0                     1                     2                     3    |
|                                                                             |
|o14 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|| a2-b c3 | <-- image | a2-b c3 | <-- image | a2-b c3 | <-- image | a2-b c3 ||
|                                                                             |
|                4                     5                     6                |
|-----------------------------------------------------------------------------|
|<-- image | a2-b c3 |<-- ...                                                 |
|                                                                             |
|    7                                                                        |
+-----------------------------------------------------------------------------+
|i15 : C4 = simplicialModule( (S/I), 8, Degeneracy => true)                   |
|                                                                             |
|      /      S     \1     /      S     \1     /      S     \1                |
|o15 = |------------|  <-- |------------|  <-- |------------|  <--            |
|      |  2       3 |      |  2       3 |      |  2       3 |                 |
|      \(a  - b, c )/      \(a  - b, c )/      \(a  - b, c )/                 |
|                                                                             |
|      0                   1                   2                              |
|                                                                             |
|o15 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|/      S     \1     /      S     \1     /      S     \1     /      S     \1  |
||------------|  <-- |------------|  <-- |------------|  <-- |------------|   |
||  2       3 |      |  2       3 |      |  2       3 |      |  2       3 |   |
|\(a  - b, c )/      \(a  - b, c )/      \(a  - b, c )/      \(a  - b, c )/   |
|                                                                             |
|3                   4                   5                   6                |
|-----------------------------------------------------------------------------|
|    /      S     \1     /      S     \1                                      |
|<-- |------------|  <-- |------------| <-- ...                               |
|    |  2       3 |      |  2       3 |                                       |
|    \(a  - b, c )/      \(a  - b, c )/                                       |
|                                                                             |
|    7                   8                                                    |
+-----------------------------------------------------------------------------+
|i16 : (ring C3, ring C4)                                                     |
|                                                                             |
|                S                                                            |
|o16 = (S, ------------)                                                      |
|            2       3                                                        |
|          (a  - b, c )                                                       |
|                                                                             |
|o16 : Sequence                                                               |
+-----------------------------------------------------------------------------+

The zero simplicial module over a ring S is most conveniently created by giving
the zero module.

+--------------------------------------------------------------+
|i17 : C5 = simplicialModule(S^0, 8, Degeneracy => true)       |
|                                                              |
|o17 = 0 <-- 0 <-- 0 <-- 0 <-- 0 <-- 0 <-- 0 <-- 0 <-- 0<-- ...|
|                                                              |
|      0     1     2     3     4     5     6     7     8       |
|                                                              |
|o17 : SimplicialModule                                        |
+--------------------------------------------------------------+
|i18 : C5 == 0                                                 |
|                                                              |
|o18 = true                                                    |
+--------------------------------------------------------------+
|i19 : dd^C5 == 0                                              |
|                                                              |
|o19 = true                                                    |
+--------------------------------------------------------------+
|i20 : ss^C5 == 0                                              |
|                                                              |
|o20 = true                                                    |
+--------------------------------------------------------------+
|i21 : C5_0                                                    |
|                                                              |
|o21 = 0                                                       |
|                                                              |
|o21 : S-module                                                |
+--------------------------------------------------------------+

See also
========

  * Making simplicial modules -- information
    about the basic constructors
  * isWellDefined(SimplicialModule),
    -- whether a simplicial module is well-defined
  * simplicialModule(HashTable,HashTable,HashTable,ZZ),
    -- make a simplicial module

Ways to use this method:
========================

  * "simplicialModule(Ideal,ZZ)"
  * simplicialModule(Module,ZZ), --
    make a simplicial module associated to a complex concentrated in degree 0
  * "simplicialModule(Ring,ZZ)"
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:583:0.

SimplicialModuleMap -- the class of all maps between simplicial modules
***********************************************************************

Description
===========

A map of simplicial modules $f \colon C \rightarrow D$ of degree $d$ is a
sequence of maps $f_i \colon C_i \rightarrow D_{d+i}$. No relationship between
the maps $f_i$ and the face/degeneracy maps of either $C$ or $D$ is assumed. If
a simplicial module map is obtained as the image of a morphism of complexes
under the Dold-Kan functor, the key complexMap will be stored for more
efficient normalization computations.

The usual algebraic operations are available: addition, subtraction, scalar
multiplication, and composition. The identity map from a simplicial module to
itself can be produced with id (Macaulay2Doc). An attempt to add
(subtract, or compare) a ring element to a simplicial module map will result in
the ring element being multiplied by the appropriate identity map.

See also
========

  * SimplicialModule -- the class of all simplicial
    modules
  * Making maps between simplicial modules
    -- information about the basic constructors
  * arithmetic with simplicial module maps
    -- perform arithmetic operations on simplicial module maps

Functions and methods returning a Map of Simplicial Modules:
============================================================

  * "Number + SimplicialModuleMap" -- see arithmetic with simplicial module maps,
    -- perform arithmetic
    operations on simplicial module maps
  * "Number - SimplicialModuleMap" -- see arithmetic with simplicial module maps,
    -- perform arithmetic
    operations on simplicial module maps
  * "RingElement + SimplicialModuleMap" -- see arithmetic with simplicial module maps,
    -- perform arithmetic
    operations on simplicial module maps
  * "RingElement - SimplicialModuleMap" -- see arithmetic with simplicial module maps,
    -- perform arithmetic
    operations on simplicial module maps
  * "SimplicialModuleMap + Number" -- see arithmetic with simplicial module maps,
    -- perform arithmetic
    operations on simplicial module maps
  * "SimplicialModuleMap + RingElement" -- see arithmetic with simplicial module maps,
    -- perform arithmetic
    operations on simplicial module maps
  * "SimplicialModuleMap - Number" -- see arithmetic with simplicial module maps,
    -- perform arithmetic
    operations on simplicial module maps
  * "SimplicialModuleMap - RingElement" -- see arithmetic with simplicial module maps,
    -- perform arithmetic
    operations on simplicial module maps
  * "SimplicialModuleMap - SimplicialModuleMap" -- see arithmetic with simplicial module maps,
    -- perform
    arithmetic operations on simplicial module maps
  * basis(List,SimplicialModuleMap) (missing documentation)
  * basis(ZZ,SimplicialModuleMap) (missing documentation)
  * "extPower(ZZ,SimplicialModuleMap)" -- see extPower, --
    compute the Dold-Puppe extension of the exterior power functor to
    simplicial modules
  * "Symbol ^ SimplicialModule" -- see face/degeneracy maps of a simplicial module,
    -- access
    the face and degeneracy maps of a simplicial module
  * HH SimplicialModuleMap (missing documentation)
  * inducedMap(SimplicialModule,SimplicialModule)
    -- make the map of
    simplicial modules induced at each term by the identity map
  * map(SimplicialModule,SimplicialModule,HashTable)
    -- make a map of
    simplicial modules
  * map(SimplicialModule,SimplicialModule,SimplicialModuleMap)
    -- make
    a new map of simplicial modules, induced from an existing one
  * map(SimplicialModule,SimplicialModule,ZZ)
    -- make the zero map or
    identity between simplicial modules
  * "minimalPresentation(SimplicialModuleMap)" -- see
    minimalPresentation(SimplicialModule),
    -- minimal presentation of all
    terms in a simplicial module
  * "prune(SimplicialModuleMap)" -- see
    minimalPresentation(SimplicialModule),
    -- minimal presentation of all
    terms in a simplicial module
  * randomSimplicialMap(SimplicialModule,SimplicialModule)
    -- a random
    map of simplicial modules
  * "Ring ** SimplicialModuleMap" -- see RingMap ** SimplicialModuleMap,
    -- tensor a map of simplicial modules
    along a ring map
  * RingMap ** SimplicialModuleMap,
    -- tensor a map of simplicial modules along a ring map
  * "SimplicialModuleMap ** Ring" -- see RingMap ** SimplicialModuleMap,
    -- tensor a map of simplicial modules
    along a ring map
  * "SimplicialModuleMap ** RingMap" -- see RingMap ** SimplicialModuleMap,
    -- tensor a map of
    simplicial modules along a ring map
  * "tensor(RingMap,SimplicialModuleMap)" -- see RingMap ** SimplicialModuleMap,
    -- tensor a map of
    simplicial modules along a ring map
  * "tensor(SimplicialModuleMap,RingMap)" -- see RingMap ** SimplicialModuleMap,
    -- tensor a map of
    simplicial modules along a ring map
  * RingMap SimplicialModuleMap -- apply a
    ring map to a map of simplicial modules
  * "schurMap(List,SimplicialModuleMap)" -- see schurMap, --
    construct the Dold-Puppe extension of the Schur functor
  * "Matrix ** SimplicialModule" -- see SimplicialModule ** Matrix,
    -- create the tensor product of a
    simplicial module and a map of modules
  * SimplicialModule ** Matrix -- create
    the tensor product of a simplicial module and a map of modules
  * "SimplicialModule ^ Array" -- see SimplicialModule _ Array,
    -- the canonical inclusion or projection map of
    a direct sum
  * SimplicialModule _ Array -- the
    canonical inclusion or projection map of a direct sum
  * simplicialModule(ComplexMap) (missing documentation)
  * simplicialModule(ComplexMap,ZZ) (missing documentation)
  * "Complex ** SimplicialModuleMap" -- see SimplicialModuleMap ** SimplicialModuleMap,
    -- the
    map of simplicial modules between tensor simplicial modules
  * "ComplexMap ** SimplicialModuleMap" -- see SimplicialModuleMap ** SimplicialModuleMap,
    -- the
    map of simplicial modules between tensor simplicial modules
  * "Module ** SimplicialModuleMap" -- see SimplicialModuleMap ** SimplicialModuleMap,
    -- the
    map of simplicial modules between tensor simplicial modules
  * "SimplicialModule ** SimplicialModuleMap" -- see SimplicialModuleMap ** SimplicialModuleMap,
    --
    the map of simplicial modules between tensor simplicial modules
  * "SimplicialModuleMap ** Complex" -- see SimplicialModuleMap ** SimplicialModuleMap,
    -- the
    map of simplicial modules between tensor simplicial modules
  * "SimplicialModuleMap ** ComplexMap" -- see SimplicialModuleMap ** SimplicialModuleMap,
    -- the
    map of simplicial modules between tensor simplicial modules
  * "SimplicialModuleMap ** Module" -- see SimplicialModuleMap ** SimplicialModuleMap,
    -- the
    map of simplicial modules between tensor simplicial modules
  * "SimplicialModuleMap ** SimplicialModule" -- see SimplicialModuleMap ** SimplicialModuleMap,
    --
    the map of simplicial modules between tensor simplicial modules
  * SimplicialModuleMap ** SimplicialModuleMap
    -- the map of simplicial modules between tensor
    simplicial modules
  * "tensor(SimplicialModuleMap,SimplicialModuleMap)" -- see
    SimplicialModuleMap ** SimplicialModuleMap,
    -- the map of simplicial modules between tensor
    simplicial modules
  * SimplicialModuleMap ++ SimplicialModuleMap
    -- direct sum of simplicial module maps
  * SimplicialModuleMap ^ Array -- the
    composition with the canonical inclusion or projection map
  * "SimplicialModuleMap _ Array" -- see SimplicialModuleMap ^ Array,
    -- the composition with the canonical
    inclusion or projection map
  * SimplicialModuleMap ^ ZZ -- the n-fold
    composition
  * SimplicialModuleMap | SimplicialModuleMap
    -- join or concatenate maps horizontally
  * SimplicialModuleMap || SimplicialModuleMap
    -- join or concatenate maps vertically
  * truncate(List,SimplicialModuleMap)
    -- truncation of a simplicial
    module map at a specified degree or set of degrees
  * "truncate(ZZ,SimplicialModuleMap)" -- see
    truncate(List,SimplicialModuleMap),
    -- truncation of a simplicial
    module map at a specified degree or set of degrees

Methods that use a Map of Simplicial Modules:
=============================================

  * "- SimplicialModuleMap" -- see arithmetic with simplicial module maps,
    -- perform arithmetic
    operations on simplicial module maps
  * "Number * SimplicialModuleMap" -- see arithmetic with simplicial module maps,
    -- perform arithmetic
    operations on simplicial module maps
  * "RingElement * SimplicialModuleMap" -- see arithmetic with simplicial module maps,
    -- perform arithmetic
    operations on simplicial module maps
  * "SimplicialModuleMap * Number" -- see arithmetic with simplicial module maps,
    -- perform arithmetic
    operations on simplicial module maps
  * "SimplicialModuleMap * RingElement" -- see arithmetic with simplicial module maps,
    -- perform arithmetic
    operations on simplicial module maps
  * "SimplicialModuleMap + SimplicialModuleMap" -- see arithmetic with simplicial module maps,
    -- perform
    arithmetic operations on simplicial module maps
  * coimage(SimplicialModuleMap), --
    make the coimage of a map of simplicial modules
  * cokernel(SimplicialModuleMap), --
    make the cokernel of a map of simplicial modules
  * components(SimplicialModuleMap),
    -- list the components of a direct sum
  * degree(SimplicialModuleMap) -- get
    the degree of a map of simplicial modules
  * expression(SimplicialModuleMap) (missing documentation)
  * image(SimplicialModuleMap) -- make
    the image of a map of simplicial modules
  * isCommutative(SimplicialModuleMap)
    -- whether a simplicial module map
    commutes with the face/degeneracy maps
  * isHomogeneous(SimplicialModuleMap)
    -- whether a map of simplicial
    modules is homogeneous
  * isShortExactSequence(SimplicialModuleMap,SimplicialModuleMap),
    --
    whether a pair of simplicial module maps forms a short exact sequence
  * isSimplicialMorphism(SimplicialModuleMap)
    -- whether a simplicial
    module map is a morphism of simplicial modules
  * isWellDefined(SimplicialModuleMap)
    -- whether a map of simplicial
    modules is well-defined
  * kernel(SimplicialModuleMap) -- make
    the kernel of a map of simplicial modules
  * net(SimplicialModuleMap) (missing documentation)
  * "normalize(SimplicialModuleMap)" -- see
    normalize(SimplicialModule,ZZ), --
    normalization functor from simplicial modules to nonnegatively-graded chain
    complexes
  * "normalize(SimplicialModuleMap,ZZ)" -- see
    normalize(SimplicialModule,ZZ), --
    normalization functor from simplicial modules to nonnegatively-graded chain
    complexes
  * "ring(SimplicialModuleMap)" -- see ring(SimplicialModule),
    -- access the ring of a simplicial module or a
    simplicial module map
  * "SimplicialModuleMap Array" -- see SimplicialModule Array,
    -- shift a simplicial module or simplicial module
    map
  * SimplicialModuleMap * SimplicialModuleMap
    -- composition of homomorphisms of simplicial modules
  * "directSum(SimplicialModuleMap)" -- see SimplicialModuleMap ++ SimplicialModuleMap,
    -- direct
    sum of simplicial module maps
  * SimplicialModuleMap == SimplicialModuleMap
    -- whether two simplicial module maps are equal
  * "SimplicialModuleMap == ZZ" -- see SimplicialModuleMap == SimplicialModuleMap,
    -- whether
    two simplicial module maps are equal
  * "ZZ == SimplicialModuleMap" -- see SimplicialModuleMap == SimplicialModuleMap,
    -- whether
    two simplicial module maps are equal
  * "SimplicialModuleMap _ Sequence" -- see SimplicialModuleMap _ ZZ,
    -- access individual matrices in a simplicial
    module map
  * SimplicialModuleMap _ ZZ -- access
    individual matrices in a simplicial module map
  * source(SimplicialModuleMap) -- get
    the source of a map of simplicial modules
  * target(SimplicialModuleMap) -- get
    the target of a map of simplicial modules
  * "topDegree(SimplicialModuleMap)" -- see topDegree, -- top
    degree of a simplicial module

For the programmer
==================

The object SimplicialModuleMap, is a type (Macaulay2Doc),
with ancestor classes HashTable (Macaulay2Doc),
< Thing (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1356:0.

SimplicialModuleMap * SimplicialModuleMap -- composition of homomorphisms of simplicial modules
***********************************************************************************************

  * Operator: * (Macaulay2Doc),
  * Usage:
        f = h * g
  * Inputs:
      * h, a Map of Simplicial Modules, if a ring
        element or integer, then we multiply the ring element by the
        appropriate identity map
      * g, a Map of Simplicial Modules,
  * Outputs:
      * f, a Map of Simplicial Modules, the
        composition of $g$ followed by $h$

Description
===========

If $g_i : C_i \rightarrow D_{d+i}$, and $h_j : D_j \rightarrow E_{e+j}$, then
the composition corresponds to $f_i := h_{d+i} * g_i : C_i \rightarrow
E_{i+d+e}$.  In particular, the degree of the composition $f$ is the sum of the
degrees of $g$ and $h$.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[a..d]                                                        |
|                                                                             |
|o1 = R                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution coker vars R, Degeneracy => true)   |
|                                                                             |
|      1      5      15      35      70                                       |
|o2 = R  <-- R  <-- R   <-- R   <-- R  <-- ...                                |
|                                                                             |
|     0      1      2       3       4                                         |
|                                                                             |
|o2 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i3 : 3 * dd^C                                                                |
|                                                                             |
|o3 = 0                                                                       |
|                                                                             |
|o3 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+
|i4 : 0 * dd^C                                                                |
|                                                                             |
|o4 = 0                                                                       |
|                                                                             |
|o4 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+
|i5 : dd^C * ss^C --if C is a well-defined simplicial object, these should all|
|                                                                             |
|               1             1                                               |
|o5 = (0, 0) : R  <--------- R  : (0, 0)                                      |
|                    | 1 |                                                    |
|                                                                             |
|               5                         5                                   |
|     (1, 0) : R  <--------------------- R  : (1, 0)                          |
|                    {0} | 1 0 0 0 0 |                                        |
|                    {1} | 0 1 0 0 0 |                                        |
|                    {1} | 0 0 1 0 0 |                                        |
|                    {1} | 0 0 0 1 0 |                                        |
|                    {1} | 0 0 0 0 1 |                                        |
|                                                                             |
|               5                         5                                   |
|     (1, 1) : R  <--------------------- R  : (1, 1)                          |
|                    {0} | 1 0 0 0 0 |                                        |
|                    {1} | 0 1 0 0 0 |                                        |
|                    {1} | 0 0 1 0 0 |                                        |
|                    {1} | 0 0 0 1 0 |                                        |
|                    {1} | 0 0 0 0 1 |                                        |
|                                                                             |
|               15                                             15             |
|     (2, 0) : R   <----------------------------------------- R   : (2, 0)    |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                   |
|                                                                             |
|               15                                             15             |
|     (2, 1) : R   <----------------------------------------- R   : (2, 1)    |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                   |
|                                                                             |
|               15                                             15             |
|     (2, 2) : R   <----------------------------------------- R   : (2, 2)    |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                   |
|                     {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                   |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                   |
|                                                                             |
|               35                                                            |
|     (3, 0) : R   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|               35                                                            |
|     (3, 1) : R   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|               35                                                            |
|     (3, 2) : R   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|               35                                                            |
|     (3, 3) : R   <----------------------------------------------------------|
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                     {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|o5 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
| be identity maps                                                            |
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|                         35                                                  |
|----------------------- R   : (3, 0)                                         |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|1 0 0 0 0 0 0 0 0 0 |                                                        |
|0 1 0 0 0 0 0 0 0 0 |                                                        |
|0 0 1 0 0 0 0 0 0 0 |                                                        |
|0 0 0 1 0 0 0 0 0 0 |                                                        |
|0 0 0 0 1 0 0 0 0 0 |                                                        |
|0 0 0 0 0 1 0 0 0 0 |                                                        |
|0 0 0 0 0 0 1 0 0 0 |                                                        |
|0 0 0 0 0 0 0 1 0 0 |                                                        |
|0 0 0 0 0 0 0 0 1 0 |                                                        |
|0 0 0 0 0 0 0 0 0 1 |                                                        |
|                                                                             |
|                         35                                                  |
|----------------------- R   : (3, 1)                                         |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|1 0 0 0 0 0 0 0 0 0 |                                                        |
|0 1 0 0 0 0 0 0 0 0 |                                                        |
|0 0 1 0 0 0 0 0 0 0 |                                                        |
|0 0 0 1 0 0 0 0 0 0 |                                                        |
|0 0 0 0 1 0 0 0 0 0 |                                                        |
|0 0 0 0 0 1 0 0 0 0 |                                                        |
|0 0 0 0 0 0 1 0 0 0 |                                                        |
|0 0 0 0 0 0 0 1 0 0 |                                                        |
|0 0 0 0 0 0 0 0 1 0 |                                                        |
|0 0 0 0 0 0 0 0 0 1 |                                                        |
|                                                                             |
|                         35                                                  |
|----------------------- R   : (3, 2)                                         |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|1 0 0 0 0 0 0 0 0 0 |                                                        |
|0 1 0 0 0 0 0 0 0 0 |                                                        |
|0 0 1 0 0 0 0 0 0 0 |                                                        |
|0 0 0 1 0 0 0 0 0 0 |                                                        |
|0 0 0 0 1 0 0 0 0 0 |                                                        |
|0 0 0 0 0 1 0 0 0 0 |                                                        |
|0 0 0 0 0 0 1 0 0 0 |                                                        |
|0 0 0 0 0 0 0 1 0 0 |                                                        |
|0 0 0 0 0 0 0 0 1 0 |                                                        |
|0 0 0 0 0 0 0 0 0 1 |                                                        |
|                                                                             |
|                         35                                                  |
|----------------------- R   : (3, 3)                                         |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|0 0 0 0 0 0 0 0 0 0 |                                                        |
|1 0 0 0 0 0 0 0 0 0 |                                                        |
|0 1 0 0 0 0 0 0 0 0 |                                                        |
|0 0 1 0 0 0 0 0 0 0 |                                                        |
|0 0 0 1 0 0 0 0 0 0 |                                                        |
|0 0 0 0 1 0 0 0 0 0 |                                                        |
|0 0 0 0 0 1 0 0 0 0 |                                                        |
|0 0 0 0 0 0 1 0 0 0 |                                                        |
|0 0 0 0 0 0 0 1 0 0 |                                                        |
|0 0 0 0 0 0 0 0 1 0 |                                                        |
|0 0 0 0 0 0 0 0 0 1 |                                                        |
+-----------------------------------------------------------------------------+

See also
========

  * Making maps between simplicial modules
    -- information about the basic constructors
  * arithmetic with simplicial module maps
    -- perform arithmetic operations on simplicial module maps

Ways to use this method:
========================

  * SimplicialModuleMap * SimplicialModuleMap
    -- composition of homomorphisms of simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1974:0.

SimplicialModuleMap ** SimplicialModuleMap -- the map of simplicial modules between tensor simplicial modules
*************************************************************************************************************

  * Operator: ** (Macaulay2Doc),
  * Usage:
        h = f ** g
        h = tensor(f, g)
  * Inputs:
      * f, a Map of Simplicial Modules,
      * g, a Map of Simplicial Modules,
  * Outputs:
      * h, a Map of Simplicial Modules,

Description
===========

The maps $f : C \to D$ and $g : E \to F$ of simplicial modules induces the map
$h = f \otimes g : C \otimes E \to D \otimes F$ defined by $c \otimes e \mapsto
f(c) \otimes g(e)$.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a..c]                                                        |
|                                                                             |
|o1 = S                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution coker vars S, Degeneracy => true)   |
|                                                                             |
|      1      4      10      20                                               |
|o2 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o2 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i3 : D = simplicialModule((freeResolution coker matrix{{a^2,a*b,b^3}})[-1],  |
|                                                                             |
|            1      5      14                                                 |
|o3 = 0 <-- S  <-- S  <-- S  <-- ...                                          |
|                                                                             |
|     0     1      2      3                                                   |
|                                                                             |
|o3 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i4 : f = randomSimplicialMap(D,C)                                            |
|                                                                             |
|                   1                                                         |
|o4 = 0 : 0 <----- S  : 0                                                     |
|              0                                                              |
|                                                                             |
|          1                                                  4               |
|     1 : S  <---------------------------------------------- S  : 1           |
|               | 0 24a-36b-30c -29a+19b+19c -10a-29b-8c |                    |
|                                                                             |
|          5                                                                  |
|     2 : S  <----------------------------------------------------------------|
|               {0} | 0 24a-36b-30c -29a+19b+19c -10a-29b-8c 0           0    |
|               {0} | 0 0           0            0           24a-36b-30c -29a+|
|               {2} | 0 0           0            0           0           0    |
|               {2} | 0 0           0            0           0           0    |
|               {3} | 0 0           0            0           0           0    |
|                                                                             |
|          14                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {0} | 0 24a-36b-30c -29a+19b+19c -10a-29b-8c 0           0   |
|                {0} | 0 0           0            0           24a-36b-30c -29a|
|                {0} | 0 0           0            0           0           0   |
|                {2} | 0 0           0            0           0           0   |
|                {2} | 0 0           0            0           0           0   |
|                {3} | 0 0           0            0           0           0   |
|                {2} | 0 0           0            0           0           0   |
|                {2} | 0 0           0            0           0           0   |
|                {3} | 0 0           0            0           0           0   |
|                {2} | 0 0           0            0           0           0   |
|                {2} | 0 0           0            0           0           0   |
|                {3} | 0 0           0            0           0           0   |
|                {3} | 0 0           0            0           0           0   |
|                {4} | 0 0           0            0           0           0   |
|                                                                             |
|o4 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                     10                                      |
|----------------------------------- S   : 2                                  |
|        0           0   0   0   |                                            |
|19b+19c -10a-29b-8c 0   0   0   |                                            |
|        0           -22 -24 -16 |                                            |
|        0           -29 -38 39  |                                            |
|        0           0   0   0   |                                            |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|         0           0           0            0           0   0   0   0   0  |
|+19b+19c -10a-29b-8c 0           0            0           0   0   0   0   0  |
|         0           24a-36b-30c -29a+19b+19c -10a-29b-8c 0   0   0   0   0  |
|         0           0           0            0           -22 -24 -16 0   0  |
|         0           0           0            0           -29 -38 39  0   0  |
|         0           0           0            0           0   0   0   0   0  |
|         0           0           0            0           0   0   0   -22 -24|
|         0           0           0            0           0   0   0   -29 -38|
|         0           0           0            0           0   0   0   0   0  |
|         0           0           0            0           0   0   0   0   0  |
|         0           0           0            0           0   0   0   0   0  |
|         0           0           0            0           0   0   0   0   0  |
|         0           0           0            0           0   0   0   0   0  |
|         0           0           0            0           0   0   0   0   0  |
|-----------------------------------------------------------------------------|
|                         20                                                  |
|----------------------- S   : 3                                              |
| 0   0   0   0   0  |                                                        |
| 0   0   0   0   0  |                                                        |
| 0   0   0   0   0  |                                                        |
| 0   0   0   0   0  |                                                        |
| 0   0   0   0   0  |                                                        |
| 0   0   0   0   0  |                                                        |
| -16 0   0   0   0  |                                                        |
| 39  0   0   0   0  |                                                        |
| 0   0   0   0   0  |                                                        |
| 0   -22 -24 -16 0  |                                                        |
| 0   -29 -38 39  0  |                                                        |
| 0   0   0   0   0  |                                                        |
| 0   0   0   0   21 |                                                        |
| 0   0   0   0   0  |                                                        |
+-----------------------------------------------------------------------------+
|i5 : E = simplicialModule((dual C.complex)[-3], Degeneracy => true)          |
|                                                                             |
|      1      4      10      20                                               |
|o5 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o5 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i6 : F = simplicialModule((dual D.complex)[-3], 3, Degeneracy => true)       |
|                                                                             |
|      2      5      9      14                                                |
|o6 = S  <-- S  <-- S  <-- S  <-- ...                                         |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o6 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i7 : g = randomSimplicialMap(F,E)                                            |
|                                                                             |
|          2                            1                                     |
|o7 = 0 : S  <------------------------ S  : 0                                 |
|               {-3} | 34          |                                          |
|               {-4} | 19a-47b-39c |                                          |
|                                                                             |
|          5                                                                 4|
|     1 : S  <------------------------------------------------------------- S |
|               {-3} | 34          0            0          0            |     |
|               {-4} | 19a-47b-39c 0            0          0            |     |
|               {-2} | 0           -18          -47        45           |     |
|               {-2} | 0           -13          38         -34          |     |
|               {-3} | 0           -43a-15b-28c 2a+16b+22c -48a-47b+47c |     |
|                                                                             |
|          9                                                                  |
|     2 : S  <----------------------------------------------------------------|
|               {-3} | 34          0            0          0            0     |
|               {-4} | 19a-47b-39c 0            0          0            0     |
|               {-2} | 0           -18          -47        45           0     |
|               {-2} | 0           -13          38         -34          0     |
|               {-3} | 0           -43a-15b-28c 2a+16b+22c -48a-47b+47c 0     |
|               {-2} | 0           0            0          0            -18   |
|               {-2} | 0           0            0          0            -13   |
|               {-3} | 0           0            0          0            -43a-1|
|               {0}  | 0           0            0          0            0     |
|                                                                             |
|          14                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {-3} | 34          0            0          0            0    |
|                {-4} | 19a-47b-39c 0            0          0            0    |
|                {-2} | 0           -18          -47        45           0    |
|                {-2} | 0           -13          38         -34          0    |
|                {-3} | 0           -43a-15b-28c 2a+16b+22c -48a-47b+47c 0    |
|                {-2} | 0           0            0          0            -18  |
|                {-2} | 0           0            0          0            -13  |
|                {-3} | 0           0            0          0            -43a-|
|                {-2} | 0           0            0          0            0    |
|                {-2} | 0           0            0          0            0    |
|                {-3} | 0           0            0          0            0    |
|                {0}  | 0           0            0          0            0    |
|                {0}  | 0           0            0          0            0    |
|                {0}  | 0           0            0          0            0    |
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
| : 1                                                                         |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                          10                                 |
|---------------------------------------- S   : 2                             |
|       0          0            0 0 0 |                                       |
|       0          0            0 0 0 |                                       |
|       0          0            0 0 0 |                                       |
|       0          0            0 0 0 |                                       |
|       0          0            0 0 0 |                                       |
|       -47        45           0 0 0 |                                       |
|       38         -34          0 0 0 |                                       |
|5b-28c 2a+16b+22c -48a-47b+47c 0 0 0 |                                       |
|       0          0            0 0 0 |                                       |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|        0          0            0            0          0            0 0 0 0 |
|        0          0            0            0          0            0 0 0 0 |
|        0          0            0            0          0            0 0 0 0 |
|        0          0            0            0          0            0 0 0 0 |
|        0          0            0            0          0            0 0 0 0 |
|        -47        45           0            0          0            0 0 0 0 |
|        38         -34          0            0          0            0 0 0 0 |
|15b-28c 2a+16b+22c -48a-47b+47c 0            0          0            0 0 0 0 |
|        0          0            -18          -47        45           0 0 0 0 |
|        0          0            -13          38         -34          0 0 0 0 |
|        0          0            -43a-15b-28c 2a+16b+22c -48a-47b+47c 0 0 0 0 |
|        0          0            0            0          0            0 0 0 0 |
|        0          0            0            0          0            0 0 0 0 |
|        0          0            0            0          0            0 0 0 0 |
|-----------------------------------------------------------------------------|
|                 20                                                          |
|--------------- S   : 3                                                      |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
|0 0 0 0 0 0 |                                                                |
+-----------------------------------------------------------------------------+
|i8 : h = f ** g;                                                             |
+-----------------------------------------------------------------------------+
|i9 : assert isWellDefined h                                                  |
+-----------------------------------------------------------------------------+
|i10 : assert(prune source h == C ** E)                                       |
+-----------------------------------------------------------------------------+
|i11 : assert(prune target h == D ** F)                                       |
+-----------------------------------------------------------------------------+

If one argument is a SimplicialModule or Module, then the identity map of the
corresponding complex is used.

+------------------------------+
|i12 : fE = f ** E;            |
+------------------------------+
|i13 : assert(fE == f ** id_E) |
+------------------------------+
|i14 : k = coker vars S        |
|                              |
|o14 = cokernel | a b c |      |
|                              |
|                             1|
|o14 : S-module, quotient of S |
+------------------------------+
|i15 : gk = g ** k;            |
+------------------------------+

This routine is functorial.

+-----------------------------------------------------------------------------+
|i16 : D' = simplicialModule((freeResolution coker matrix{{a^2,a*b,c^3}})[-1],|
|                                                                             |
|             1      5      15                                                |
|o16 = 0 <-- S  <-- S  <-- S  <-- ...                                         |
|                                                                             |
|      0     1      2      3                                                  |
|                                                                             |
|o16 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|3, Degeneracy => true)                                                       |
+-----------------------------------------------------------------------------+
|i17 : f' = randomSimplicialMap(D', D)                                        |
|                                                                             |
|o17 = 0 : 0 <----- 0 : 0                                                     |
|               0                                                             |
|                                                                             |
|           1              1                                                  |
|      1 : S  <---------- S  : 1                                              |
|                | 19 |                                                       |
|                                                                             |
|           5                                          5                      |
|      2 : S  <-------------------------------------- S  : 2                  |
|                {0} | 19 0  0   0   0            |                           |
|                {0} | 0  19 0   0   0            |                           |
|                {2} | 0  0  -16 15  39a+43b-17c  |                           |
|                {2} | 0  0  7   -23 -11a+48b+36c |                           |
|                {3} | 0  0  0   0   35           |                           |
|                                                                             |
|           15                                                                |
|      3 : S   <--------------------------------------------------------------|
|                 {0} | 19 0  0  0   0   0            0   0   0            0  |
|                 {0} | 0  19 0  0   0   0            0   0   0            0  |
|                 {0} | 0  0  19 0   0   0            0   0   0            0  |
|                 {2} | 0  0  0  -16 15  39a+43b-17c  0   0   0            0  |
|                 {2} | 0  0  0  7   -23 -11a+48b+36c 0   0   0            0  |
|                 {3} | 0  0  0  0   0   35           0   0   0            0  |
|                 {2} | 0  0  0  0   0   0            -16 15  39a+43b-17c  0  |
|                 {2} | 0  0  0  0   0   0            7   -23 -11a+48b+36c 0  |
|                 {3} | 0  0  0  0   0   0            0   0   35           0  |
|                 {2} | 0  0  0  0   0   0            0   0   0            -16|
|                 {2} | 0  0  0  0   0   0            0   0   0            7  |
|                 {3} | 0  0  0  0   0   0            0   0   0            0  |
|                 {3} | 0  0  0  0   0   0            0   0   0            0  |
|                 {5} | 0  0  0  0   0   0            0   0   0            0  |
|                 {5} | 0  0  0  0   0   0            0   0   0            0  |
|                                                                             |
|o17 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                                       14                                    |
|------------------------------------- S   : 3                                |
| 0   0            0  0            |                                          |
| 0   0            0  0            |                                          |
| 0   0            0  0            |                                          |
| 0   0            0  0            |                                          |
| 0   0            0  0            |                                          |
| 0   0            0  0            |                                          |
| 0   0            0  0            |                                          |
| 0   0            0  0            |                                          |
| 0   0            0  0            |                                          |
| 15  39a+43b-17c  0  0            |                                          |
| -23 -11a+48b+36c 0  0            |                                          |
| 0   35           0  0            |                                          |
| 0   0            11 -38a+33b+40c |                                          |
| 0   0            0  0            |                                          |
| 0   0            0  0            |                                          |
+-----------------------------------------------------------------------------+
|i18 : (f' * f) ** g == (f' ** g) * (f ** id_E)                               |
|                                                                             |
|o18 = true                                                                   |
+-----------------------------------------------------------------------------+
|i19 : (f' * f) ** g == (f' ** id_F) * (f ** g)                               |
|                                                                             |
|o19 = true                                                                   |
+-----------------------------------------------------------------------------+
|i20 : F' = simplicialModule(dual (freeResolution coker                       |
|                                                                             |
|       1      5      13      26                                              |
|o20 = S  <-- S  <-- S   <-- S  <-- ...                                       |
|                                                                             |
|      0      1      2       3                                                |
|                                                                             |
|o20 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|matrix{{a^2,a*b,a*c,b^3}})[-3], Degeneracy => true)                          |
+-----------------------------------------------------------------------------+
|i21 : g' = randomSimplicialMap(F', F)                                        |
|                                                                             |
|           1                              2                                  |
|o21 = 0 : S  <-------------------------- S  : 0                              |
|                {-4} | 11a+46b-28c 1 |                                       |
|                                                                             |
|           5                                                                 |
|      1 : S  <---------------------------------------------------------------|
|                {-4} | 11a+46b-28c 1 0                              0        |
|                {-3} | 0           0 -3a+22b-47c                    27a-22b+3|
|                {-3} | 0           0 -23a-7b+2c                     -9a-32b-2|
|                {-3} | 0           0 29a-47b+15c                    24a-30b-4|
|                {-4} | 0           0 -37a2-13ab+30b2-10ac-18bc+39c2 -15a2+39a|
|                                                                             |
|           13                                                                |
|      2 : S   <--------------------------------------------------------------|
|                 {-4} | 11a+46b-28c 1 0                              0       |
|                 {-3} | 0           0 -3a+22b-47c                    27a-22b+|
|                 {-3} | 0           0 -23a-7b+2c                     -9a-32b-|
|                 {-3} | 0           0 29a-47b+15c                    24a-30b-|
|                 {-4} | 0           0 -37a2-13ab+30b2-10ac-18bc+39c2 -15a2+39|
|                 {-3} | 0           0 0                              0       |
|                 {-3} | 0           0 0                              0       |
|                 {-3} | 0           0 0                              0       |
|                 {-4} | 0           0 0                              0       |
|                 {-2} | 0           0 0                              0       |
|                 {-2} | 0           0 0                              0       |
|                 {-2} | 0           0 0                              0       |
|                 {-3} | 0           0 0                              0       |
|                                                                             |
|           26                                                                |
|      3 : S   <--------------------------------------------------------------|
|                 {-4} | 11a+46b-28c 1 0                              0       |
|                 {-3} | 0           0 -3a+22b-47c                    27a-22b+|
|                 {-3} | 0           0 -23a-7b+2c                     -9a-32b-|
|                 {-3} | 0           0 29a-47b+15c                    24a-30b-|
|                 {-4} | 0           0 -37a2-13ab+30b2-10ac-18bc+39c2 -15a2+39|
|                 {-3} | 0           0 0                              0       |
|                 {-3} | 0           0 0                              0       |
|                 {-3} | 0           0 0                              0       |
|                 {-4} | 0           0 0                              0       |
|                 {-3} | 0           0 0                              0       |
|                 {-3} | 0           0 0                              0       |
|                 {-3} | 0           0 0                              0       |
|                 {-4} | 0           0 0                              0       |
|                 {-2} | 0           0 0                              0       |
|                 {-2} | 0           0 0                              0       |
|                 {-2} | 0           0 0                              0       |
|                 {-3} | 0           0 0                              0       |
|                 {-2} | 0           0 0                              0       |
|                 {-2} | 0           0 0                              0       |
|                 {-2} | 0           0 0                              0       |
|                 {-3} | 0           0 0                              0       |
|                 {-2} | 0           0 0                              0       |
|                 {-2} | 0           0 0                              0       |
|                 {-2} | 0           0 0                              0       |
|                 {-3} | 0           0 0                              0       |
|                 {0}  | 0           0 0                              0       |
|                                                                             |
|o21 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                                  5                                          |
|-------------------------------- S  : 1                                      |
|                 0           |                                               |
|2c               -19         |                                               |
|0c               17          |                                               |
|8c               -20         |                                               |
|b+33b2-49bc-33c2 44a-39b+36c |                                               |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                  0           0                              0               |
|32c               -19         0                              0               |
|20c               17          0                              0               |
|48c               -20         0                              0               |
|ab+33b2-49bc-33c2 44a-39b+36c 0                              0               |
|                  0           -3a+22b-47c                    27a-22b+32c     |
|                  0           -23a-7b+2c                     -9a-32b-20c     |
|                  0           29a-47b+15c                    24a-30b-48c     |
|                  0           -37a2-13ab+30b2-10ac-18bc+39c2 -15a2+39ab+33b2-|
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                  0           0                              0               |
|32c               -19         0                              0               |
|20c               17          0                              0               |
|48c               -20         0                              0               |
|ab+33b2-49bc-33c2 44a-39b+36c 0                              0               |
|                  0           -3a+22b-47c                    27a-22b+32c     |
|                  0           -23a-7b+2c                     -9a-32b-20c     |
|                  0           29a-47b+15c                    24a-30b-48c     |
|                  0           -37a2-13ab+30b2-10ac-18bc+39c2 -15a2+39ab+33b2-|
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|                  0           0                              0               |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|          0           0                                                    | |
|          0           0                                                    | |
|          0           0                                                    | |
|          0           0                                                    | |
|          0           0                                                    | |
|          -19         0                                                    | |
|          17          0                                                    | |
|          -20         0                                                    | |
|49bc-33c2 44a-39b+36c 0                                                    | |
|          0           9a2-39ab+13b2+4ac-26bc+22c2                          | |
|          0           -49a2-11ab+43b2-8ac-8bc+36c2                         | |
|          0           -3a2-22ab+41b2-30ac+16bc-28c2                        | |
|          0           -6a3+35a2b-35ab2+3b3-9a2c+6abc-31b2c+40ac2+25bc2-2c3 | |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|          0           0                              0                       |
|          0           0                              0                       |
|          0           0                              0                       |
|          0           0                              0                       |
|          0           0                              0                       |
|          -19         0                              0                       |
|          17          0                              0                       |
|          -20         0                              0                       |
|49bc-33c2 44a-39b+36c 0                              0                       |
|          0           -3a+22b-47c                    27a-22b+32c             |
|          0           -23a-7b+2c                     -9a-32b-20c             |
|          0           29a-47b+15c                    24a-30b-48c             |
|          0           -37a2-13ab+30b2-10ac-18bc+39c2 -15a2+39ab+33b2-49bc-33c|
|          0           0                              0                       |
|          0           0                              0                       |
|          0           0                              0                       |
|          0           0                              0                       |
|          0           0                              0                       |
|          0           0                              0                       |
|          0           0                              0                       |
|          0           0                              0                       |
|          0           0                              0                       |
|          0           0                              0                       |
|          0           0                              0                       |
|          0           0                              0                       |
|          0           0                              0                       |
|-----------------------------------------------------------------------------|
|   9                                                                         |
|- S  : 2                                                                     |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|  0           0                                                    0         |
|  0           0                                                    0         |
|  0           0                                                    0         |
|  0           0                                                    0         |
|  0           0                                                    0         |
|  0           0                                                    0         |
|  0           0                                                    0         |
|  0           0                                                    0         |
|  0           0                                                    0         |
|  -19         0                                                    0         |
|  17          0                                                    0         |
|  -20         0                                                    0         |
|2 44a-39b+36c 0                                                    0         |
|  0           9a2-39ab+13b2+4ac-26bc+22c2                          0         |
|  0           -49a2-11ab+43b2-8ac-8bc+36c2                         0         |
|  0           -3a2-22ab+41b2-30ac+16bc-28c2                        0         |
|  0           -6a3+35a2b-35ab2+3b3-9a2c+6abc-31b2c+40ac2+25bc2-2c3 0         |
|  0           0                                                    9a2-39ab+1|
|  0           0                                                    -49a2-11ab|
|  0           0                                                    -3a2-22ab+|
|  0           0                                                    -6a3+35a2b|
|  0           0                                                    0         |
|  0           0                                                    0         |
|  0           0                                                    0         |
|  0           0                                                    0         |
|  0           0                                                    0         |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|                                           0                                 |
|3b2+4ac-26bc+22c2                          0                                 |
|+43b2-8ac-8bc+36c2                         0                                 |
|41b2-30ac+16bc-28c2                        0                                 |
|-35ab2+3b3-9a2c+6abc-31b2c+40ac2+25bc2-2c3 0                                 |
|                                           9a2-39ab+13b2+4ac-26bc+22c2       |
|                                           -49a2-11ab+43b2-8ac-8bc+36c2      |
|                                           -3a2-22ab+41b2-30ac+16bc-28c2     |
|                                           -6a3+35a2b-35ab2+3b3-9a2c+6abc-31b|
|                                           0                                 |
|-----------------------------------------------------------------------------|
|                        14                                                   |
|---------------------- S   : 3                                               |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|2c+40ac2+25bc2-2c3 |                                                         |
|                   |                                                         |
+-----------------------------------------------------------------------------+
|i22 : f ** (g' * g) == (f ** g') * (id_C ** g)                               |
|                                                                             |
|o22 = true                                                                   |
+-----------------------------------------------------------------------------+
|i23 : f ** (g' * g) == (id_D ** g') * (f ** g)                               |
|                                                                             |
|o23 = true                                                                   |
+-----------------------------------------------------------------------------+

See also
========

  * SimplicialModule ** SimplicialModule
    -- tensor product of simplicial modules
  * randomSimplicialMap(SimplicialModule,SimplicialModule)
    -- a random
    map of simplicial modules

Ways to use this method:
========================

  * "Complex ** SimplicialModuleMap"
  * "ComplexMap ** SimplicialModuleMap"
  * "Module ** SimplicialModuleMap"
  * "SimplicialModule ** SimplicialModuleMap"
  * "SimplicialModuleMap ** Complex"
  * "SimplicialModuleMap ** ComplexMap"
  * "SimplicialModuleMap ** Module"
  * "SimplicialModuleMap ** SimplicialModule"
  * SimplicialModuleMap ** SimplicialModuleMap
    -- the map of simplicial modules between tensor
    simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2291:0.

SimplicialModuleMap ++ SimplicialModuleMap -- direct sum of simplicial module maps
**********************************************************************************

  * Operator: ++ (Macaulay2Doc),
  * Usage:
        h = f ++ g
        h = directSum(f,g,...)
        h = directSum(name1 => f, name2 => g, ...)
  * Inputs:
      * f, a Map of Simplicial Modules,
      * g, a Map of Simplicial Modules,
  * Outputs:
      * h, a Map of Simplicial Modules, that is the
        direct sum of the input simplicial module maps

Description
===========

The direct sum of two simplicial module maps is a simplicial module map from
the direct sum of the sources to the direct sum of the targets.

First, we define some non-trivial maps of simplicial modules.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[a..d];                                                       |
+-----------------------------------------------------------------------------+
|i2 : C1 = simplicialModule((freeResolution coker matrix{{a,b,c}})[1],        |
|                                                                             |
|      3      6      10                                                       |
|o2 = R  <-- R  <-- R  <-- ...                                                |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i3 : C2 = simplicialModule(freeResolution coker matrix{{a*b,a*c,b*c}},       |
|                                                                             |
|      1      4      9                                                        |
|o3 = R  <-- R  <-- R <-- ...                                                 |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o3 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i4 : D1 = simplicialModule((freeResolution coker matrix{{a,b,c}}),2,         |
|                                                                             |
|      1      4      10                                                       |
|o4 = R  <-- R  <-- R  <-- ...                                                |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o4 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i5 : D2 = simplicialModule(freeResolution coker matrix{{a^2, b^2, c^2}}[-1], |
|                                                                             |
|            1      5                                                         |
|o5 = 0 <-- R  <-- R <-- ...                                                  |
|                                                                             |
|     0     1      2                                                          |
|                                                                             |
|o5 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|2, Degeneracy => true)                                                       |
+-----------------------------------------------------------------------------+
|i6 : f = randomSimplicialMap(D1, C1, Cycle => true)                          |
|                                                                             |
|          1                                                        3         |
|o6 = 0 : R  <---------------------------------------------------- R  : 0     |
|               | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d |              |
|                                                                             |
|          4                                                                  |
|     1 : R  <----------------------------------------------------------------|
|               {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0          |
|               {1} | 0               0              0             -48a+3b-10c|
|               {1} | 0               0              0             46a+17b-29c|
|               {1} | 0               0              0             -18a+21b   |
|                                                                             |
|          10                                                                 |
|     2 : R   <---------------------------------------------------------------|
|                {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0         |
|                {1} | 0               0              0             -48a+3b-10|
|                {1} | 0               0              0             46a+17b-29|
|                {1} | 0               0              0             -18a+21b  |
|                {1} | 0               0              0             0         |
|                {1} | 0               0              0             0         |
|                {1} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                                                                             |
|o6 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                           6                                 |
|----------------------------------------- R  : 1                             |
|     0               0                |                                      |
|-29d -5a+36b+14c+29d 19b-34c          |                                      |
|-24d -39a+39c        -24a-3b-24c+29d  |                                      |
|     -21a-22b-8c-24d -19a-32b+28c+29d |                                      |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|      0               0                0               0               0     |
|c-29d -5a+36b+14c+29d 19b-34c          0               0               0     |
|c-24d -39a+39c        -24a-3b-24c+29d  0               0               0     |
|      -21a-22b-8c-24d -19a-32b+28c+29d 0               0               0     |
|      0               0                -48a+3b-10c-29d -5a+36b+14c+29d 19b-34|
|      0               0                46a+17b-29c-24d -39a+39c        -24a-3|
|      0               0                -18a+21b        -21a-22b-8c-24d -19a-3|
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|-----------------------------------------------------------------------------|
|                                10                                           |
|------------------------------ R   : 2                                       |
|           0               |                                                 |
|           0               |                                                 |
|           0               |                                                 |
|           0               |                                                 |
|c          0               |                                                 |
|b-24c+29d  0               |                                                 |
|2b+28c+29d 0               |                                                 |
|           24a-36b-30c-29d |                                                 |
|           19a+19b-10c-29d |                                                 |
|           -8a-22b-29c-24d |                                                 |
+-----------------------------------------------------------------------------+
|i7 : g = randomSimplicialMap(D2, C2, Cycle => true)                          |
|                                                                             |
|                   1                                                         |
|o7 = 0 : 0 <----- R  : 0                                                     |
|              0                                                              |
|                                                                             |
|          1                                                                  |
|     1 : R  <----------------------------------------------------------------|
|               | 0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc|
|                                                                             |
|          5                                                                  |
|     2 : R  <----------------------------------------------------------------|
|               {0} | 0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-|
|               {0} | 0 0                                  0                  |
|               {2} | 0 0                                  0                  |
|               {2} | 0 0                                  0                  |
|               {2} | 0 0                                  0                  |
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                                  4                          |
|------------------------------------------------ R  : 1                      |
|-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad |                               |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                           |
|               0                                 19a2+47ab-16b2-43ac-15bc-28c|
|               0                                 0                           |
|               0                                 0                           |
|               0                                 0                           |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|       0                                 0                                 0 |
|2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0 |
|       0                                 0                                 -7|
|       0                                 0                                 -4|
|       0                                 0                                 -4|
|-----------------------------------------------------------------------------|
|                                    9                                        |
|---------------------------------- R  : 2                                    |
|               0               |                                             |
|               0               |                                             |
|b+19c          -38a+5b-16c-22d |                                             |
|5a+34b+32c+47d 30a-34b-48c-47d |                                             |
|3a+8b-28c-47d  -39a-23b        |                                             |
+-----------------------------------------------------------------------------+
+-----------------------------------------------------------------------------+
|i8 : h = f ++ g                                                              |
|                                                                             |
|          1                                                          4       |
|o8 = 0 : R  <------------------------------------------------------ R  : 0   |
|               | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0 |            |
|                                                                             |
|          5                                                                  |
|     1 : R  <----------------------------------------------------------------|
|               {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0          |
|               {1} | 0               0              0             -48a+3b-10c|
|               {1} | 0               0              0             46a+17b-29c|
|               {1} | 0               0              0             -18a+21b   |
|               {0} | 0               0              0             0          |
|                                                                             |
|          15                                                                 |
|     2 : R   <---------------------------------------------------------------|
|                {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0         |
|                {1} | 0               0              0             -48a+3b-10|
|                {1} | 0               0              0             46a+17b-29|
|                {1} | 0               0              0             -18a+21b  |
|                {1} | 0               0              0             0         |
|                {1} | 0               0              0             0         |
|                {1} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {0} | 0               0              0             0         |
|                {0} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                                                                             |
|o8 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|     0               0                0 0                                  0 |
|-29d -5a+36b+14c+29d 19b-34c          0 0                                  0 |
|-24d -39a+39c        -24a-3b-24c+29d  0 0                                  0 |
|     -21a-22b-8c-24d -19a-32b+28c+29d 0 0                                  0 |
|     0               0                0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|      0               0                0               0               0     |
|c-29d -5a+36b+14c+29d 19b-34c          0               0               0     |
|c-24d -39a+39c        -24a-3b-24c+29d  0               0               0     |
|      -21a-22b-8c-24d -19a-32b+28c+29d 0               0               0     |
|      0               0                -48a+3b-10c-29d -5a+36b+14c+29d 19b-34|
|      0               0                46a+17b-29c-24d -39a+39c        -24a-3|
|      0               0                -18a+21b        -21a-22b-8c-24d -19a-3|
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|-----------------------------------------------------------------------------|
|                                                                       10    |
|--------------------------------------------------------------------- R   : 1|
|                                0                                 |          |
|                                0                                 |          |
|                                0                                 |          |
|                                0                                 |          |
|2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad |          |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|           0               0 0                                  0            |
|           0               0 0                                  0            |
|           0               0 0                                  0            |
|           0               0 0                                  0            |
|c          0               0 0                                  0            |
|b-24c+29d  0               0 0                                  0            |
|2b+28c+29d 0               0 0                                  0            |
|           24a-36b-30c-29d 0 0                                  0            |
|           19a+19b-10c-29d 0 0                                  0            |
|           -8a-22b-29c-24d 0 0                                  0            |
|           0               0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2|
|           0               0 0                                  0            |
|           0               0 0                                  0            |
|           0               0 0                                  0            |
|           0               0 0                                  0            |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|                     0                                 0                     |
|                     0                                 0                     |
|                     0                                 0                     |
|                     0                                 0                     |
|                     0                                 0                     |
|                     0                                 0                     |
|                     0                                 0                     |
|                     0                                 0                     |
|                     0                                 0                     |
|                     0                                 0                     |
|+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                     |
|                     0                                 19a2+47ab-16b2-43ac-15|
|                     0                                 0                     |
|                     0                                 0                     |
|                     0                                 0                     |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|             0                                 0                             |
|             0                                 0                             |
|             0                                 0                             |
|             0                                 0                             |
|             0                                 0                             |
|             0                                 0                             |
|             0                                 0                             |
|             0                                 0                             |
|             0                                 0                             |
|             0                                 0                             |
|             0                                 0                             |
|bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+2|
|             0                                 0                             |
|             0                                 0                             |
|             0                                 0                             |
|-----------------------------------------------------------------------------|
|                                          19                                 |
|---------------------------------------- R   : 2                             |
|    0                0               |                                       |
|    0                0               |                                       |
|    0                0               |                                       |
|    0                0               |                                       |
|    0                0               |                                       |
|    0                0               |                                       |
|    0                0               |                                       |
|    0                0               |                                       |
|    0                0               |                                       |
|    0                0               |                                       |
|    0                0               |                                       |
|2ad 0                0               |                                       |
|    -7b+19c          -38a+5b-16c-22d |                                       |
|    -45a+34b+32c+47d 30a-34b-48c-47d |                                       |
|    -43a+8b-28c-47d  -39a-23b        |                                       |
+-----------------------------------------------------------------------------+
|i9 : assert isWellDefined h                                                  |
+-----------------------------------------------------------------------------+
|i10 : assert isSimplicialMorphism h                                          |
+-----------------------------------------------------------------------------+

The direct sum of any sequence of simplicial module maps can be computed as
follows.

+-----------------------------------------------------------------------------+
|i11 : directSum(f, g, f[2])                                                  |
|                                                                             |
|           4                                                                 |
|o11 = 0 : R  <---------------------------------------------------------------|
|                {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0 0       |
|                {2} | 0               0              0             0 24a-36b-|
|                {2} | 0               0              0             0 19a+19b-|
|                {2} | 0               0              0             0 -8a-22b-|
|                                                                             |
|           8                                                                 |
|      1 : R  <---------------------------------------------------------------|
|                {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0         |
|                {1} | 0               0              0             -48a+3b-10|
|                {1} | 0               0              0             46a+17b-29|
|                {1} | 0               0              0             -18a+21b  |
|                {0} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                                                                             |
|           18                                                                |
|      2 : R   <--------------------------------------------------------------|
|                 {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0        |
|                 {1} | 0               0              0             -48a+3b-1|
|                 {1} | 0               0              0             46a+17b-2|
|                 {1} | 0               0              0             -18a+21b |
|                 {1} | 0               0              0             0        |
|                 {1} | 0               0              0             0        |
|                 {1} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {0} | 0               0              0             0        |
|                 {0} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                                                                             |
|o11 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|             5                                                               |
|----------- R  : 0                                                           |
|        |                                                                    |
|30c-29d |                                                                    |
|10c-29d |                                                                    |
|29c-24d |                                                                    |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|      0               0                0 0                                  0|
|c-29d -5a+36b+14c+29d 19b-34c          0 0                                  0|
|c-24d -39a+39c        -24a-3b-24c+29d  0 0                                  0|
|      -21a-22b-8c-24d -19a-32b+28c+29d 0 0                                  0|
|      0               0                0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7|
|      0               0                0 0                                  0|
|      0               0                0 0                                  0|
|      0               0                0 0                                  0|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|       0               0                0               0               0    |
|0c-29d -5a+36b+14c+29d 19b-34c          0               0               0    |
|9c-24d -39a+39c        -24a-3b-24c+29d  0               0               0    |
|       -21a-22b-8c-24d -19a-32b+28c+29d 0               0               0    |
|       0               0                -48a+3b-10c-29d -5a+36b+14c+29d 19b-3|
|       0               0                46a+17b-29c-24d -39a+39c        -24a-|
|       0               0                -18a+21b        -21a-22b-8c-24d -19a-|
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|                                 0                                 0         |
|                                 0                                 0         |
|                                 0                                 0         |
|                                 0                                 0         |
|a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0         |
|                                 0                                 24a-36b-30|
|                                 0                                 19a+19b-10|
|                                 0                                 -8a-22b-29|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|4c          0               0 0                                  0           |
|3b-24c+29d  0               0 0                                  0           |
|32b+28c+29d 0               0 0                                  0           |
|            24a-36b-30c-29d 0 0                                  0           |
|            19a+19b-10c-29d 0 0                                  0           |
|            -8a-22b-29c-24d 0 0                                  0           |
|            0               0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b|
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|-----------------------------------------------------------------------------|
|           11                                                                |
|--------- R   : 1                                                            |
|      |                                                                      |
|      |                                                                      |
|      |                                                                      |
|      |                                                                      |
|      |                                                                      |
|c-29d |                                                                      |
|c-29d |                                                                      |
|c-24d |                                                                      |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                    |
|                      0                                 19a2+47ab-16b2-43ac-1|
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|5bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+|
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|-----------------------------------------------------------------------------|
|                                                           20                |
|--------------------------------------------------------- R   : 2            |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|22ad 0                0               0               |                      |
|     -7b+19c          -38a+5b-16c-22d 0               |                      |
|     -45a+34b+32c+47d 30a-34b-48c-47d 0               |                      |
|     -43a+8b-28c-47d  -39a-23b        0               |                      |
|     0                0               24a-36b-30c-29d |                      |
|     0                0               19a+19b-10c-29d |                      |
|     0                0               -8a-22b-29c-24d |                      |
+-----------------------------------------------------------------------------+
|i12 : h2 = directSum(peanut => f, butter => g, jelly => f[2])                |
|                                                                             |
|           4                                                                 |
|o12 = 0 : R  <---------------------------------------------------------------|
|                {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0 0       |
|                {2} | 0               0              0             0 24a-36b-|
|                {2} | 0               0              0             0 19a+19b-|
|                {2} | 0               0              0             0 -8a-22b-|
|                                                                             |
|           8                                                                 |
|      1 : R  <---------------------------------------------------------------|
|                {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0         |
|                {1} | 0               0              0             -48a+3b-10|
|                {1} | 0               0              0             46a+17b-29|
|                {1} | 0               0              0             -18a+21b  |
|                {0} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                                                                             |
|           18                                                                |
|      2 : R   <--------------------------------------------------------------|
|                 {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0        |
|                 {1} | 0               0              0             -48a+3b-1|
|                 {1} | 0               0              0             46a+17b-2|
|                 {1} | 0               0              0             -18a+21b |
|                 {1} | 0               0              0             0        |
|                 {1} | 0               0              0             0        |
|                 {1} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {0} | 0               0              0             0        |
|                 {0} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                 {2} | 0               0              0             0        |
|                                                                             |
|o12 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|             5                                                               |
|----------- R  : 0                                                           |
|        |                                                                    |
|30c-29d |                                                                    |
|10c-29d |                                                                    |
|29c-24d |                                                                    |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|      0               0                0 0                                  0|
|c-29d -5a+36b+14c+29d 19b-34c          0 0                                  0|
|c-24d -39a+39c        -24a-3b-24c+29d  0 0                                  0|
|      -21a-22b-8c-24d -19a-32b+28c+29d 0 0                                  0|
|      0               0                0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7|
|      0               0                0 0                                  0|
|      0               0                0 0                                  0|
|      0               0                0 0                                  0|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|       0               0                0               0               0    |
|0c-29d -5a+36b+14c+29d 19b-34c          0               0               0    |
|9c-24d -39a+39c        -24a-3b-24c+29d  0               0               0    |
|       -21a-22b-8c-24d -19a-32b+28c+29d 0               0               0    |
|       0               0                -48a+3b-10c-29d -5a+36b+14c+29d 19b-3|
|       0               0                46a+17b-29c-24d -39a+39c        -24a-|
|       0               0                -18a+21b        -21a-22b-8c-24d -19a-|
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|       0               0                0               0               0    |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|                                 0                                 0         |
|                                 0                                 0         |
|                                 0                                 0         |
|                                 0                                 0         |
|a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0         |
|                                 0                                 24a-36b-30|
|                                 0                                 19a+19b-10|
|                                 0                                 -8a-22b-29|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|4c          0               0 0                                  0           |
|3b-24c+29d  0               0 0                                  0           |
|32b+28c+29d 0               0 0                                  0           |
|            24a-36b-30c-29d 0 0                                  0           |
|            19a+19b-10c-29d 0 0                                  0           |
|            -8a-22b-29c-24d 0 0                                  0           |
|            0               0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b|
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|            0               0 0                                  0           |
|-----------------------------------------------------------------------------|
|           11                                                                |
|--------- R   : 1                                                            |
|      |                                                                      |
|      |                                                                      |
|      |                                                                      |
|      |                                                                      |
|      |                                                                      |
|c-29d |                                                                      |
|c-29d |                                                                      |
|c-24d |                                                                      |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                    |
|                      0                                 19a2+47ab-16b2-43ac-1|
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|                      0                                 0                    |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|5bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+|
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|              0                                 0                            |
|-----------------------------------------------------------------------------|
|                                                           20                |
|--------------------------------------------------------- R   : 2            |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|     0                0               0               |                      |
|22ad 0                0               0               |                      |
|     -7b+19c          -38a+5b-16c-22d 0               |                      |
|     -45a+34b+32c+47d 30a-34b-48c-47d 0               |                      |
|     -43a+8b-28c-47d  -39a-23b        0               |                      |
|     0                0               24a-36b-30c-29d |                      |
|     0                0               19a+19b-10c-29d |                      |
|     0                0               -8a-22b-29c-24d |                      |
+-----------------------------------------------------------------------------+
|i13 : indices h2                                                             |
|                                                                             |
|o13 = {peanut, butter, jelly}                                                |
|                                                                             |
|o13 : List                                                                   |
+-----------------------------------------------------------------------------+
|i14 : h2_[butter,jelly]                                                      |
|                                                                             |
|           4                                 2                               |
|o14 = 0 : R  <----------------------------- R  : 0                           |
|                {0} | 0 0               |                                    |
|                {2} | 0 24a-36b-30c-29d |                                    |
|                {2} | 0 19a+19b-10c-29d |                                    |
|                {2} | 0 -8a-22b-29c-24d |                                    |
|                                                                             |
|           8                                                                 |
|      1 : R  <---------------------------------------------------------------|
|                {0} | 0 0                                  0                 |
|                {1} | 0 0                                  0                 |
|                {1} | 0 0                                  0                 |
|                {1} | 0 0                                  0                 |
|                {0} | 0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac|
|                {2} | 0 0                                  0                 |
|                {2} | 0 0                                  0                 |
|                {2} | 0 0                                  0                 |
|                                                                             |
|           18                                                                |
|      2 : R   <--------------------------------------------------------------|
|                 {0} | 0 0                                  0                |
|                 {1} | 0 0                                  0                |
|                 {1} | 0 0                                  0                |
|                 {1} | 0 0                                  0                |
|                 {1} | 0 0                                  0                |
|                 {1} | 0 0                                  0                |
|                 {1} | 0 0                                  0                |
|                 {2} | 0 0                                  0                |
|                 {2} | 0 0                                  0                |
|                 {2} | 0 0                                  0                |
|                 {0} | 0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47a|
|                 {0} | 0 0                                  0                |
|                 {2} | 0 0                                  0                |
|                 {2} | 0 0                                  0                |
|                 {2} | 0 0                                  0                |
|                 {2} | 0 0                                  0                |
|                 {2} | 0 0                                  0                |
|                 {2} | 0 0                                  0                |
|                                                                             |
|o14 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                                                                       5     |
|--------------------------------------------------------------------- R  : 1 |
|                0                                 0               |          |
|                0                                 0               |          |
|                0                                 0               |          |
|                0                                 0               |          |
|-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0               |          |
|                0                                 24a-36b-30c-29d |          |
|                0                                 19a+19b-10c-29d |          |
|                0                                 -8a-22b-29c-24d |          |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                 0                                 0                         |
|                 0                                 0                         |
|                 0                                 0                         |
|                 0                                 0                         |
|                 0                                 0                         |
|                 0                                 0                         |
|                 0                                 0                         |
|                 0                                 0                         |
|                 0                                 0                         |
|                 0                                 0                         |
|c-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                         |
|                 0                                 19a2+47ab-16b2-43ac-15bc-2|
|                 0                                 0                         |
|                 0                                 0                         |
|                 0                                 0                         |
|                 0                                 0                         |
|                 0                                 0                         |
|                 0                                 0                         |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|         0                                 0                                 |
|         0                                 0                                 |
|         0                                 0                                 |
|         0                                 0                                 |
|         0                                 0                                 |
|         0                                 0                                 |
|         0                                 0                                 |
|         0                                 0                                 |
|         0                                 0                                 |
|         0                                 0                                 |
|         0                                 0                                 |
|8c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad |
|         0                                 0                                 |
|         0                                 0                                 |
|         0                                 0                                 |
|         0                                 0                                 |
|         0                                 0                                 |
|         0                                 0                                 |
|-----------------------------------------------------------------------------|
|                                                      10                     |
|---------------------------------------------------- R   : 2                 |
|0                0               0               |                           |
|0                0               0               |                           |
|0                0               0               |                           |
|0                0               0               |                           |
|0                0               0               |                           |
|0                0               0               |                           |
|0                0               0               |                           |
|0                0               0               |                           |
|0                0               0               |                           |
|0                0               0               |                           |
|0                0               0               |                           |
|0                0               0               |                           |
|-7b+19c          -38a+5b-16c-22d 0               |                           |
|-45a+34b+32c+47d 30a-34b-48c-47d 0               |                           |
|-43a+8b-28c-47d  -39a-23b        0               |                           |
|0                0               24a-36b-30c-29d |                           |
|0                0               19a+19b-10c-29d |                           |
|0                0               -8a-22b-29c-24d |                           |
+-----------------------------------------------------------------------------+
|i15 : assert(source oo == C2 ++ C1[2])                                       |
+-----------------------------------------------------------------------------+

One can easily obtain the compositions with canonical injections and
surjections.

+--------------------+
|i16 : h_[0]^[0] == f|
|                    |
|o16 = true          |
+--------------------+
|i17 : h_[1]^[1] == g|
|                    |
|o17 = true          |
+--------------------+
|i18 : h_[0]^[1] == 0|
|                    |
|o18 = true          |
+--------------------+
|i19 : h_[1]^[0] == 0|
|                    |
|o19 = true          |
+--------------------+
+---------------------------------+
|i20 : h_[0] == h * (C1 ++ C2)_[0]|
|                                 |
|o20 = true                       |
+---------------------------------+
|i21 : h_[1] == h * (C1 ++ C2)_[1]|
|                                 |
|o21 = true                       |
+---------------------------------+
|i22 : h^[0] == (D1 ++ D2)^[0] * h|
|                                 |
|o22 = true                       |
+---------------------------------+
|i23 : h^[1] == (D1 ++ D2)^[1] * h|
|                                 |
|o23 = true                       |
+---------------------------------+

See also
========

  * SimplicialModule ++ SimplicialModule
    -- direct sum of simplicial modules
  * SimplicialModuleMap ** SimplicialModuleMap
    -- the map of simplicial modules between tensor
    simplicial modules
  * SimplicialModuleMap _ Array -- the
    composition with the canonical inclusion or projection map

Ways to use this method:
========================

  * SimplicialModuleMap ++ SimplicialModuleMap
    -- direct sum of simplicial module maps
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2729:0.

SimplicialModuleMap == SimplicialModuleMap -- whether two simplicial module maps are equal
******************************************************************************************

  * Operator: == (Macaulay2Doc),
  * Usage:
        f == g
        f == 0
        f == 1
  * Inputs:
      * f, a Map of Simplicial Modules, or 0, or 1.
      * g, a Map of Simplicial Modules, or 0, or 1.
  * Outputs:
      * a Boolean value (Macaulay2Doc), that is true when f and
        g are equal

Description
===========

Two simplicial module maps are equal if they have the same source, the same
target, and $f_i = g_i$ for all $i$.

+--------------------------------------------------------------------------+
|i1 : S = ZZ/101[a..c]                                                     |
|                                                                          |
|o1 = S                                                                    |
|                                                                          |
|o1 : PolynomialRing                                                       |
+--------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution coker vars S, Degeneracy => true)|
|                                                                          |
|      1      4      10      20                                            |
|o2 = S  <-- S  <-- S   <-- S  <-- ...                                     |
|                                                                          |
|     0      1      2       3                                              |
|                                                                          |
|o2 : SimplicialModule                                                     |
+--------------------------------------------------------------------------+
|i3 : f = id_C                                                             |
|                                                                          |
|          1             1                                                 |
|o3 = 0 : S  <--------- S  : 0                                             |
|               | 1 |                                                      |
|                                                                          |
|          4                       4                                       |
|     1 : S  <------------------- S  : 1                                   |
|               {0} | 1 0 0 0 |                                            |
|               {1} | 0 1 0 0 |                                            |
|               {1} | 0 0 1 0 |                                            |
|               {1} | 0 0 0 1 |                                            |
|                                                                          |
|          10                                   10                         |
|     2 : S   <------------------------------- S   : 2                     |
|                {0} | 1 0 0 0 0 0 0 0 0 0 |                               |
|                {1} | 0 1 0 0 0 0 0 0 0 0 |                               |
|                {1} | 0 0 1 0 0 0 0 0 0 0 |                               |
|                {1} | 0 0 0 1 0 0 0 0 0 0 |                               |
|                {1} | 0 0 0 0 1 0 0 0 0 0 |                               |
|                {1} | 0 0 0 0 0 1 0 0 0 0 |                               |
|                {1} | 0 0 0 0 0 0 1 0 0 0 |                               |
|                {2} | 0 0 0 0 0 0 0 1 0 0 |                               |
|                {2} | 0 0 0 0 0 0 0 0 1 0 |                               |
|                {2} | 0 0 0 0 0 0 0 0 0 1 |                               |
|                                                                          |
|          20                                                       20     |
|     3 : S   <--------------------------------------------------- S   : 3 |
|                {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |           |
|                {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |           |
|                {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |           |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |           |
|                                                                          |
|o3 : SimplicialModuleMap                                                  |
+--------------------------------------------------------------------------+
|i4 : assert(f == 1)                                                       |
+--------------------------------------------------------------------------+

A simplicial module map is equal to zero if all the maps are zero. This could
require computation to determine if something that is superficially not zero is
in fact zero.

+--------------------------+
|i5 : assert(0 * id_C == 0)|
+--------------------------+
+-----------------------------------------------------------------------------+
|i6 : g = randomSimplicialMap(C, C)                                           |
|                                                                             |
|          1              1                                                   |
|o6 = 0 : S  <---------- S  : 0                                               |
|               | 24 |                                                        |
|                                                                             |
|          4                              4                                   |
|     1 : S  <-------------------------- S  : 1                               |
|               {0} | 24 0   0   0   |                                        |
|               {1} | 0  -36 19  -29 |                                        |
|               {1} | 0  -30 19  -8  |                                        |
|               {1} | 0  -29 -10 -22 |                                        |
|                                                                             |
|          10                                                      10         |
|     2 : S   <-------------------------------------------------- S   : 2     |
|                {0} | 24 0   0   0   0   0   0   0   0   0   |               |
|                {1} | 0  -36 19  -29 0   0   0   0   0   0   |               |
|                {1} | 0  -30 19  -8  0   0   0   0   0   0   |               |
|                {1} | 0  -29 -10 -22 0   0   0   0   0   0   |               |
|                {1} | 0  0   0   0   -36 19  -29 0   0   0   |               |
|                {1} | 0  0   0   0   -30 19  -8  0   0   0   |               |
|                {1} | 0  0   0   0   -29 -10 -22 0   0   0   |               |
|                {2} | 0  0   0   0   0   0   0   -29 -16 34  |               |
|                {2} | 0  0   0   0   0   0   0   -24 39  19  |               |
|                {2} | 0  0   0   0   0   0   0   -38 21  -47 |               |
|                                                                             |
|          20                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {0} | 24 0   0   0   0   0   0   0   0   0   0   0   0   0   |
|                {1} | 0  -36 19  -29 0   0   0   0   0   0   0   0   0   0   |
|                {1} | 0  -30 19  -8  0   0   0   0   0   0   0   0   0   0   |
|                {1} | 0  -29 -10 -22 0   0   0   0   0   0   0   0   0   0   |
|                {1} | 0  0   0   0   -36 19  -29 0   0   0   0   0   0   0   |
|                {1} | 0  0   0   0   -30 19  -8  0   0   0   0   0   0   0   |
|                {1} | 0  0   0   0   -29 -10 -22 0   0   0   0   0   0   0   |
|                {1} | 0  0   0   0   0   0   0   -36 19  -29 0   0   0   0   |
|                {1} | 0  0   0   0   0   0   0   -30 19  -8  0   0   0   0   |
|                {1} | 0  0   0   0   0   0   0   -29 -10 -22 0   0   0   0   |
|                {2} | 0  0   0   0   0   0   0   0   0   0   -29 -16 34  0   |
|                {2} | 0  0   0   0   0   0   0   0   0   0   -24 39  19  0   |
|                {2} | 0  0   0   0   0   0   0   0   0   0   -38 21  -47 0   |
|                {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -29 |
|                {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -24 |
|                {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -38 |
|                {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   |
|                {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   |
|                {2} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   |
|                {3} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   |
|                                                                             |
|o6 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                             20                                              |
|--------------------------- S   : 3                                          |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|-16 34  0   0   0   0   |                                                    |
|39  19  0   0   0   0   |                                                    |
|21  -47 0   0   0   0   |                                                    |
|0   0   -29 -16 34  0   |                                                    |
|0   0   -24 39  19  0   |                                                    |
|0   0   -38 21  -47 0   |                                                    |
|0   0   0   0   0   -39 |                                                    |
+-----------------------------------------------------------------------------+
|i7 : h = inducedMap(coker g, target g)                                       |
|                                                                             |
|                                 1                                           |
|o7 = 0 : cokernel | 24 | <----- S  : 0                                       |
|                            0                                                |
|                                                                             |
|                                                 4                           |
|     1 : cokernel {0} | 24 0   0   0   | <----- S  : 1                       |
|                  {1} | 0  -36 19  -29 |    0                                |
|                  {1} | 0  -30 19  -8  |                                     |
|                  {1} | 0  -29 -10 -22 |                                     |
|                                                                             |
|                                                                             |
|     2 : cokernel {0} | 24 0   0   0   0   0   0   0   0   0                 |
|                  {1} | 0  -36 19  -29 0   0   0   0   0   0                 |
|                  {1} | 0  -30 19  -8  0   0   0   0   0   0                 |
|                  {1} | 0  -29 -10 -22 0   0   0   0   0   0                 |
|                  {1} | 0  0   0   0   -36 19  -29 0   0   0                 |
|                  {1} | 0  0   0   0   -30 19  -8  0   0   0                 |
|                  {1} | 0  0   0   0   -29 -10 -22 0   0   0                 |
|                  {2} | 0  0   0   0   0   0   0   -29 -16 34                |
|                  {2} | 0  0   0   0   0   0   0   -24 39  19                |
|                  {2} | 0  0   0   0   0   0   0   -38 21  -47               |
|                                                                             |
|                                                                             |
|     3 : cokernel {0} | 24 0   0   0   0   0   0   0   0   0                 |
|                  {1} | 0  -36 19  -29 0   0   0   0   0   0                 |
|                  {1} | 0  -30 19  -8  0   0   0   0   0   0                 |
|                  {1} | 0  -29 -10 -22 0   0   0   0   0   0                 |
|                  {1} | 0  0   0   0   -36 19  -29 0   0   0                 |
|                  {1} | 0  0   0   0   -30 19  -8  0   0   0                 |
|                  {1} | 0  0   0   0   -29 -10 -22 0   0   0                 |
|                  {1} | 0  0   0   0   0   0   0   -36 19  -29               |
|                  {1} | 0  0   0   0   0   0   0   -30 19  -8                |
|                  {1} | 0  0   0   0   0   0   0   -29 -10 -22               |
|                  {2} | 0  0   0   0   0   0   0   0   0   0                 |
|                  {2} | 0  0   0   0   0   0   0   0   0   0                 |
|                  {2} | 0  0   0   0   0   0   0   0   0   0                 |
|                  {2} | 0  0   0   0   0   0   0   0   0   0                 |
|                  {2} | 0  0   0   0   0   0   0   0   0   0                 |
|                  {2} | 0  0   0   0   0   0   0   0   0   0                 |
|                  {2} | 0  0   0   0   0   0   0   0   0   0                 |
|                  {2} | 0  0   0   0   0   0   0   0   0   0                 |
|                  {2} | 0  0   0   0   0   0   0   0   0   0                 |
|                  {3} | 0  0   0   0   0   0   0   0   0   0                 |
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|          10                                                                 |
|| <----- S   : 2                                                             |
||    0                                                                       |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
|                                                                             |
|                                                  20                         |
|0   0   0   0   0   0   0   0   0   0   | <----- S   : 3                     |
|0   0   0   0   0   0   0   0   0   0   |    0                               |
|0   0   0   0   0   0   0   0   0   0   |                                    |
|0   0   0   0   0   0   0   0   0   0   |                                    |
|0   0   0   0   0   0   0   0   0   0   |                                    |
|0   0   0   0   0   0   0   0   0   0   |                                    |
|0   0   0   0   0   0   0   0   0   0   |                                    |
|0   0   0   0   0   0   0   0   0   0   |                                    |
|0   0   0   0   0   0   0   0   0   0   |                                    |
|0   0   0   0   0   0   0   0   0   0   |                                    |
|-29 -16 34  0   0   0   0   0   0   0   |                                    |
|-24 39  19  0   0   0   0   0   0   0   |                                    |
|-38 21  -47 0   0   0   0   0   0   0   |                                    |
|0   0   0   -29 -16 34  0   0   0   0   |                                    |
|0   0   0   -24 39  19  0   0   0   0   |                                    |
|0   0   0   -38 21  -47 0   0   0   0   |                                    |
|0   0   0   0   0   0   -29 -16 34  0   |                                    |
|0   0   0   0   0   0   -24 39  19  0   |                                    |
|0   0   0   0   0   0   -38 21  -47 0   |                                    |
|0   0   0   0   0   0   0   0   0   -39 |                                    |
+-----------------------------------------------------------------------------+
|i8 : assert(h == 0)                                                          |
+-----------------------------------------------------------------------------+

Testing whether a map is equal to 1 is a shorthand for determining if the
simplicial module map is the identity. Although the matrices may appear to be
the identity, the map is not the identity when the source and target are not
equal.

+-----------------------------------------------------------------------------+
|i9 : g = randomSimplicialMap(C, C, InternalDegree=>1, Cycle=>true)           |
|                                                                             |
|          1                       1                                          |
|o9 = 0 : S  <------------------- S  : 0                                      |
|               | -33a-3b+13c |                                               |
|                                                                             |
|          4                                                             4    |
|     1 : S  <--------------------------------------------------------- S  : 1|
|               {0} | -33a-3b+13c 0            0        0           |         |
|               {1} | 0           -33a+25b-41c -37b+34c -45b+33c    |         |
|               {1} | 0           -28a+16c     4a-3b+4c 45a-37c     |         |
|               {1} | 0           -47a-16b     -34a+9b  35a+34b+13c |         |
|                                                                             |
|          10                                                                 |
|     2 : S   <---------------------------------------------------------------|
|                {0} | -33a-3b+13c 0            0        0           0        |
|                {1} | 0           -33a+25b-41c -37b+34c -45b+33c    0        |
|                {1} | 0           -28a+16c     4a-3b+4c 45a-37c     0        |
|                {1} | 0           -47a-16b     -34a+9b  35a+34b+13c 0        |
|                {1} | 0           0            0        0           -33a+25b-|
|                {1} | 0           0            0        0           -28a+16c |
|                {1} | 0           0            0        0           -47a-16b |
|                {2} | 0           0            0        0           0        |
|                {2} | 0           0            0        0           0        |
|                {2} | 0           0            0        0           0        |
|                                                                             |
|          20                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {0} | -33a-3b+13c 0            0        0           0        |
|                {1} | 0           -33a+25b-41c -37b+34c -45b+33c    0        |
|                {1} | 0           -28a+16c     4a-3b+4c 45a-37c     0        |
|                {1} | 0           -47a-16b     -34a+9b  35a+34b+13c 0        |
|                {1} | 0           0            0        0           -33a+25b-|
|                {1} | 0           0            0        0           -28a+16c |
|                {1} | 0           0            0        0           -47a-16b |
|                {1} | 0           0            0        0           0        |
|                {1} | 0           0            0        0           0        |
|                {1} | 0           0            0        0           0        |
|                {2} | 0           0            0        0           0        |
|                {2} | 0           0            0        0           0        |
|                {2} | 0           0            0        0           0        |
|                {2} | 0           0            0        0           0        |
|                {2} | 0           0            0        0           0        |
|                {2} | 0           0            0        0           0        |
|                {2} | 0           0            0        0           0        |
|                {2} | 0           0            0        0           0        |
|                {2} | 0           0            0        0           0        |
|                {3} | 0           0            0        0           0        |
|                                                                             |
|o9 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                                                 10          |
|--------------------------------------------------------------- S   : 2      |
|    0        0           0          0           0           |                |
|    0        0           0          0           0           |                |
|    0        0           0          0           0           |                |
|    0        0           0          0           0           |                |
|41c -37b+34c -45b+33c    0          0           0           |                |
|    4a-3b+4c 45a-37c     0          0           0           |                |
|    -34a+9b  35a+34b+13c 0          0           0           |                |
|    0        0           4a+25b-43c 45a+38c     45b-22c     |                |
|    0        0           -34a+2b    35a-13b-41c -48b+34c    |                |
|    0        0           -47a+16b   47a+16c     -18a+34b+4c |                |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|    0        0           0            0        0           0          0      |
|    0        0           0            0        0           0          0      |
|    0        0           0            0        0           0          0      |
|    0        0           0            0        0           0          0      |
|41c -37b+34c -45b+33c    0            0        0           0          0      |
|    4a-3b+4c 45a-37c     0            0        0           0          0      |
|    -34a+9b  35a+34b+13c 0            0        0           0          0      |
|    0        0           -33a+25b-41c -37b+34c -45b+33c    0          0      |
|    0        0           -28a+16c     4a-3b+4c 45a-37c     0          0      |
|    0        0           -47a-16b     -34a+9b  35a+34b+13c 0          0      |
|    0        0           0            0        0           4a+25b-43c 45a+38c|
|    0        0           0            0        0           -34a+2b    35a-13b|
|    0        0           0            0        0           -47a+16b   47a+16c|
|    0        0           0            0        0           0          0      |
|    0        0           0            0        0           0          0      |
|    0        0           0            0        0           0          0      |
|    0        0           0            0        0           0          0      |
|    0        0           0            0        0           0          0      |
|    0        0           0            0        0           0          0      |
|    0        0           0            0        0           0          0      |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|     0           0          0           0           0          0           0 |
|     0           0          0           0           0          0           0 |
|     0           0          0           0           0          0           0 |
|     0           0          0           0           0          0           0 |
|     0           0          0           0           0          0           0 |
|     0           0          0           0           0          0           0 |
|     0           0          0           0           0          0           0 |
|     0           0          0           0           0          0           0 |
|     0           0          0           0           0          0           0 |
|     0           0          0           0           0          0           0 |
|     45b-22c     0          0           0           0          0           0 |
|-41c -48b+34c    0          0           0           0          0           0 |
|     -18a+34b+4c 0          0           0           0          0           0 |
|     0           4a+25b-43c 45a+38c     45b-22c     0          0           0 |
|     0           -34a+2b    35a-13b-41c -48b+34c    0          0           0 |
|     0           -47a+16b   47a+16c     -18a+34b+4c 0          0           0 |
|     0           0          0           0           4a+25b-43c 45a+38c     45|
|     0           0          0           0           -34a+2b    35a-13b-41c -4|
|     0           0          0           0           -47a+16b   47a+16c     -1|
|     0           0          0           0           0          0           0 |
|-----------------------------------------------------------------------------|
|                            20                                               |
|-------------------------- S   : 3                                           |
|          0            |                                                     |
|          0            |                                                     |
|          0            |                                                     |
|          0            |                                                     |
|          0            |                                                     |
|          0            |                                                     |
|          0            |                                                     |
|          0            |                                                     |
|          0            |                                                     |
|          0            |                                                     |
|          0            |                                                     |
|          0            |                                                     |
|          0            |                                                     |
|          0            |                                                     |
|          0            |                                                     |
|          0            |                                                     |
|b-22c     0            |                                                     |
|8b+34c    0            |                                                     |
|8a+34b+4c 0            |                                                     |
|          -18a-13b-43c |                                                     |
+-----------------------------------------------------------------------------+
|i10 : h = inducedMap(coker g, target g)                                      |
|                                                                             |
|                                               1                             |
|o10 = 0 : cokernel | -33a-3b+13c | <--------- S  : 0                         |
|                                      | 1 |                                  |
|                                                                             |
|                                                                             |
|      1 : cokernel {0} | -33a-3b+13c 0            0        0                 |
|                   {1} | 0           -33a+25b-41c -37b+34c -45b+33c          |
|                   {1} | 0           -28a+16c     4a-3b+4c 45a-37c           |
|                   {1} | 0           -47a-16b     -34a+9b  35a+34b+13c       |
|                                                                             |
|                                                                             |
|                                                                             |
|      2 : cokernel {0} | -33a-3b+13c 0            0        0                 |
|                   {1} | 0           -33a+25b-41c -37b+34c -45b+33c          |
|                   {1} | 0           -28a+16c     4a-3b+4c 45a-37c           |
|                   {1} | 0           -47a-16b     -34a+9b  35a+34b+13c       |
|                   {1} | 0           0            0        0                 |
|                   {1} | 0           0            0        0                 |
|                   {1} | 0           0            0        0                 |
|                   {2} | 0           0            0        0                 |
|                   {2} | 0           0            0        0                 |
|                   {2} | 0           0            0        0                 |
|                                                                             |
|                                                                             |
|                                                                             |
|      3 : cokernel {0} | -33a-3b+13c 0            0        0                 |
|                   {1} | 0           -33a+25b-41c -37b+34c -45b+33c          |
|                   {1} | 0           -28a+16c     4a-3b+4c 45a-37c           |
|                   {1} | 0           -47a-16b     -34a+9b  35a+34b+13c       |
|                   {1} | 0           0            0        0                 |
|                   {1} | 0           0            0        0                 |
|                   {1} | 0           0            0        0                 |
|                   {1} | 0           0            0        0                 |
|                   {1} | 0           0            0        0                 |
|                   {1} | 0           0            0        0                 |
|                   {2} | 0           0            0        0                 |
|                   {2} | 0           0            0        0                 |
|                   {2} | 0           0            0        0                 |
|                   {2} | 0           0            0        0                 |
|                   {2} | 0           0            0        0                 |
|                   {2} | 0           0            0        0                 |
|                   {2} | 0           0            0        0                 |
|                   {2} | 0           0            0        0                 |
|                   {2} | 0           0            0        0                 |
|                   {3} | 0           0            0        0                 |
|                                                                             |
|                                                                             |
|o10 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                        4                                                    |
|| <------------------- S  : 1                                                |
||    {0} | 1 0 0 0 |                                                         |
||    {1} | 0 1 0 0 |                                                         |
||    {1} | 0 0 1 0 |                                                         |
|     {1} | 0 0 0 1 |                                                         |
|                                                                             |
|                                                                             |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|-33a+25b-41c -37b+34c -45b+33c                                               |
|-28a+16c     4a-3b+4c 45a-37c                                                |
|-47a-16b     -34a+9b  35a+34b+13c                                            |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|                                                                             |
|                                                                             |
|                                                                             |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|-33a+25b-41c -37b+34c -45b+33c                                               |
|-28a+16c     4a-3b+4c 45a-37c                                                |
|-47a-16b     -34a+9b  35a+34b+13c                                            |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|0          0           0           | <-------------------------------        |
|0          0           0           |    {0} | 1 0 0 0 0 0 0 0 0 0 |          |
|0          0           0           |    {1} | 0 1 0 0 0 0 0 0 0 0 |          |
|0          0           0           |    {1} | 0 0 1 0 0 0 0 0 0 0 |          |
|0          0           0           |    {1} | 0 0 0 1 0 0 0 0 0 0 |          |
|0          0           0           |    {1} | 0 0 0 0 1 0 0 0 0 0 |          |
|0          0           0           |    {1} | 0 0 0 0 0 1 0 0 0 0 |          |
|4a+25b-43c 45a+38c     45b-22c     |    {1} | 0 0 0 0 0 0 1 0 0 0 |          |
|-34a+2b    35a-13b-41c -48b+34c    |    {2} | 0 0 0 0 0 0 0 1 0 0 |          |
|-47a+16b   47a+16c     -18a+34b+4c |    {2} | 0 0 0 0 0 0 0 0 1 0 |          |
|                                        {2} | 0 0 0 0 0 0 0 0 0 1 |          |
|                                                                             |
|                                                                             |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|-33a+25b-41c -37b+34c -45b+33c    0          0           0                   |
|-28a+16c     4a-3b+4c 45a-37c     0          0           0                   |
|-47a-16b     -34a+9b  35a+34b+13c 0          0           0                   |
|0            0        0           4a+25b-43c 45a+38c     45b-22c             |
|0            0        0           -34a+2b    35a-13b-41c -48b+34c            |
|0            0        0           -47a+16b   47a+16c     -18a+34b+4c         |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|                                                                             |
|-----------------------------------------------------------------------------|
|  10                                                                         |
| S   : 2                                                                     |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|4a+25b-43c 45a+38c     45b-22c     0          0           0                  |
|-34a+2b    35a-13b-41c -48b+34c    0          0           0                  |
|-47a+16b   47a+16c     -18a+34b+4c 0          0           0                  |
|0          0           0           4a+25b-43c 45a+38c     45b-22c            |
|0          0           0           -34a+2b    35a-13b-41c -48b+34c           |
|0          0           0           -47a+16b   47a+16c     -18a+34b+4c        |
|0          0           0           0          0           0                  |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                     20      |
|0            | <--------------------------------------------------- S   : 3  |
|0            |    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |            |
|-18a-13b-43c |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |            |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |            |
+-----------------------------------------------------------------------------+
|i11 : assert(h != 1)                                                         |
+-----------------------------------------------------------------------------+

Testing for equality is not the same as testing for isomorphism. In particular,
different presentations of a simplicial module need not be equal.

+-----------------------------------------------------------------------------+
|i12 : D = prune image g                                                      |
|                                                                             |
|       1      4      10      20                                              |
|o12 = S  <-- S  <-- S   <-- S  <-- ...                                       |
|                                                                             |
|      0      1      2       3                                                |
|                                                                             |
|o12 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i13 : p = D.cache.pruningMap                                                 |
|                                                                             |
|                                                1                            |
|o13 = 0 : image | -33a-3b+13c | <------------- S  : 0                        |
|                                   {1} | 1 |                                 |
|                                                                             |
|                                                                             |
|      1 : image {0} | -33a-3b+13c 0            0        0                    |
|                {1} | 0           -33a+25b-41c -37b+34c -45b+33c             |
|                {1} | 0           -28a+16c     4a-3b+4c 45a-37c              |
|                {1} | 0           -47a-16b     -34a+9b  35a+34b+13c          |
|                                                                             |
|                                                                             |
|                                                                             |
|      2 : image {0} | -33a-3b+13c 0            0        0                    |
|                {1} | 0           -33a+25b-41c -37b+34c -45b+33c             |
|                {1} | 0           -28a+16c     4a-3b+4c 45a-37c              |
|                {1} | 0           -47a-16b     -34a+9b  35a+34b+13c          |
|                {1} | 0           0            0        0                    |
|                {1} | 0           0            0        0                    |
|                {1} | 0           0            0        0                    |
|                {2} | 0           0            0        0                    |
|                {2} | 0           0            0        0                    |
|                {2} | 0           0            0        0                    |
|                                                                             |
|                                                                             |
|                                                                             |
|      3 : image {0} | -33a-3b+13c 0            0        0                    |
|                {1} | 0           -33a+25b-41c -37b+34c -45b+33c             |
|                {1} | 0           -28a+16c     4a-3b+4c 45a-37c              |
|                {1} | 0           -47a-16b     -34a+9b  35a+34b+13c          |
|                {1} | 0           0            0        0                    |
|                {1} | 0           0            0        0                    |
|                {1} | 0           0            0        0                    |
|                {1} | 0           0            0        0                    |
|                {1} | 0           0            0        0                    |
|                {1} | 0           0            0        0                    |
|                {2} | 0           0            0        0                    |
|                {2} | 0           0            0        0                    |
|                {2} | 0           0            0        0                    |
|                {2} | 0           0            0        0                    |
|                {2} | 0           0            0        0                    |
|                {2} | 0           0            0        0                    |
|                {2} | 0           0            0        0                    |
|                {2} | 0           0            0        0                    |
|                {2} | 0           0            0        0                    |
|                {3} | 0           0            0        0                    |
|                                                                             |
|                                                                             |
|o13 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                        4                                                    |
|| <------------------- S  : 1                                                |
||    {0} | 1 0 0 0 |                                                         |
||    {1} | 0 1 0 0 |                                                         |
||    {1} | 0 0 1 0 |                                                         |
|     {1} | 0 0 0 1 |                                                         |
|                                                                             |
|                                                                             |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|-33a+25b-41c -37b+34c -45b+33c                                               |
|-28a+16c     4a-3b+4c 45a-37c                                                |
|-47a-16b     -34a+9b  35a+34b+13c                                            |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|                                                                             |
|                                                                             |
|                                                                             |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|-33a+25b-41c -37b+34c -45b+33c                                               |
|-28a+16c     4a-3b+4c 45a-37c                                                |
|-47a-16b     -34a+9b  35a+34b+13c                                            |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|0            0        0                                                      |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|0          0           0           | <-------------------------------        |
|0          0           0           |    {0} | 1 0 0 0 0 0 0 0 0 0 |          |
|0          0           0           |    {1} | 0 1 0 0 0 0 0 0 0 0 |          |
|0          0           0           |    {1} | 0 0 1 0 0 0 0 0 0 0 |          |
|0          0           0           |    {1} | 0 0 0 1 0 0 0 0 0 0 |          |
|0          0           0           |    {1} | 0 0 0 0 1 0 0 0 0 0 |          |
|0          0           0           |    {1} | 0 0 0 0 0 1 0 0 0 0 |          |
|4a+25b-43c 45a+38c     45b-22c     |    {1} | 0 0 0 0 0 0 1 0 0 0 |          |
|-34a+2b    35a-13b-41c -48b+34c    |    {2} | 0 0 0 0 0 0 0 1 0 0 |          |
|-47a+16b   47a+16c     -18a+34b+4c |    {2} | 0 0 0 0 0 0 0 0 1 0 |          |
|                                        {2} | 0 0 0 0 0 0 0 0 0 1 |          |
|                                                                             |
|                                                                             |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|-33a+25b-41c -37b+34c -45b+33c    0          0           0                   |
|-28a+16c     4a-3b+4c 45a-37c     0          0           0                   |
|-47a-16b     -34a+9b  35a+34b+13c 0          0           0                   |
|0            0        0           4a+25b-43c 45a+38c     45b-22c             |
|0            0        0           -34a+2b    35a-13b-41c -48b+34c            |
|0            0        0           -47a+16b   47a+16c     -18a+34b+4c         |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|0            0        0           0          0           0                   |
|                                                                             |
|-----------------------------------------------------------------------------|
|  10                                                                         |
| S   : 2                                                                     |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|0          0           0           0          0           0                  |
|4a+25b-43c 45a+38c     45b-22c     0          0           0                  |
|-34a+2b    35a-13b-41c -48b+34c    0          0           0                  |
|-47a+16b   47a+16c     -18a+34b+4c 0          0           0                  |
|0          0           0           4a+25b-43c 45a+38c     45b-22c            |
|0          0           0           -34a+2b    35a-13b-41c -48b+34c           |
|0          0           0           -47a+16b   47a+16c     -18a+34b+4c        |
|0          0           0           0          0           0                  |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                     20      |
|0            | <--------------------------------------------------- S   : 3  |
|0            |    {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |            |
|0            |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |            |
|-18a-13b-43c |    {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |            |
|                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |            |
+-----------------------------------------------------------------------------+
|i14 : p == 1                                                                 |
|                                                                             |
|o14 = false                                                                  |
+-----------------------------------------------------------------------------+
|i15 : assert(coker p == 0 and ker p == 0)                                    |
+-----------------------------------------------------------------------------+
|i16 : assert(prune p == 1)                                                   |
+-----------------------------------------------------------------------------+

See also
========

  * SimplicialModule == SimplicialModule
    -- whether two simplicial modules are equal
  * SimplicialModuleMap Array -- shift a
    simplicial module or simplicial module map
  * randomSimplicialMap
    -- a random
    map of simplicial modules
  * prune(SimplicialModule),
    -- minimal presentation of all terms in a simplicial module

Ways to use this method:
========================

  * SimplicialModuleMap == SimplicialModuleMap
    -- whether two simplicial module maps are equal
  * "SimplicialModuleMap == ZZ"
  * "ZZ == SimplicialModuleMap"
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2088:0.

SimplicialModuleMap ^ Array -- the composition with the canonical inclusion or projection map
*********************************************************************************************

  * Operator: ^ (Macaulay2Doc),
  * Usage:
        i = f_[name]
        p = f^[name]
  * Inputs:
      * f, a Map of Simplicial Modules,
      * name, an array (Macaulay2Doc),
  * Outputs:
      * a Map of Simplicial Modules, i is the
        composition of f with the canonical inclusion and p is the composition
        of the canonical projection with f

Description
===========

The direct sum is an n-ary operator with projection and inclusion maps from
each component satisfying appropriate identities.

One can access these maps as follows.  First, we define some non-trivial maps
of simplicial modules.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[a..d];                                                       |
+-----------------------------------------------------------------------------+
|i2 : C1 = simplicialModule((freeResolution coker matrix{{a,b,c}})[1], 3,     |
|                                                                             |
|      3      6      10      15                                               |
|o2 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i3 : C2 = simplicialModule(freeResolution coker matrix{{a*b,a*c,b*c}}, 3,    |
|                                                                             |
|      1      4      9      16                                                |
|o3 = R  <-- R  <-- R  <-- R  <-- ...                                         |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o3 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i4 : D1 = simplicialModule((freeResolution coker matrix{{a,b,c}}), Degeneracy|
|                                                                             |
|      1      4      10      20                                               |
|o4 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o4 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|=> true)                                                                     |
+-----------------------------------------------------------------------------+
|i5 : D2 = simplicialModule(freeResolution coker matrix{{a^2, b^2, c^2}}[-1], |
|                                                                             |
|            1      5      15                                                 |
|o5 = 0 <-- R  <-- R  <-- R  <-- ...                                          |
|                                                                             |
|     0     1      2      3                                                   |
|                                                                             |
|o5 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|3, Degeneracy => true)                                                       |
+-----------------------------------------------------------------------------+
|i6 : f = randomSimplicialMap(D1, C1, Cycle => true)                          |
|                                                                             |
|          1                                                        3         |
|o6 = 0 : R  <---------------------------------------------------- R  : 0     |
|               | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d |              |
|                                                                             |
|          4                                                                  |
|     1 : R  <----------------------------------------------------------------|
|               {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0          |
|               {1} | 0               0              0             -48a+3b-10c|
|               {1} | 0               0              0             46a+17b-29c|
|               {1} | 0               0              0             -18a+21b   |
|                                                                             |
|          10                                                                 |
|     2 : R   <---------------------------------------------------------------|
|                {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0         |
|                {1} | 0               0              0             -48a+3b-10|
|                {1} | 0               0              0             46a+17b-29|
|                {1} | 0               0              0             -18a+21b  |
|                {1} | 0               0              0             0         |
|                {1} | 0               0              0             0         |
|                {1} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                                                                             |
|          20                                                                 |
|     3 : R   <---------------------------------------------------------------|
|                {0} | -46a+17b-8c-24d 48a+6b+28c+29d 5a+3b-39c-29d 0         |
|                {1} | 0               0              0             -48a+3b-10|
|                {1} | 0               0              0             46a+17b-29|
|                {1} | 0               0              0             -18a+21b  |
|                {1} | 0               0              0             0         |
|                {1} | 0               0              0             0         |
|                {1} | 0               0              0             0         |
|                {1} | 0               0              0             0         |
|                {1} | 0               0              0             0         |
|                {1} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {2} | 0               0              0             0         |
|                {3} | 0               0              0             0         |
|                                                                             |
|o6 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                           6                                 |
|----------------------------------------- R  : 1                             |
|     0               0                |                                      |
|-29d -5a+36b+14c+29d 19b-34c          |                                      |
|-24d -39a+39c        -24a-3b-24c+29d  |                                      |
|     -21a-22b-8c-24d -19a-32b+28c+29d |                                      |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|      0               0                0               0               0     |
|c-29d -5a+36b+14c+29d 19b-34c          0               0               0     |
|c-24d -39a+39c        -24a-3b-24c+29d  0               0               0     |
|      -21a-22b-8c-24d -19a-32b+28c+29d 0               0               0     |
|      0               0                -48a+3b-10c-29d -5a+36b+14c+29d 19b-34|
|      0               0                46a+17b-29c-24d -39a+39c        -24a-3|
|      0               0                -18a+21b        -21a-22b-8c-24d -19a-3|
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|      0               0                0               0               0     |
|c-29d -5a+36b+14c+29d 19b-34c          0               0               0     |
|c-24d -39a+39c        -24a-3b-24c+29d  0               0               0     |
|      -21a-22b-8c-24d -19a-32b+28c+29d 0               0               0     |
|      0               0                -48a+3b-10c-29d -5a+36b+14c+29d 19b-34|
|      0               0                46a+17b-29c-24d -39a+39c        -24a-3|
|      0               0                -18a+21b        -21a-22b-8c-24d -19a-3|
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|      0               0                0               0               0     |
|-----------------------------------------------------------------------------|
|                                10                                           |
|------------------------------ R   : 2                                       |
|           0               |                                                 |
|           0               |                                                 |
|           0               |                                                 |
|           0               |                                                 |
|c          0               |                                                 |
|b-24c+29d  0               |                                                 |
|2b+28c+29d 0               |                                                 |
|           24a-36b-30c-29d |                                                 |
|           19a+19b-10c-29d |                                                 |
|           -8a-22b-29c-24d |                                                 |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|           0               0               0                0               0|
|           0               0               0                0               0|
|           0               0               0                0               0|
|           0               0               0                0               0|
|c          0               0               0                0               0|
|b-24c+29d  0               0               0                0               0|
|2b+28c+29d 0               0               0                0               0|
|           -48a+3b-10c-29d -5a+36b+14c+29d 19b-34c          0               0|
|           46a+17b-29c-24d -39a+39c        -24a-3b-24c+29d  0               0|
|           -18a+21b        -21a-22b-8c-24d -19a-32b+28c+29d 0               0|
|           0               0               0                24a-36b-30c-29d 0|
|           0               0               0                19a+19b-10c-29d 0|
|           0               0               0                -8a-22b-29c-24d 0|
|           0               0               0                0               2|
|           0               0               0                0               1|
|           0               0               0                0               -|
|           0               0               0                0               0|
|           0               0               0                0               0|
|           0               0               0                0               0|
|           0               0               0                0               0|
|-----------------------------------------------------------------------------|
|                                    15                                       |
|---------------------------------- R   : 3                                   |
|               0               |                                             |
|               0               |                                             |
|               0               |                                             |
|               0               |                                             |
|               0               |                                             |
|               0               |                                             |
|               0               |                                             |
|               0               |                                             |
|               0               |                                             |
|               0               |                                             |
|               0               |                                             |
|               0               |                                             |
|               0               |                                             |
|4a-36b-30c-29d 0               |                                             |
|9a+19b-10c-29d 0               |                                             |
|8a-22b-29c-24d 0               |                                             |
|               24a-36b-30c-29d |                                             |
|               19a+19b-10c-29d |                                             |
|               -8a-22b-29c-24d |                                             |
|               0               |                                             |
+-----------------------------------------------------------------------------+
|i7 : g = randomSimplicialMap(D2, C2, Cycle => true)                          |
|                                                                             |
|                   1                                                         |
|o7 = 0 : 0 <----- R  : 0                                                     |
|              0                                                              |
|                                                                             |
|          1                                                                  |
|     1 : R  <----------------------------------------------------------------|
|               | 0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc|
|                                                                             |
|          5                                                                  |
|     2 : R  <----------------------------------------------------------------|
|               {0} | 0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-|
|               {0} | 0 0                                  0                  |
|               {2} | 0 0                                  0                  |
|               {2} | 0 0                                  0                  |
|               {2} | 0 0                                  0                  |
|                                                                             |
|          15                                                                 |
|     3 : R   <---------------------------------------------------------------|
|                {0} | 0 19a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac|
|                {0} | 0 0                                  0                 |
|                {0} | 0 0                                  0                 |
|                {2} | 0 0                                  0                 |
|                {2} | 0 0                                  0                 |
|                {2} | 0 0                                  0                 |
|                {2} | 0 0                                  0                 |
|                {2} | 0 0                                  0                 |
|                {2} | 0 0                                  0                 |
|                {2} | 0 0                                  0                 |
|                {2} | 0 0                                  0                 |
|                {2} | 0 0                                  0                 |
|                {4} | 0 0                                  0                 |
|                {4} | 0 0                                  0                 |
|                {4} | 0 0                                  0                 |
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                                  4                          |
|------------------------------------------------ R  : 1                      |
|-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad |                               |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                           |
|               0                                 19a2+47ab-16b2-43ac-15bc-28c|
|               0                                 0                           |
|               0                                 0                           |
|               0                                 0                           |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0                          |
|                0                                 19a2+47ab-16b2-43ac-15bc-28|
|                0                                 0                          |
|                0                                 0                          |
|                0                                 0                          |
|                0                                 0                          |
|                0                                 0                          |
|                0                                 0                          |
|                0                                 0                          |
|                0                                 0                          |
|                0                                 0                          |
|                0                                 0                          |
|                0                                 0                          |
|                0                                 0                          |
|                0                                 0                          |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|       0                                 0                                 0 |
|2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0 |
|       0                                 0                                 -7|
|       0                                 0                                 -4|
|       0                                 0                                 -4|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|        0                                 0                                 0|
|c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+15b2+16ac+47bc+39c2+22ad 0|
|        0                                 0                                 1|
|        0                                 0                                 0|
|        0                                 0                                 0|
|        0                                 0                                 0|
|        0                                 0                                 0|
|        0                                 0                                 0|
|        0                                 0                                 0|
|        0                                 0                                 0|
|        0                                 0                                 0|
|        0                                 0                                 0|
|        0                                 0                                 0|
|        0                                 0                                 0|
|        0                                 0                                 0|
|-----------------------------------------------------------------------------|
|                                    9                                        |
|---------------------------------- R  : 2                                    |
|               0               |                                             |
|               0               |                                             |
|b+19c          -38a+5b-16c-22d |                                             |
|5a+34b+32c+47d 30a-34b-48c-47d |                                             |
|3a+8b-28c-47d  -39a-23b        |                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                  0                                 0        |
|                                  0                                 0        |
|9a2+47ab-16b2-43ac-15bc-28c2-47cd 7a2+45ab-34b2+47ac-48bc-23c2-47bd 38a2+2ab+|
|                                  0                                 0        |
|                                  0                                 0        |
|                                  0                                 0        |
|                                  0                                 0        |
|                                  0                                 0        |
|                                  0                                 0        |
|                                  0                                 0        |
|                                  0                                 0        |
|                                  0                                 0        |
|                                  0                                 0        |
|                                  0                                 0        |
|                                  0                                 0        |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|                         0                0               0                0 |
|                         0                0               0                0 |
|15b2+16ac+47bc+39c2+22ad 0                0               0                0 |
|                         -7b+19c          -38a+5b-16c-22d 0                0 |
|                         -45a+34b+32c+47d 30a-34b-48c-47d 0                0 |
|                         -43a+8b-28c-47d  -39a-23b        0                0 |
|                         0                0               -7b+19c          -3|
|                         0                0               -45a+34b+32c+47d 30|
|                         0                0               -43a+8b-28c-47d  -3|
|                         0                0               0                0 |
|                         0                0               0                0 |
|                         0                0               0                0 |
|                         0                0               0                0 |
|                         0                0               0                0 |
|                         0                0               0                0 |
|-----------------------------------------------------------------------------|
|                                                    16                       |
|-------------------------------------------------- R   : 3                   |
|              0                0               |                             |
|              0                0               |                             |
|              0                0               |                             |
|              0                0               |                             |
|              0                0               |                             |
|              0                0               |                             |
|8a+5b-16c-22d 0                0               |                             |
|a-34b-48c-47d 0                0               |                             |
|9a-23b        0                0               |                             |
|              -7b+19c          -38a+5b-16c-22d |                             |
|              -45a+34b+32c+47d 30a-34b-48c-47d |                             |
|              -43a+8b-28c-47d  -39a-23b        |                             |
|              0                0               |                             |
|              0                0               |                             |
|              0                0               |                             |
+-----------------------------------------------------------------------------+
+----------------+
|i8 : h = f ++ g;|
+----------------+

The four basic maps are the inclusion from each summand of the source and the
projection to each summand of the target.

+---------------------------------+
|i9 : h_[0] == h * (C1 ++ C2)_[0] |
|                                 |
|o9 = true                        |
+---------------------------------+
|i10 : h_[1] == h * (C1 ++ C2)_[1]|
|                                 |
|o10 = true                       |
+---------------------------------+
|i11 : h^[0] == (D1 ++ D2)^[0] * h|
|                                 |
|o11 = true                       |
+---------------------------------+
|i12 : h^[1] == (D1 ++ D2)^[1] * h|
|                                 |
|o12 = true                       |
+---------------------------------+

These can be combined to obtain the blocks of the map of simplicial modules.

+--------------------+
|i13 : h_[0]^[0] == f|
|                    |
|o13 = true          |
+--------------------+
|i14 : h_[1]^[1] == g|
|                    |
|o14 = true          |
+--------------------+
|i15 : h_[0]^[1] == 0|
|                    |
|o15 = true          |
+--------------------+
|i16 : h_[1]^[0] == 0|
|                    |
|o16 = true          |
+--------------------+

The default names for the components are the non-negative integers.  However,
one can choose any name.

+-------------------------------------------+
|i17 : h = (chicken => f) ++ (nuggets => g);|
+-------------------------------------------+
|i18 : indices h                            |
|                                           |
|o18 = {chicken, nuggets}                   |
|                                           |
|o18 : List                                 |
+-------------------------------------------+
|i19 : h_[chicken]^[chicken] == f           |
|                                           |
|o19 = true                                 |
+-------------------------------------------+
|i20 : h_[nuggets]^[nuggets] == g           |
|                                           |
|o20 = true                                 |
+-------------------------------------------+

See also
========

  * SimplicialModule ++ SimplicialModule
    -- direct sum of simplicial modules
  * SimplicialModule ^ Array -- the
    canonical inclusion or projection map of a direct sum
  * SimplicialModule _ Array -- the
    canonical inclusion or projection map of a direct sum
  * directSum(SimplicialModule), --
    direct sum of simplicial modules
  * components(SimplicialModule), --
    list the components of a direct sum
  * indices (Macaulay2Doc) -- indices of a polynomial; also
    components for a direct sum

Ways to use this method:
========================

  * SimplicialModuleMap ^ Array -- the
    composition with the canonical inclusion or projection map
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2962:0.

SimplicialModuleMap ^ ZZ -- the n-fold composition
**************************************************

  * Operator: ^ (Macaulay2Doc),
  * Usage:
        f^n
  * Inputs:
      * f, a Map of Simplicial Modules, whose
        source and target are the same simplicial module
      * n, an integer (Macaulay2Doc),
  * Outputs:
      * a Map of Simplicial Modules, the
        composition of $f$ with itself $n$ times.

Description
===========

A simplicial module map $f : C \to C$ can be composed with itself. This method
produces these new maps of simplicial modules.

The face/degeneracy map always composes with itself to give the zero map.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a..c];                                                       |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution coker matrix{{a^2, b^2, c^2}},      |
|                                                                             |
|      1      4      10      20                                               |
|o2 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i3 : f = dd^C                                                                |
|                                                                             |
|               1                      4                                      |
|o3 = (0, 0) : S  <------------------ S  : (1, 0)                             |
|                    | 1 a2 b2 c2 |                                           |
|                                                                             |
|               1                   4                                         |
|     (0, 1) : S  <--------------- S  : (1, 1)                                |
|                    | 1 0 0 0 |                                              |
|                                                                             |
|               4                                            10               |
|     (1, 0) : S  <---------------------------------------- S   : (2, 0)      |
|                    {0} | 1 a2 b2 c2 0 0 0 0   0   0   |                     |
|                    {2} | 0 0  0  0  1 0 0 -b2 -c2 0   |                     |
|                    {2} | 0 0  0  0  0 1 0 a2  0   -c2 |                     |
|                    {2} | 0 0  0  0  0 0 1 0   a2  b2  |                     |
|                                                                             |
|               4                                   10                        |
|     (1, 1) : S  <------------------------------- S   : (2, 1)               |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 |                              |
|                    {2} | 0 1 0 0 1 0 0 0 0 0 |                              |
|                    {2} | 0 0 1 0 0 1 0 0 0 0 |                              |
|                    {2} | 0 0 0 1 0 0 1 0 0 0 |                              |
|                                                                             |
|               4                                   10                        |
|     (1, 2) : S  <------------------------------- S   : (2, 2)               |
|                    {0} | 1 0 0 0 0 0 0 0 0 0 |                              |
|                    {2} | 0 1 0 0 0 0 0 0 0 0 |                              |
|                    {2} | 0 0 1 0 0 0 0 0 0 0 |                              |
|                    {2} | 0 0 0 1 0 0 0 0 0 0 |                              |
|                                                                             |
|               10                                                            |
|     (2, 0) : S   <----------------------------------------------------------|
|                     {0} | 1 a2 b2 c2 0 0 0 0 0 0 0   0   0   0   0   0   0 0|
|                     {2} | 0 0  0  0  1 0 0 0 0 0 -b2 -c2 0   0   0   0   0 0|
|                     {2} | 0 0  0  0  0 1 0 0 0 0 a2  0   -c2 0   0   0   0 0|
|                     {2} | 0 0  0  0  0 0 1 0 0 0 0   a2  b2  0   0   0   0 0|
|                     {2} | 0 0  0  0  0 0 0 1 0 0 0   0   0   -b2 -c2 0   0 0|
|                     {2} | 0 0  0  0  0 0 0 0 1 0 0   0   0   a2  0   -c2 0 0|
|                     {2} | 0 0  0  0  0 0 0 0 0 1 0   0   0   0   a2  b2  0 0|
|                     {4} | 0 0  0  0  0 0 0 0 0 0 0   0   0   0   0   0   1 0|
|                     {4} | 0 0  0  0  0 0 0 0 0 0 0   0   0   0   0   0   0 1|
|                     {4} | 0 0  0  0  0 0 0 0 0 0 0   0   0   0   0   0   0 0|
|                                                                             |
|               10                                                       20   |
|     (2, 1) : S   <--------------------------------------------------- S   : |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |         |
|                     {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 |         |
|                     {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 |         |
|                     {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 |         |
|                                                                             |
|               10                                                       20   |
|     (2, 2) : S   <--------------------------------------------------- S   : |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 |         |
|                     {4} | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 |         |
|                     {4} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 |         |
|                     {4} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 |         |
|                                                                             |
|               10                                                       20   |
|     (2, 3) : S   <--------------------------------------------------- S   : |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |         |
|                     {4} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |         |
|                     {4} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |         |
|                     {4} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |         |
|                                                                             |
|o3 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|            20                                                               |
|---------- S   : (3, 0)                                                      |
| 0 0   |                                                                     |
| 0 0   |                                                                     |
| 0 0   |                                                                     |
| 0 0   |                                                                     |
| 0 0   |                                                                     |
| 0 0   |                                                                     |
| 0 0   |                                                                     |
| 0 c2  |                                                                     |
| 0 -b2 |                                                                     |
| 1 a2  |                                                                     |
|                                                                             |
|                                                                             |
|(3, 1)                                                                       |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|(3, 2)                                                                       |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|(3, 3)                                                                       |
+-----------------------------------------------------------------------------+
|i4 : f^2                                                                     |
|                                                                             |
|o4 = 0                                                                       |
|                                                                             |
|o4 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+
|i5 : assert(source f == target f)                                            |
+-----------------------------------------------------------------------------+
|i6 : assert(degree f == -1)                                                  |
+-----------------------------------------------------------------------------+
|i7 : assert(degree f^2 == -2)                                                |
+-----------------------------------------------------------------------------+
+-----------------------------------------------------------------------------+
|i8 : g = randomSimplicialMap(C, C)                                           |
|                                                                             |
|          1              1                                                   |
|o8 = 0 : S  <---------- S  : 0                                               |
|               | 24 |                                                        |
|                                                                             |
|          4                              4                                   |
|     1 : S  <-------------------------- S  : 1                               |
|               {0} | 24 0   0   0   |                                        |
|               {2} | 0  -36 19  -29 |                                        |
|               {2} | 0  -30 19  -8  |                                        |
|               {2} | 0  -29 -10 -22 |                                        |
|                                                                             |
|          10                                                      10         |
|     2 : S   <-------------------------------------------------- S   : 2     |
|                {0} | 24 0   0   0   0   0   0   0   0   0   |               |
|                {2} | 0  -36 19  -29 0   0   0   0   0   0   |               |
|                {2} | 0  -30 19  -8  0   0   0   0   0   0   |               |
|                {2} | 0  -29 -10 -22 0   0   0   0   0   0   |               |
|                {2} | 0  0   0   0   -36 19  -29 0   0   0   |               |
|                {2} | 0  0   0   0   -30 19  -8  0   0   0   |               |
|                {2} | 0  0   0   0   -29 -10 -22 0   0   0   |               |
|                {4} | 0  0   0   0   0   0   0   -29 -16 34  |               |
|                {4} | 0  0   0   0   0   0   0   -24 39  19  |               |
|                {4} | 0  0   0   0   0   0   0   -38 21  -47 |               |
|                                                                             |
|          20                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {0} | 24 0   0   0   0   0   0   0   0   0   0   0   0   0   |
|                {2} | 0  -36 19  -29 0   0   0   0   0   0   0   0   0   0   |
|                {2} | 0  -30 19  -8  0   0   0   0   0   0   0   0   0   0   |
|                {2} | 0  -29 -10 -22 0   0   0   0   0   0   0   0   0   0   |
|                {2} | 0  0   0   0   -36 19  -29 0   0   0   0   0   0   0   |
|                {2} | 0  0   0   0   -30 19  -8  0   0   0   0   0   0   0   |
|                {2} | 0  0   0   0   -29 -10 -22 0   0   0   0   0   0   0   |
|                {2} | 0  0   0   0   0   0   0   -36 19  -29 0   0   0   0   |
|                {2} | 0  0   0   0   0   0   0   -30 19  -8  0   0   0   0   |
|                {2} | 0  0   0   0   0   0   0   -29 -10 -22 0   0   0   0   |
|                {4} | 0  0   0   0   0   0   0   0   0   0   -29 -16 34  0   |
|                {4} | 0  0   0   0   0   0   0   0   0   0   -24 39  19  0   |
|                {4} | 0  0   0   0   0   0   0   0   0   0   -38 21  -47 0   |
|                {4} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -29 |
|                {4} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -24 |
|                {4} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -38 |
|                {4} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   |
|                {4} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   |
|                {4} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   |
|                {6} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   |
|                                                                             |
|o8 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                             20                                              |
|--------------------------- S   : 3                                          |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|0   0   0   0   0   0   |                                                    |
|-16 34  0   0   0   0   |                                                    |
|39  19  0   0   0   0   |                                                    |
|21  -47 0   0   0   0   |                                                    |
|0   0   -29 -16 34  0   |                                                    |
|0   0   -24 39  19  0   |                                                    |
|0   0   -38 21  -47 0   |                                                    |
|0   0   0   0   0   -39 |                                                    |
+-----------------------------------------------------------------------------+
|i9 : g^2                                                                     |
|                                                                             |
|          1               1                                                  |
|o9 = 0 : S  <----------- S  : 0                                              |
|               | -30 |                                                       |
|                                                                             |
|          4                               4                                  |
|     1 : S  <--------------------------- S  : 1                              |
|               {0} | -30 0   0   0   |                                       |
|               {2} | 0   -49 -33 15  |                                       |
|               {2} | 0   35  -28 -15 |                                       |
|               {2} | 0   -38 -16 -9  |                                       |
|                                                                             |
|          10                                                      10         |
|     2 : S   <-------------------------------------------------- S   : 2     |
|                {0} | -30 0   0   0   0   0   0   0   0   0  |               |
|                {2} | 0   -49 -33 15  0   0   0   0   0   0  |               |
|                {2} | 0   35  -28 -15 0   0   0   0   0   0  |               |
|                {2} | 0   -38 -16 -9  0   0   0   0   0   0  |               |
|                {2} | 0   0   0   0   -49 -33 15  0   0   0  |               |
|                {2} | 0   0   0   0   35  -28 -15 0   0   0  |               |
|                {2} | 0   0   0   0   -38 -16 -9  0   0   0  |               |
|                {4} | 0   0   0   0   0   0   0   34  49  41 |               |
|                {4} | 0   0   0   0   0   0   0   48  -19 42 |               |
|                {4} | 0   0   0   0   0   0   0   -40 36  3  |               |
|                                                                             |
|          20                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {0} | -30 0   0   0   0   0   0   0   0   0   0   0   0  0   |
|                {2} | 0   -49 -33 15  0   0   0   0   0   0   0   0   0  0   |
|                {2} | 0   35  -28 -15 0   0   0   0   0   0   0   0   0  0   |
|                {2} | 0   -38 -16 -9  0   0   0   0   0   0   0   0   0  0   |
|                {2} | 0   0   0   0   -49 -33 15  0   0   0   0   0   0  0   |
|                {2} | 0   0   0   0   35  -28 -15 0   0   0   0   0   0  0   |
|                {2} | 0   0   0   0   -38 -16 -9  0   0   0   0   0   0  0   |
|                {2} | 0   0   0   0   0   0   0   -49 -33 15  0   0   0  0   |
|                {2} | 0   0   0   0   0   0   0   35  -28 -15 0   0   0  0   |
|                {2} | 0   0   0   0   0   0   0   -38 -16 -9  0   0   0  0   |
|                {4} | 0   0   0   0   0   0   0   0   0   0   34  49  41 0   |
|                {4} | 0   0   0   0   0   0   0   0   0   0   48  -19 42 0   |
|                {4} | 0   0   0   0   0   0   0   0   0   0   -40 36  3  0   |
|                {4} | 0   0   0   0   0   0   0   0   0   0   0   0   0  34  |
|                {4} | 0   0   0   0   0   0   0   0   0   0   0   0   0  48  |
|                {4} | 0   0   0   0   0   0   0   0   0   0   0   0   0  -40 |
|                {4} | 0   0   0   0   0   0   0   0   0   0   0   0   0  0   |
|                {4} | 0   0   0   0   0   0   0   0   0   0   0   0   0  0   |
|                {4} | 0   0   0   0   0   0   0   0   0   0   0   0   0  0   |
|                {6} | 0   0   0   0   0   0   0   0   0   0   0   0   0  0   |
|                                                                             |
|o9 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                         20                                                  |
|----------------------- S   : 3                                              |
|0   0  0   0   0  0 |                                                        |
|0   0  0   0   0  0 |                                                        |
|0   0  0   0   0  0 |                                                        |
|0   0  0   0   0  0 |                                                        |
|0   0  0   0   0  0 |                                                        |
|0   0  0   0   0  0 |                                                        |
|0   0  0   0   0  0 |                                                        |
|0   0  0   0   0  0 |                                                        |
|0   0  0   0   0  0 |                                                        |
|0   0  0   0   0  0 |                                                        |
|0   0  0   0   0  0 |                                                        |
|0   0  0   0   0  0 |                                                        |
|0   0  0   0   0  0 |                                                        |
|49  41 0   0   0  0 |                                                        |
|-19 42 0   0   0  0 |                                                        |
|36  3  0   0   0  0 |                                                        |
|0   0  34  49  41 0 |                                                        |
|0   0  48  -19 42 0 |                                                        |
|0   0  -40 36  3  0 |                                                        |
|0   0  0   0   0  6 |                                                        |
+-----------------------------------------------------------------------------+
|i10 : g^3                                                                    |
|                                                                             |
|           1               1                                                 |
|o10 = 0 : S  <----------- S  : 0                                             |
|                | -13 |                                                      |
|                                                                             |
|           4                              4                                  |
|      1 : S  <-------------------------- S  : 1                              |
|                {0} | -13 0   0   0  |                                       |
|                {2} | 0   -4  9   42 |                                       |
|                {2} | 0   15  -20 44 |                                       |
|                {2} | 0   -12 -27 14 |                                       |
|                                                                             |
|           10                                                     10         |
|      2 : S   <------------------------------------------------- S   : 2     |
|                 {0} | -13 0   0   0  0   0   0  0   0   0   |               |
|                 {2} | 0   -4  9   42 0   0   0  0   0   0   |               |
|                 {2} | 0   15  -20 44 0   0   0  0   0   0   |               |
|                 {2} | 0   -12 -27 14 0   0   0  0   0   0   |               |
|                 {2} | 0   0   0   0  -4  9   42 0   0   0   |               |
|                 {2} | 0   0   0   0  15  -20 44 0   0   0   |               |
|                 {2} | 0   0   0   0  -12 -27 14 0   0   0   |               |
|                 {4} | 0   0   0   0  0   0   0  17  6   -42 |               |
|                 {4} | 0   0   0   0  0   0   0  -7  -21 4   |               |
|                 {4} | 0   0   0   0  0   0   0  -20 -14 -9  |               |
|                                                                             |
|           20                                                                |
|      3 : S   <--------------------------------------------------------------|
|                 {0} | -13 0   0   0  0   0   0  0   0   0  0   0   0   0   0|
|                 {2} | 0   -4  9   42 0   0   0  0   0   0  0   0   0   0   0|
|                 {2} | 0   15  -20 44 0   0   0  0   0   0  0   0   0   0   0|
|                 {2} | 0   -12 -27 14 0   0   0  0   0   0  0   0   0   0   0|
|                 {2} | 0   0   0   0  -4  9   42 0   0   0  0   0   0   0   0|
|                 {2} | 0   0   0   0  15  -20 44 0   0   0  0   0   0   0   0|
|                 {2} | 0   0   0   0  -12 -27 14 0   0   0  0   0   0   0   0|
|                 {2} | 0   0   0   0  0   0   0  -4  9   42 0   0   0   0   0|
|                 {2} | 0   0   0   0  0   0   0  15  -20 44 0   0   0   0   0|
|                 {2} | 0   0   0   0  0   0   0  -12 -27 14 0   0   0   0   0|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  17  6   -42 0   0|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  -7  -21 4   0   0|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  -20 -14 -9  0   0|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   17  6|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   -7  -|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   -20 -|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0|
|                 {6} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0|
|                                                                             |
|o10 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                            20                                               |
|-------------------------- S   : 3                                           |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   -42 0   0   0   0   |                                                     |
|21 4   0   0   0   0   |                                                     |
|14 -9  0   0   0   0   |                                                     |
|   0   17  6   -42 0   |                                                     |
|   0   -7  -21 4   0   |                                                     |
|   0   -20 -14 -9  0   |                                                     |
|   0   0   0   0   -32 |                                                     |
+-----------------------------------------------------------------------------+

The zero-th power returns the identity map

+----------------------+
|i11 : f^0 == id_(C[1])|
|                      |
|o11 = false           |
+----------------------+
|i12 : g^0 == id_C     |
|                      |
|o12 = true            |
+----------------------+

When $n$ is negative, the result is the $n$-fold power of the inverse
simplicial module map, if it exists.

+-----------------------------------------------------------------------------+
|i13 : h = randomSimplicialMap(C, C)                                          |
|                                                                             |
|           1               1                                                 |
|o13 = 0 : S  <----------- S  : 0                                             |
|                | -18 |                                                      |
|                                                                             |
|           4                              4                                  |
|      1 : S  <-------------------------- S  : 1                              |
|                {0} | -18 0   0   0  |                                       |
|                {2} | 0   -13 -28 2  |                                       |
|                {2} | 0   -43 -47 16 |                                       |
|                {2} | 0   -15 38  22 |                                       |
|                                                                             |
|           10                                                     10         |
|      2 : S   <------------------------------------------------- S   : 2     |
|                 {0} | -18 0   0   0  0   0   0  0   0   0   |               |
|                 {2} | 0   -13 -28 2  0   0   0  0   0   0   |               |
|                 {2} | 0   -43 -47 16 0   0   0  0   0   0   |               |
|                 {2} | 0   -15 38  22 0   0   0  0   0   0   |               |
|                 {2} | 0   0   0   0  -13 -28 2  0   0   0   |               |
|                 {2} | 0   0   0   0  -43 -47 16 0   0   0   |               |
|                 {2} | 0   0   0   0  -15 38  22 0   0   0   |               |
|                 {4} | 0   0   0   0  0   0   0  45  -47 -16 |               |
|                 {4} | 0   0   0   0  0   0   0  -34 47  7   |               |
|                 {4} | 0   0   0   0  0   0   0  -48 19  15  |               |
|                                                                             |
|           20                                                                |
|      3 : S   <--------------------------------------------------------------|
|                 {0} | -18 0   0   0  0   0   0  0   0   0  0   0   0   0   0|
|                 {2} | 0   -13 -28 2  0   0   0  0   0   0  0   0   0   0   0|
|                 {2} | 0   -43 -47 16 0   0   0  0   0   0  0   0   0   0   0|
|                 {2} | 0   -15 38  22 0   0   0  0   0   0  0   0   0   0   0|
|                 {2} | 0   0   0   0  -13 -28 2  0   0   0  0   0   0   0   0|
|                 {2} | 0   0   0   0  -43 -47 16 0   0   0  0   0   0   0   0|
|                 {2} | 0   0   0   0  -15 38  22 0   0   0  0   0   0   0   0|
|                 {2} | 0   0   0   0  0   0   0  -13 -28 2  0   0   0   0   0|
|                 {2} | 0   0   0   0  0   0   0  -43 -47 16 0   0   0   0   0|
|                 {2} | 0   0   0   0  0   0   0  -15 38  22 0   0   0   0   0|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  45  -47 -16 0   0|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  -34 47  7   0   0|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  -48 19  15  0   0|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   45  -|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   -34 4|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   -48 1|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0|
|                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0|
|                 {6} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0|
|                                                                             |
|o13 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                            20                                               |
|-------------------------- S   : 3                                           |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|   0   0   0   0   0   |                                                     |
|47 -16 0   0   0   0   |                                                     |
|7  7   0   0   0   0   |                                                     |
|9  15  0   0   0   0   |                                                     |
|   0   45  -47 -16 0   |                                                     |
|   0   -34 47  7   0   |                                                     |
|   0   -48 19  15  0   |                                                     |
|   0   0   0   0   -23 |                                                     |
+-----------------------------------------------------------------------------+
|i14 : h^-1                                                                   |
|                                                                             |
|           1              1                                                  |
|o14 = 0 : S  <---------- S  : 0                                              |
|                | 28 |                                                       |
|                                                                             |
|           4                            4                                    |
|      1 : S  <------------------------ S  : 1                                |
|                {0} | 28 0  0  0   |                                         |
|                {2} | 0  35 -7 -44 |                                         |
|                {2} | 0  13 -5 30  |                                         |
|                {2} | 0  6  36 33  |                                         |
|                                                                             |
|           10                                                  10            |
|      2 : S   <---------------------------------------------- S   : 2        |
|                 {0} | 28 0  0  0   0  0  0   0   0   0   |                  |
|                 {2} | 0  35 -7 -44 0  0  0   0   0   0   |                  |
|                 {2} | 0  13 -5 30  0  0  0   0   0   0   |                  |
|                 {2} | 0  6  36 33  0  0  0   0   0   0   |                  |
|                 {2} | 0  0  0  0   35 -7 -44 0   0   0   |                  |
|                 {2} | 0  0  0  0   13 -5 30  0   0   0   |                  |
|                 {2} | 0  0  0  0   6  36 33  0   0   0   |                  |
|                 {4} | 0  0  0  0   0  0  0   2   18  -13 |                  |
|                 {4} | 0  0  0  0   0  0  0   -34 -48 40  |                  |
|                 {4} | 0  0  0  0   0  0  0   36  -23 29  |                  |
|                                                                             |
|           20                                                                |
|      3 : S   <--------------------------------------------------------------|
|                 {0} | 28 0  0  0   0  0  0   0  0  0   0   0   0   0   0   0|
|                 {2} | 0  35 -7 -44 0  0  0   0  0  0   0   0   0   0   0   0|
|                 {2} | 0  13 -5 30  0  0  0   0  0  0   0   0   0   0   0   0|
|                 {2} | 0  6  36 33  0  0  0   0  0  0   0   0   0   0   0   0|
|                 {2} | 0  0  0  0   35 -7 -44 0  0  0   0   0   0   0   0   0|
|                 {2} | 0  0  0  0   13 -5 30  0  0  0   0   0   0   0   0   0|
|                 {2} | 0  0  0  0   6  36 33  0  0  0   0   0   0   0   0   0|
|                 {2} | 0  0  0  0   0  0  0   35 -7 -44 0   0   0   0   0   0|
|                 {2} | 0  0  0  0   0  0  0   13 -5 30  0   0   0   0   0   0|
|                 {2} | 0  0  0  0   0  0  0   6  36 33  0   0   0   0   0   0|
|                 {4} | 0  0  0  0   0  0  0   0  0  0   2   18  -13 0   0   0|
|                 {4} | 0  0  0  0   0  0  0   0  0  0   -34 -48 40  0   0   0|
|                 {4} | 0  0  0  0   0  0  0   0  0  0   36  -23 29  0   0   0|
|                 {4} | 0  0  0  0   0  0  0   0  0  0   0   0   0   2   18  -|
|                 {4} | 0  0  0  0   0  0  0   0  0  0   0   0   0   -34 -48 4|
|                 {4} | 0  0  0  0   0  0  0   0  0  0   0   0   0   36  -23 2|
|                 {4} | 0  0  0  0   0  0  0   0  0  0   0   0   0   0   0   0|
|                 {4} | 0  0  0  0   0  0  0   0  0  0   0   0   0   0   0   0|
|                 {4} | 0  0  0  0   0  0  0   0  0  0   0   0   0   0   0   0|
|                 {6} | 0  0  0  0   0  0  0   0  0  0   0   0   0   0   0   0|
|                                                                             |
|o14 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                        20                                                   |
|---------------------- S   : 3                                               |
|   0   0   0   0   |                                                         |
|   0   0   0   0   |                                                         |
|   0   0   0   0   |                                                         |
|   0   0   0   0   |                                                         |
|   0   0   0   0   |                                                         |
|   0   0   0   0   |                                                         |
|   0   0   0   0   |                                                         |
|   0   0   0   0   |                                                         |
|   0   0   0   0   |                                                         |
|   0   0   0   0   |                                                         |
|   0   0   0   0   |                                                         |
|   0   0   0   0   |                                                         |
|   0   0   0   0   |                                                         |
|13 0   0   0   0   |                                                         |
|0  0   0   0   0   |                                                         |
|9  0   0   0   0   |                                                         |
|   2   18  -13 0   |                                                         |
|   -34 -48 40  0   |                                                         |
|   36  -23 29  0   |                                                         |
|   0   0   0   -22 |                                                         |
+-----------------------------------------------------------------------------+
|i15 : assert(h * h^-1 == id_C)                                               |
+-----------------------------------------------------------------------------+
|i16 : h^-4                                                                   |
|                                                                             |
|           1               1                                                 |
|o16 = 0 : S  <----------- S  : 0                                             |
|                | -30 |                                                      |
|                                                                             |
|           4                               4                                 |
|      1 : S  <--------------------------- S  : 1                             |
|                {0} | -30 0   0   0   |                                      |
|                {2} | 0   -30 -39 -6  |                                      |
|                {2} | 0   -9  -15 -23 |                                      |
|                {2} | 0   0   -49 39  |                                      |
|                                                                             |
|           10                                                      10        |
|      2 : S   <-------------------------------------------------- S   : 2    |
|                 {0} | -30 0   0   0   0   0   0   0  0   0   |              |
|                 {2} | 0   -30 -39 -6  0   0   0   0  0   0   |              |
|                 {2} | 0   -9  -15 -23 0   0   0   0  0   0   |              |
|                 {2} | 0   0   -49 39  0   0   0   0  0   0   |              |
|                 {2} | 0   0   0   0   -30 -39 -6  0  0   0   |              |
|                 {2} | 0   0   0   0   -9  -15 -23 0  0   0   |              |
|                 {2} | 0   0   0   0   0   -49 39  0  0   0   |              |
|                 {4} | 0   0   0   0   0   0   0   40 -37 -41 |              |
|                 {4} | 0   0   0   0   0   0   0   38 -13 -2  |              |
|                 {4} | 0   0   0   0   0   0   0   15 7   42  |              |
|                                                                             |
|           20                                                                |
|      3 : S   <--------------------------------------------------------------|
|                 {0} | -30 0   0   0   0   0   0   0   0   0   0  0   0   0  |
|                 {2} | 0   -30 -39 -6  0   0   0   0   0   0   0  0   0   0  |
|                 {2} | 0   -9  -15 -23 0   0   0   0   0   0   0  0   0   0  |
|                 {2} | 0   0   -49 39  0   0   0   0   0   0   0  0   0   0  |
|                 {2} | 0   0   0   0   -30 -39 -6  0   0   0   0  0   0   0  |
|                 {2} | 0   0   0   0   -9  -15 -23 0   0   0   0  0   0   0  |
|                 {2} | 0   0   0   0   0   -49 39  0   0   0   0  0   0   0  |
|                 {2} | 0   0   0   0   0   0   0   -30 -39 -6  0  0   0   0  |
|                 {2} | 0   0   0   0   0   0   0   -9  -15 -23 0  0   0   0  |
|                 {2} | 0   0   0   0   0   0   0   0   -49 39  0  0   0   0  |
|                 {4} | 0   0   0   0   0   0   0   0   0   0   40 -37 -41 0  |
|                 {4} | 0   0   0   0   0   0   0   0   0   0   38 -13 -2  0  |
|                 {4} | 0   0   0   0   0   0   0   0   0   0   15 7   42  0  |
|                 {4} | 0   0   0   0   0   0   0   0   0   0   0  0   0   40 |
|                 {4} | 0   0   0   0   0   0   0   0   0   0   0  0   0   38 |
|                 {4} | 0   0   0   0   0   0   0   0   0   0   0  0   0   15 |
|                 {4} | 0   0   0   0   0   0   0   0   0   0   0  0   0   0  |
|                 {4} | 0   0   0   0   0   0   0   0   0   0   0  0   0   0  |
|                 {4} | 0   0   0   0   0   0   0   0   0   0   0  0   0   0  |
|                 {6} | 0   0   0   0   0   0   0   0   0   0   0  0   0   0  |
|                                                                             |
|o16 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|                           20                                                |
|------------------------- S   : 3                                            |
|0   0   0  0   0   0  |                                                      |
|0   0   0  0   0   0  |                                                      |
|0   0   0  0   0   0  |                                                      |
|0   0   0  0   0   0  |                                                      |
|0   0   0  0   0   0  |                                                      |
|0   0   0  0   0   0  |                                                      |
|0   0   0  0   0   0  |                                                      |
|0   0   0  0   0   0  |                                                      |
|0   0   0  0   0   0  |                                                      |
|0   0   0  0   0   0  |                                                      |
|0   0   0  0   0   0  |                                                      |
|0   0   0  0   0   0  |                                                      |
|0   0   0  0   0   0  |                                                      |
|-37 -41 0  0   0   0  |                                                      |
|-13 -2  0  0   0   0  |                                                      |
|7   42  0  0   0   0  |                                                      |
|0   0   40 -37 -41 0  |                                                      |
|0   0   38 -13 -2  0  |                                                      |
|0   0   15 7   42  0  |                                                      |
|0   0   0  0   0   37 |                                                      |
+-----------------------------------------------------------------------------+
|i17 : assert(h^-4 * h^4 == id_C)                                             |
+-----------------------------------------------------------------------------+

See also
========

  * Matrix ^ ZZ (Macaulay2Doc) -- power

Ways to use this method:
========================

  * SimplicialModuleMap ^ ZZ -- the n-fold
    composition
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2025:0.

SimplicialModuleMap _ ZZ -- access individual matrices in a simplicial module map
*********************************************************************************

  * Operator: _ (Macaulay2Doc),
  * Usage:
        f_i
  * Inputs:
      * f, a Map of Simplicial Modules,
      * i, an integer (Macaulay2Doc), the degree index or a sequence
  * Outputs:
      * a matrix (Macaulay2Doc), the i-th map

Description
===========

A simplicial module map $f : C \to D$ of degree $d$ is a sequence of maps $f_i
: C_i \to D_{i+d}$. This method allows one to access the individual $f_i$.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a..c];                                                       |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution coker matrix{{a^2, b^2, c^2}},      |
|                                                                             |
|      1      4      10      20                                               |
|o2 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i3 : D = simplicialModule(freeResolution coker vars S, Degeneracy => true)   |
|                                                                             |
|      1      4      10      20                                               |
|o3 = S  <-- S  <-- S   <-- S  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : f = randomSimplicialMap(D, C)                                           |
|                                                                             |
|          1              1                                                   |
|o4 = 0 : S  <---------- S  : 0                                               |
|               | 24 |                                                        |
|                                                                             |
|          4                                                         4        |
|     1 : S  <----------------------------------------------------- S  : 1    |
|               {0} | 24 0            0            0            |             |
|               {1} | 0  -36a-30b-29c -29a-24b-38c -39a-18b-13c |             |
|               {1} | 0  19a+19b-10c  -16a+39b+21c -43a-15b-28c |             |
|               {1} | 0  -29a-8b-22c  34a+19b-47c  -47a+38b+2c  |             |
|                                                                             |
|          10                                                                 |
|     2 : S   <---------------------------------------------------------------|
|                {0} | 24 0            0            0            0            |
|                {1} | 0  -36a-30b-29c -29a-24b-38c -39a-18b-13c 0            |
|                {1} | 0  19a+19b-10c  -16a+39b+21c -43a-15b-28c 0            |
|                {1} | 0  -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0            |
|                {1} | 0  0            0            0            -36a-30b-29c |
|                {1} | 0  0            0            0            19a+19b-10c  |
|                {1} | 0  0            0            0            -29a-8b-22c  |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                                                                             |
|          20                                                                 |
|     3 : S   <---------------------------------------------------------------|
|                {0} | 24 0            0            0            0            |
|                {1} | 0  -36a-30b-29c -29a-24b-38c -39a-18b-13c 0            |
|                {1} | 0  19a+19b-10c  -16a+39b+21c -43a-15b-28c 0            |
|                {1} | 0  -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0            |
|                {1} | 0  0            0            0            -36a-30b-29c |
|                {1} | 0  0            0            0            19a+19b-10c  |
|                {1} | 0  0            0            0            -29a-8b-22c  |
|                {1} | 0  0            0            0            0            |
|                {1} | 0  0            0            0            0            |
|                {1} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {2} | 0  0            0            0            0            |
|                {3} | 0  0            0            0            0            |
|                                                                             |
|o4 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|0            0            0                             0                    |
|0            0            0                             0                    |
|0            0            0                             0                    |
|0            0            0                             0                    |
|-29a-24b-38c -39a-18b-13c 0                             0                    |
|-16a+39b+21c -43a-15b-28c 0                             0                    |
|34a+19b-47c  -47a+38b+2c  0                             0                    |
|0            0            16a2+22ab-34b2+45ac-48bc-47c2 35a2+11ab+33b2-38ac+4|
|0            0            47a2+19ab+7b2-16ac+15bc-23c2  46a2-28ab-3b2+ac+22bc|
|0            0            39a2+43ab-11b2-17ac+48bc+36c2 -23a2-7ab+29b2+2ac-47|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|-29a-24b-38c -39a-18b-13c 0            0            0            0           |
|-16a+39b+21c -43a-15b-28c 0            0            0            0           |
|34a+19b-47c  -47a+38b+2c  0            0            0            0           |
|0            0            -36a-30b-29c -29a-24b-38c -39a-18b-13c 0           |
|0            0            19a+19b-10c  -16a+39b+21c -43a-15b-28c 0           |
|0            0            -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0           |
|0            0            0            0            0            16a2+22ab-34|
|0            0            0            0            0            47a2+19ab+7b|
|0            0            0            0            0            39a2+43ab-11|
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|0            0            0            0            0            0           |
|-----------------------------------------------------------------------------|
|                                             10                              |
|------------------------------------------- S   : 2                          |
|         0                              |                                    |
|         0                              |                                    |
|         0                              |                                    |
|         0                              |                                    |
|         0                              |                                    |
|         0                              |                                    |
|         0                              |                                    |
|0bc+11c2 -37a2-13ab+30b2-10ac-18bc+39c2 |                                    |
|-47c2    27a2-22ab-9b2+32ac-32bc-20c2   |                                    |
|bc+15c2  24a2-30ab-15b2-48ac+39bc       |                                    |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                  0                             0                            |
|                  0                             0                            |
|                  0                             0                            |
|                  0                             0                            |
|                  0                             0                            |
|                  0                             0                            |
|                  0                             0                            |
|                  0                             0                            |
|                  0                             0                            |
|                  0                             0                            |
|b2+45ac-48bc-47c2 35a2+11ab+33b2-38ac+40bc+11c2 -37a2-13ab+30b2-10ac-18bc+39c|
|2-16ac+15bc-23c2  46a2-28ab-3b2+ac+22bc-47c2    27a2-22ab-9b2+32ac-32bc-20c2 |
|b2-17ac+48bc+36c2 -23a2-7ab+29b2+2ac-47bc+15c2  24a2-30ab-15b2-48ac+39bc     |
|                  0                             0                            |
|                  0                             0                            |
|                  0                             0                            |
|                  0                             0                            |
|                  0                             0                            |
|                  0                             0                            |
|                  0                             0                            |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|  0                             0                             0              |
|  0                             0                             0              |
|  0                             0                             0              |
|  0                             0                             0              |
|  0                             0                             0              |
|  0                             0                             0              |
|  0                             0                             0              |
|  0                             0                             0              |
|  0                             0                             0              |
|  0                             0                             0              |
|2 0                             0                             0              |
|  0                             0                             0              |
|  0                             0                             0              |
|  16a2+22ab-34b2+45ac-48bc-47c2 35a2+11ab+33b2-38ac+40bc+11c2 -37a2-13ab+30b2|
|  47a2+19ab+7b2-16ac+15bc-23c2  46a2-28ab-3b2+ac+22bc-47c2    27a2-22ab-9b2+3|
|  39a2+43ab-11b2-17ac+48bc+36c2 -23a2-7ab+29b2+2ac-47bc+15c2  24a2-30ab-15b2-|
|  0                             0                             0              |
|  0                             0                             0              |
|  0                             0                             0              |
|  0                             0                             0              |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|                0                             0                             0|
|                0                             0                             0|
|                0                             0                             0|
|                0                             0                             0|
|                0                             0                             0|
|                0                             0                             0|
|                0                             0                             0|
|                0                             0                             0|
|                0                             0                             0|
|                0                             0                             0|
|                0                             0                             0|
|                0                             0                             0|
|                0                             0                             0|
|-10ac-18bc+39c2 0                             0                             0|
|2ac-32bc-20c2   0                             0                             0|
|48ac+39bc       0                             0                             0|
|                16a2+22ab-34b2+45ac-48bc-47c2 35a2+11ab+33b2-38ac+40bc+11c2 -|
|                47a2+19ab+7b2-16ac+15bc-23c2  46a2-28ab-3b2+ac+22bc-47c2    2|
|                39a2+43ab-11b2-17ac+48bc+36c2 -23a2-7ab+29b2+2ac-47bc+15c2  2|
|                0                             0                             0|
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|                              0                                              |
|                              0                                              |
|                              0                                              |
|                              0                                              |
|                              0                                              |
|                              0                                              |
|                              0                                              |
|                              0                                              |
|                              0                                              |
|                              0                                              |
|                              0                                              |
|                              0                                              |
|                              0                                              |
|                              0                                              |
|                              0                                              |
|                              0                                              |
|37a2-13ab+30b2-10ac-18bc+39c2 0                                              |
|7a2-22ab-9b2+32ac-32bc-20c2   0                                              |
|4a2-30ab-15b2-48ac+39bc       0                                              |
|                              33a3-49a2b-19ab2+44b3-33a2c+17abc-39b2c-20ac2+3|
|-----------------------------------------------------------------------------|
|              20                                                             |
|------------ S   : 3                                                         |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|         |                                                                   |
|6bc2+9c3 |                                                                   |
+-----------------------------------------------------------------------------+
|i5 : f_2                                                                     |
|                                                                             |
|o5 = {0} | 24 0            0            0            0                       |
|     {1} | 0  -36a-30b-29c -29a-24b-38c -39a-18b-13c 0                       |
|     {1} | 0  19a+19b-10c  -16a+39b+21c -43a-15b-28c 0                       |
|     {1} | 0  -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0                       |
|     {1} | 0  0            0            0            -36a-30b-29c            |
|     {1} | 0  0            0            0            19a+19b-10c             |
|     {1} | 0  0            0            0            -29a-8b-22c             |
|     {2} | 0  0            0            0            0                       |
|     {2} | 0  0            0            0            0                       |
|     {2} | 0  0            0            0            0                       |
|     ------------------------------------------------------------------------|
|     0            0            0                                             |
|     0            0            0                                             |
|     0            0            0                                             |
|     0            0            0                                             |
|     -29a-24b-38c -39a-18b-13c 0                                             |
|     -16a+39b+21c -43a-15b-28c 0                                             |
|     34a+19b-47c  -47a+38b+2c  0                                             |
|     0            0            16a2+22ab-34b2+45ac-48bc-47c2                 |
|     0            0            47a2+19ab+7b2-16ac+15bc-23c2                  |
|     0            0            39a2+43ab-11b2-17ac+48bc+36c2                 |
|     ------------------------------------------------------------------------|
|     0                             0                              |          |
|     0                             0                              |          |
|     0                             0                              |          |
|     0                             0                              |          |
|     0                             0                              |          |
|     0                             0                              |          |
|     0                             0                              |          |
|     35a2+11ab+33b2-38ac+40bc+11c2 -37a2-13ab+30b2-10ac-18bc+39c2 |          |
|     46a2-28ab-3b2+ac+22bc-47c2    27a2-22ab-9b2+32ac-32bc-20c2   |          |
|     -23a2-7ab+29b2+2ac-47bc+15c2  24a2-30ab-15b2-48ac+39bc       |          |
|                                                                             |
|             10      10                                                      |
|o5 : Matrix S   <-- S                                                        |
+-----------------------------------------------------------------------------+
|i6 : f_0                                                                     |
|                                                                             |
|o6 = | 24 |                                                                  |
|                                                                             |
|             1      1                                                        |
|o6 : Matrix S  <-- S                                                         |
+-----------------------------------------------------------------------------+

The face/degeneracy maps of a simplicial module are indexed by sequences
$(a,b)$ of integers, where $b \leq a$.

+-----------------------------------------+
|i7 : dd^C_(2,0)                          |
|                                         |
|o7 = {0} | 1 a2 b2 c2 0 0 0 0   0   0   ||
|     {2} | 0 0  0  0  1 0 0 -b2 -c2 0   ||
|     {2} | 0 0  0  0  0 1 0 a2  0   -c2 ||
|     {2} | 0 0  0  0  0 0 1 0   a2  b2  ||
|                                         |
|             4      10                   |
|o7 : Matrix S  <-- S                     |
+-----------------------------------------+
|i8 : dd^C_(2,1)                          |
|                                         |
|o8 = {0} | 1 0 0 0 0 0 0 0 0 0 |         |
|     {2} | 0 1 0 0 1 0 0 0 0 0 |         |
|     {2} | 0 0 1 0 0 1 0 0 0 0 |         |
|     {2} | 0 0 0 1 0 0 1 0 0 0 |         |
|                                         |
|             4      10                   |
|o8 : Matrix S  <-- S                     |
+-----------------------------------------+
|i9 : dd^C_(2,2)                          |
|                                         |
|o9 = {0} | 1 0 0 0 0 0 0 0 0 0 |         |
|     {2} | 0 1 0 0 0 0 0 0 0 0 |         |
|     {2} | 0 0 1 0 0 0 0 0 0 0 |         |
|     {2} | 0 0 0 1 0 0 0 0 0 0 |         |
|                                         |
|             4      10                   |
|o9 : Matrix S  <-- S                     |
+-----------------------------------------+
|i10 : ss^C_(0,0)                         |
|                                         |
|o10 = {0} | 1 |                          |
|      {2} | 0 |                          |
|      {2} | 0 |                          |
|      {2} | 0 |                          |
|                                         |
|              4      1                   |
|o10 : Matrix S  <-- S                    |
+-----------------------------------------+
|i11 : ss^C_(1,0)                         |
|                                         |
|o11 = {0} | 1 0 0 0 |                    |
|      {2} | 0 0 0 0 |                    |
|      {2} | 0 0 0 0 |                    |
|      {2} | 0 0 0 0 |                    |
|      {2} | 0 1 0 0 |                    |
|      {2} | 0 0 1 0 |                    |
|      {2} | 0 0 0 1 |                    |
|      {4} | 0 0 0 0 |                    |
|      {4} | 0 0 0 0 |                    |
|      {4} | 0 0 0 0 |                    |
|                                         |
|              10      4                  |
|o11 : Matrix S   <-- S                   |
+-----------------------------------------+
|i12 : ss^C_(1,1)                         |
|                                         |
|o12 = {0} | 1 0 0 0 |                    |
|      {2} | 0 1 0 0 |                    |
|      {2} | 0 0 1 0 |                    |
|      {2} | 0 0 0 1 |                    |
|      {2} | 0 0 0 0 |                    |
|      {2} | 0 0 0 0 |                    |
|      {2} | 0 0 0 0 |                    |
|      {4} | 0 0 0 0 |                    |
|      {4} | 0 0 0 0 |                    |
|      {4} | 0 0 0 0 |                    |
|                                         |
|              10      4                  |
|o12 : Matrix S   <-- S                   |
+-----------------------------------------+

Indices that are outside of the top degree or range of sequences return 0 by
default.

+-----------------------------------------------------------------------------+
|i13 : topDegree f                                                            |
|                                                                             |
|o13 = 3                                                                      |
+-----------------------------------------------------------------------------+
|i14 : f_-1                                                                   |
|                                                                             |
|o14 = 0                                                                      |
|                                                                             |
|o14 : Matrix 0 <-- 0                                                         |
+-----------------------------------------------------------------------------+
|i15 : f_3                                                                    |
|                                                                             |
|o15 = {0} | 24 0            0            0            0                      |
|      {1} | 0  -36a-30b-29c -29a-24b-38c -39a-18b-13c 0                      |
|      {1} | 0  19a+19b-10c  -16a+39b+21c -43a-15b-28c 0                      |
|      {1} | 0  -29a-8b-22c  34a+19b-47c  -47a+38b+2c  0                      |
|      {1} | 0  0            0            0            -36a-30b-29c           |
|      {1} | 0  0            0            0            19a+19b-10c            |
|      {1} | 0  0            0            0            -29a-8b-22c            |
|      {1} | 0  0            0            0            0                      |
|      {1} | 0  0            0            0            0                      |
|      {1} | 0  0            0            0            0                      |
|      {2} | 0  0            0            0            0                      |
|      {2} | 0  0            0            0            0                      |
|      {2} | 0  0            0            0            0                      |
|      {2} | 0  0            0            0            0                      |
|      {2} | 0  0            0            0            0                      |
|      {2} | 0  0            0            0            0                      |
|      {2} | 0  0            0            0            0                      |
|      {2} | 0  0            0            0            0                      |
|      {2} | 0  0            0            0            0                      |
|      {3} | 0  0            0            0            0                      |
|      -----------------------------------------------------------------------|
|      0            0            0            0            0                  |
|      0            0            0            0            0                  |
|      0            0            0            0            0                  |
|      0            0            0            0            0                  |
|      -29a-24b-38c -39a-18b-13c 0            0            0                  |
|      -16a+39b+21c -43a-15b-28c 0            0            0                  |
|      34a+19b-47c  -47a+38b+2c  0            0            0                  |
|      0            0            -36a-30b-29c -29a-24b-38c -39a-18b-13c       |
|      0            0            19a+19b-10c  -16a+39b+21c -43a-15b-28c       |
|      0            0            -29a-8b-22c  34a+19b-47c  -47a+38b+2c        |
|      0            0            0            0            0                  |
|      0            0            0            0            0                  |
|      0            0            0            0            0                  |
|      0            0            0            0            0                  |
|      0            0            0            0            0                  |
|      0            0            0            0            0                  |
|      0            0            0            0            0                  |
|      0            0            0            0            0                  |
|      0            0            0            0            0                  |
|      0            0            0            0            0                  |
|      -----------------------------------------------------------------------|
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      16a2+22ab-34b2+45ac-48bc-47c2 35a2+11ab+33b2-38ac+40bc+11c2            |
|      47a2+19ab+7b2-16ac+15bc-23c2  46a2-28ab-3b2+ac+22bc-47c2               |
|      39a2+43ab-11b2-17ac+48bc+36c2 -23a2-7ab+29b2+2ac-47bc+15c2             |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      -----------------------------------------------------------------------|
|      0                              0                                       |
|      0                              0                                       |
|      0                              0                                       |
|      0                              0                                       |
|      0                              0                                       |
|      0                              0                                       |
|      0                              0                                       |
|      0                              0                                       |
|      0                              0                                       |
|      0                              0                                       |
|      -37a2-13ab+30b2-10ac-18bc+39c2 0                                       |
|      27a2-22ab-9b2+32ac-32bc-20c2   0                                       |
|      24a2-30ab-15b2-48ac+39bc       0                                       |
|      0                              16a2+22ab-34b2+45ac-48bc-47c2           |
|      0                              47a2+19ab+7b2-16ac+15bc-23c2            |
|      0                              39a2+43ab-11b2-17ac+48bc+36c2           |
|      0                              0                                       |
|      0                              0                                       |
|      0                              0                                       |
|      0                              0                                       |
|      -----------------------------------------------------------------------|
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      35a2+11ab+33b2-38ac+40bc+11c2 -37a2-13ab+30b2-10ac-18bc+39c2           |
|      46a2-28ab-3b2+ac+22bc-47c2    27a2-22ab-9b2+32ac-32bc-20c2             |
|      -23a2-7ab+29b2+2ac-47bc+15c2  24a2-30ab-15b2-48ac+39bc                 |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      -----------------------------------------------------------------------|
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      0                             0                                        |
|      16a2+22ab-34b2+45ac-48bc-47c2 35a2+11ab+33b2-38ac+40bc+11c2            |
|      47a2+19ab+7b2-16ac+15bc-23c2  46a2-28ab-3b2+ac+22bc-47c2               |
|      39a2+43ab-11b2-17ac+48bc+36c2 -23a2-7ab+29b2+2ac-47bc+15c2             |
|      0                             0                                        |
|      -----------------------------------------------------------------------|
|      0                                                                      |
|      0                                                                      |
|      0                                                                      |
|      0                                                                      |
|      0                                                                      |
|      0                                                                      |
|      0                                                                      |
|      0                                                                      |
|      0                                                                      |
|      0                                                                      |
|      0                                                                      |
|      0                                                                      |
|      0                                                                      |
|      0                                                                      |
|      0                                                                      |
|      0                                                                      |
|      -37a2-13ab+30b2-10ac-18bc+39c2                                         |
|      27a2-22ab-9b2+32ac-32bc-20c2                                           |
|      24a2-30ab-15b2-48ac+39bc                                               |
|      0                                                                      |
|      -----------------------------------------------------------------------|
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      0                                                       |              |
|      33a3-49a2b-19ab2+44b3-33a2c+17abc-39b2c-20ac2+36bc2+9c3 |              |
|                                                                             |
|              20      20                                                     |
|o15 : Matrix S   <-- S                                                       |
+-----------------------------------------------------------------------------+
|i16 : f_4                                                                    |
|                                                                             |
|o16 = 0                                                                      |
|                                                                             |
|o16 : Matrix 0 <-- 0                                                         |
+-----------------------------------------------------------------------------+
|i17 : ss^D_(2,3)                                                             |
|                                                                             |
|o17 = 0                                                                      |
|                                                                             |
|              20      10                                                     |
|o17 : Matrix S   <-- S                                                       |
+-----------------------------------------------------------------------------+
|i18 : ss^D_(4,0)                                                             |
|                                                                             |
|o18 = 0                                                                      |
|                                                                             |
|o18 : Matrix 0 <-- 0                                                         |
+-----------------------------------------------------------------------------+

See also
========

  * SimplicialModule _ ZZ -- access individual
    objects of a simplicial module

Ways to use this method:
========================

  * "SimplicialModuleMap _ Sequence"
  * SimplicialModuleMap _ ZZ -- access
    individual matrices in a simplicial module map
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1859:0.

SimplicialModuleMap | SimplicialModuleMap -- join or concatenate maps horizontally
**********************************************************************************

  * Operator: | (Macaulay2Doc),
  * Usage:
        f | g
  * Inputs:
      * f, a Map of Simplicial Modules,
      * g, a Map of Simplicial Modules,
  * Outputs:
      * a Map of Simplicial Modules,

Description
===========

Given simplicial module maps with the same target, this method constructs the
associated map from the direct sum of the sources to the target.

First, we define some non-trivial maps of simplicial modules.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[a..d];                                                       |
+-----------------------------------------------------------------------------+
|i2 : C1 = simplicialModule((freeResolution coker matrix{{a,b,c}})[1],        |
|                                                                             |
|      3      6      10                                                       |
|o2 = R  <-- R  <-- R  <-- ...                                                |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i3 : C2 = simplicialModule(freeResolution coker matrix{{a*b,a*c,b*c}},       |
|                                                                             |
|      1      4      9                                                        |
|o3 = R  <-- R  <-- R <-- ...                                                 |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o3 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i4 : D = simplicialModule(freeResolution coker matrix{{a^2,b^2,c*d}},        |
|                                                                             |
|      1      4      10      20                                               |
|o4 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o4 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i5 : f = randomSimplicialMap(D, C1)                                          |
|                                                                             |
|          1                                                           3      |
|o5 = 0 : R  <------------------------------------------------------- R  : 0  |
|               | 24a-36b-30c-29d 19a+19b-10c-29d -8a-22b-29c-24d |           |
|                                                                             |
|          4                                                                  |
|     1 : R  <----------------------------------------------------------------|
|               {0} | 24a-36b-30c-29d 19a+19b-10c-29d -8a-22b-29c-24d 0   0  0|
|               {2} | 0               0               0               -38 21 -|
|               {2} | 0               0               0               -16 34 -|
|               {2} | 0               0               0               39  19 -|
|                                                                             |
|          10                                                                 |
|     2 : R   <---------------------------------------------------------------|
|                {0} | 24a-36b-30c-29d 19a+19b-10c-29d -8a-22b-29c-24d 0   0  |
|                {2} | 0               0               0               -38 21 |
|                {2} | 0               0               0               -16 34 |
|                {2} | 0               0               0               39  19 |
|                {2} | 0               0               0               0   0  |
|                {2} | 0               0               0               0   0  |
|                {2} | 0               0               0               0   0  |
|                {4} | 0               0               0               0   0  |
|                {4} | 0               0               0               0   0  |
|                {4} | 0               0               0               0   0  |
|                                                                             |
|o5 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|        6                                                                    |
|------ R  : 1                                                                |
|   |                                                                         |
|47 |                                                                         |
|39 |                                                                         |
|18 |                                                                         |
|                                                                             |
|                      10                                                     |
|-------------------- R   : 2                                                 |
|0   0   0  0   0 |                                                           |
|-47 0   0  0   0 |                                                           |
|-39 0   0  0   0 |                                                           |
|-18 0   0  0   0 |                                                           |
|0   -38 21 -47 0 |                                                           |
|0   -16 34 -39 0 |                                                           |
|0   39  19 -18 0 |                                                           |
|0   0   0  0   0 |                                                           |
|0   0   0  0   0 |                                                           |
|0   0   0  0   0 |                                                           |
+-----------------------------------------------------------------------------+
|i6 : g = randomSimplicialMap(D, C2)                                          |
|                                                                             |
|          1               1                                                  |
|o6 = 0 : R  <----------- R  : 0                                              |
|               | -13 |                                                       |
|                                                                             |
|          4                              4                                   |
|     1 : R  <-------------------------- R  : 1                               |
|               {0} | -13 0   0   0  |                                        |
|               {2} | 0   -43 -47 16 |                                        |
|               {2} | 0   -15 38  22 |                                        |
|               {2} | 0   -28 2   45 |                                        |
|                                                                             |
|          10                                             9                   |
|     2 : R   <----------------------------------------- R  : 2               |
|                {0} | -13 0   0   0  0   0   0  0 0 |                        |
|                {2} | 0   -43 -47 16 0   0   0  0 0 |                        |
|                {2} | 0   -15 38  22 0   0   0  0 0 |                        |
|                {2} | 0   -28 2   45 0   0   0  0 0 |                        |
|                {2} | 0   0   0   0  -43 -47 16 0 0 |                        |
|                {2} | 0   0   0   0  -15 38  22 0 0 |                        |
|                {2} | 0   0   0   0  -28 2   45 0 0 |                        |
|                {4} | 0   0   0   0  0   0   0  0 0 |                        |
|                {4} | 0   0   0   0  0   0   0  0 0 |                        |
|                {4} | 0   0   0   0  0   0   0  0 0 |                        |
|                                                                             |
|o6 : SimplicialModuleMap                                                     |
+-----------------------------------------------------------------------------+
+-----------------------------------------------------------------------------+
|i7 : h = f|g                                                                 |
|                                                                             |
|          1                                                               4  |
|o7 = 0 : R  <----------------------------------------------------------- R  :|
|               | 24a-36b-30c-29d 19a+19b-10c-29d -8a-22b-29c-24d -13 |       |
|                                                                             |
|          4                                                                  |
|     1 : R  <----------------------------------------------------------------|
|               {0} | 24a-36b-30c-29d 19a+19b-10c-29d -8a-22b-29c-24d 0   0  0|
|               {2} | 0               0               0               -38 21 -|
|               {2} | 0               0               0               -16 34 -|
|               {2} | 0               0               0               39  19 -|
|                                                                             |
|          10                                                                 |
|     2 : R   <---------------------------------------------------------------|
|                {0} | 24a-36b-30c-29d 19a+19b-10c-29d -8a-22b-29c-24d 0   0  |
|                {2} | 0               0               0               -38 21 |
|                {2} | 0               0               0               -16 34 |
|                {2} | 0               0               0               39  19 |
|                {2} | 0               0               0               0   0  |
|                {2} | 0               0               0               0   0  |
|                {2} | 0               0               0               0   0  |
|                {4} | 0               0               0               0   0  |
|                {4} | 0               0               0               0   0  |
|                {4} | 0               0               0               0   0  |
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
| 0                                                                           |
|                                                                             |
|                                                                             |
|                       10                                                    |
|--------------------- R   : 1                                                |
|   -13 0   0   0  |                                                          |
|47 0   -43 -47 16 |                                                          |
|39 0   -15 38  22 |                                                          |
|18 0   -28 2   45 |                                                          |
|                                                                             |
|                                                    19                       |
|-------------------------------------------------- R   : 2                   |
|0   0   0  0   0 -13 0   0   0  0   0   0  0 0 |                             |
|-47 0   0  0   0 0   -43 -47 16 0   0   0  0 0 |                             |
|-39 0   0  0   0 0   -15 38  22 0   0   0  0 0 |                             |
|-18 0   0  0   0 0   -28 2   45 0   0   0  0 0 |                             |
|0   -38 21 -47 0 0   0   0   0  -43 -47 16 0 0 |                             |
|0   -16 34 -39 0 0   0   0   0  -15 38  22 0 0 |                             |
|0   39  19 -18 0 0   0   0   0  -28 2   45 0 0 |                             |
|0   0   0  0   0 0   0   0   0  0   0   0  0 0 |                             |
|0   0   0  0   0 0   0   0   0  0   0   0  0 0 |                             |
|0   0   0  0   0 0   0   0   0  0   0   0  0 0 |                             |
+-----------------------------------------------------------------------------+
|i8 : assert isWellDefined h                                                  |
+-----------------------------------------------------------------------------+
|i9 : assert(source h === source f ++ source g)                               |
+-----------------------------------------------------------------------------+
|i10 : assert(target h === target f)                                          |
+-----------------------------------------------------------------------------+

See also
========

  * SimplicialModule ++ SimplicialModule
    -- direct sum of simplicial modules
  * SimplicialModuleMap ++ SimplicialModuleMap
    -- direct sum of simplicial module maps
  * SimplicialModuleMap || SimplicialModuleMap
    -- join or concatenate maps vertically
  * Matrix | Matrix (Macaulay2Doc) -- join matrices
    horizontally

Ways to use this method:
========================

  * SimplicialModuleMap | SimplicialModuleMap
    -- join or concatenate maps horizontally
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2628:0.

SimplicialModuleMap || SimplicialModuleMap -- join or concatenate maps vertically
*********************************************************************************

  * Operator: || (Macaulay2Doc),
  * Usage:
        f || g
  * Inputs:
      * f, a Map of Simplicial Modules,
      * g, a Map of Simplicial Modules,
  * Outputs:
      * a Map of Simplicial Modules,

Description
===========

Given simplicial module maps with the same source, this method constructs the
associated map from the source to the direct sum of the targets.

First, we define some non-trivial maps of simplicial modules.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[a..d];                                                       |
+-----------------------------------------------------------------------------+
|i2 : D1 = simplicialModule((freeResolution coker matrix{{a,b,c}})[1],        |
|                                                                             |
|      3      6      10                                                       |
|o2 = R  <-- R  <-- R  <-- ...                                                |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i3 : D2 = simplicialModule(freeResolution coker matrix{{a*b,a*c,b*c}},       |
|                                                                             |
|      1      4      9                                                        |
|o3 = R  <-- R  <-- R <-- ...                                                 |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o3 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i4 : C = simplicialModule(freeResolution coker matrix{{a^2,b^2,c*d}},        |
|                                                                             |
|      1      4      10      20                                               |
|o4 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o4 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|Degeneracy => true)                                                          |
+-----------------------------------------------------------------------------+
|i5 : f = randomSimplicialMap(D1, C)                                          |
|                                                                             |
|          3         1                                                        |
|o5 = 0 : R  <----- R  : 0                                                    |
|               0                                                             |
|                                                                             |
|          6                             4                                    |
|     1 : R  <------------------------- R  : 1                                |
|               {1} | 0 0   0   0   |                                         |
|               {1} | 0 0   0   0   |                                         |
|               {1} | 0 0   0   0   |                                         |
|               {2} | 0 24  -29 -10 |                                         |
|               {2} | 0 -36 19  -29 |                                         |
|               {2} | 0 -30 19  -8  |                                         |
|                                                                             |
|          10                                                                 |
|     2 : R   <---------------------------------------------------------------|
|                {1} | 0 0   0   0   0   0   0   0                0           |
|                {1} | 0 0   0   0   0   0   0   0                0           |
|                {1} | 0 0   0   0   0   0   0   0                0           |
|                {2} | 0 24  -29 -10 0   0   0   0                0           |
|                {2} | 0 -36 19  -29 0   0   0   0                0           |
|                {2} | 0 -30 19  -8  0   0   0   0                0           |
|                {2} | 0 0   0   0   24  -29 -10 0                0           |
|                {2} | 0 0   0   0   -36 19  -29 0                0           |
|                {2} | 0 0   0   0   -30 19  -8  0                0           |
|                {3} | 0 0   0   0   0   0   0   -22a-29b-24c-38d -16a+39b+21c|
|                                                                             |
|o5 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                          10                                                 |
|------------------------ R   : 2                                             |
|     0               |                                                       |
|     0               |                                                       |
|     0               |                                                       |
|     0               |                                                       |
|     0               |                                                       |
|     0               |                                                       |
|     0               |                                                       |
|     0               |                                                       |
|     0               |                                                       |
|+34d 19a-47b-39c-18d |                                                       |
+-----------------------------------------------------------------------------+
|i6 : g = randomSimplicialMap(D2, C)                                          |
|                                                                             |
|          1               1                                                  |
|o6 = 0 : R  <----------- R  : 0                                              |
|               | -13 |                                                       |
|                                                                             |
|          4                              4                                   |
|     1 : R  <-------------------------- R  : 1                               |
|               {0} | -13 0   0   0  |                                        |
|               {2} | 0   -43 -47 16 |                                        |
|               {2} | 0   -15 38  22 |                                        |
|               {2} | 0   -28 2   45 |                                        |
|                                                                             |
|          9                                                                  |
|     2 : R  <----------------------------------------------------------------|
|               {0} | -13 0   0   0  0   0   0  0                0            |
|               {2} | 0   -43 -47 16 0   0   0  0                0            |
|               {2} | 0   -15 38  22 0   0   0  0                0            |
|               {2} | 0   -28 2   45 0   0   0  0                0            |
|               {2} | 0   0   0   0  -43 -47 16 0                0            |
|               {2} | 0   0   0   0  -15 38  22 0                0            |
|               {2} | 0   0   0   0  -28 2   45 0                0            |
|               {3} | 0   0   0   0  0   0   0  -34a-48b-47c+47d -23a+39b+43c-|
|               {3} | 0   0   0   0  0   0   0  19a-16b+7c+15d   -11a+48b+36c+|
|                                                                             |
|o6 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                         10                                                  |
|----------------------- R   : 2                                              |
|    0               |                                                        |
|    0               |                                                        |
|    0               |                                                        |
|    0               |                                                        |
|    0               |                                                        |
|    0               |                                                        |
|    0               |                                                        |
|17d 11a-38b+33c+40d |                                                        |
|35d 11a+46b-28c+d   |                                                        |
+-----------------------------------------------------------------------------+
+-----------------------------------------------------------------------------+
|i7 : h = f||g                                                                |
|                                                                             |
|          4                   1                                              |
|o7 = 0 : R  <--------------- R  : 0                                          |
|               {1} | 0   |                                                   |
|               {1} | 0   |                                                   |
|               {1} | 0   |                                                   |
|               {0} | -13 |                                                   |
|                                                                             |
|          10                               4                                 |
|     1 : R   <--------------------------- R  : 1                             |
|                {1} | 0   0   0   0   |                                      |
|                {1} | 0   0   0   0   |                                      |
|                {1} | 0   0   0   0   |                                      |
|                {2} | 0   24  -29 -10 |                                      |
|                {2} | 0   -36 19  -29 |                                      |
|                {2} | 0   -30 19  -8  |                                      |
|                {0} | -13 0   0   0   |                                      |
|                {2} | 0   -43 -47 16  |                                      |
|                {2} | 0   -15 38  22  |                                      |
|                {2} | 0   -28 2   45  |                                      |
|                                                                             |
|          19                                                                 |
|     2 : R   <---------------------------------------------------------------|
|                {1} | 0   0   0   0   0   0   0   0                0         |
|                {1} | 0   0   0   0   0   0   0   0                0         |
|                {1} | 0   0   0   0   0   0   0   0                0         |
|                {2} | 0   24  -29 -10 0   0   0   0                0         |
|                {2} | 0   -36 19  -29 0   0   0   0                0         |
|                {2} | 0   -30 19  -8  0   0   0   0                0         |
|                {2} | 0   0   0   0   24  -29 -10 0                0         |
|                {2} | 0   0   0   0   -36 19  -29 0                0         |
|                {2} | 0   0   0   0   -30 19  -8  0                0         |
|                {3} | 0   0   0   0   0   0   0   -22a-29b-24c-38d -16a+39b+2|
|                {0} | -13 0   0   0   0   0   0   0                0         |
|                {2} | 0   -43 -47 16  0   0   0   0                0         |
|                {2} | 0   -15 38  22  0   0   0   0                0         |
|                {2} | 0   -28 2   45  0   0   0   0                0         |
|                {2} | 0   0   0   0   -43 -47 16  0                0         |
|                {2} | 0   0   0   0   -15 38  22  0                0         |
|                {2} | 0   0   0   0   -28 2   45  0                0         |
|                {3} | 0   0   0   0   0   0   0   -34a-48b-47c+47d -23a+39b+4|
|                {3} | 0   0   0   0   0   0   0   19a-16b+7c+15d   -11a+48b+3|
|                                                                             |
|o7 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                            10                                               |
|-------------------------- R   : 2                                           |
|       0               |                                                     |
|       0               |                                                     |
|       0               |                                                     |
|       0               |                                                     |
|       0               |                                                     |
|       0               |                                                     |
|       0               |                                                     |
|       0               |                                                     |
|       0               |                                                     |
|1c+34d 19a-47b-39c-18d |                                                     |
|       0               |                                                     |
|       0               |                                                     |
|       0               |                                                     |
|       0               |                                                     |
|       0               |                                                     |
|       0               |                                                     |
|       0               |                                                     |
|3c-17d 11a-38b+33c+40d |                                                     |
|6c+35d 11a+46b-28c+d   |                                                     |
+-----------------------------------------------------------------------------+
|i8 : assert isWellDefined h                                                  |
+-----------------------------------------------------------------------------+
|i9 : assert(target h === target f ++ target g)                               |
+-----------------------------------------------------------------------------+
|i10 : assert(source h === source f)                                          |
+-----------------------------------------------------------------------------+

See also
========

  * SimplicialModule ++ SimplicialModule
    -- direct sum of simplicial modules
  * SimplicialModuleMap ++ SimplicialModuleMap
    -- direct sum of simplicial module maps
  * SimplicialModuleMap | SimplicialModuleMap
    -- join or concatenate maps horizontally
  * Matrix || Matrix (Macaulay2Doc) -- join matrices
    vertically

Ways to use this method:
========================

  * SimplicialModuleMap || SimplicialModuleMap
    -- join or concatenate maps vertically
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2666:0.

simplicialTensor -- compute the simplicial tensor product and cache direct sum indices for easy access
******************************************************************************************************

  * Usage:
        simplicialTensor(T, Options => {Degeneracy => false, TopDegree => null})
  * Inputs:
      * T, a list (Macaulay2Doc), A list of simplicial modules or
        complexes to compute the tensor product.
  * Optional inputs (Macaulay2Doc),:
      * Degeneracy,
        => a Boolean value (Macaulay2Doc), default value
        false, Default value is false. Indicates whether to include degeneracy
        maps in the construction.
      * TopDegree,
        => an integer (Macaulay2Doc), default value null, Default
        is null. If provided, specifies the top degree for the construction of
        the simplicial module.
  * Outputs:
      * a Simplicial Module, The simplicial tensor
        product of the components in T, with cached direct sum indices.

Description
===========

This function computes the simplicial tensor product of a list T, which can
consist of simplicial modules or complexes. It caches direct sum indices using
tensorwithComponents, so that the user can easily
access components of the resulting face/degeneracy maps on particular direct
summands of the tensor product.

The simplicial tensor product of complexes is built as the Dold-Kan extension
of the tensor product functor on the category of R-modules. In general, this
complex looks quite different from the standard tensor product of complexes. In
fact, the simplicial tensor product is always homotopy equivalent to the
classically defined tensor product.

+------------------------------------------+
|i1 : Q = ZZ/101[x_1,x_2];                 |
+------------------------------------------+
|i2 : K1 = complex {matrix{{x_1}}};        |
+------------------------------------------+
|i3 : K2 = complex {matrix{{x_2}}};        |
+------------------------------------------+
|i4 : T1 = K1**K2                          |
|                                          |
|      1      2      1                     |
|o4 = Q  <-- Q  <-- Q                      |
|                                          |
|     0      1      2                      |
|                                          |
|o4 : Complex                              |
+------------------------------------------+
|i5 : T1.dd                                |
|                                          |
|          1                   2           |
|o5 = 0 : Q  <--------------- Q  : 1       |
|               | x_2 x_1 |                |
|                                          |
|          2                    1          |
|     1 : Q  <---------------- Q  : 2      |
|               {1} | x_1  |               |
|               {1} | -x_2 |               |
|                                          |
|o5 : ComplexMap                           |
+------------------------------------------+
|i6 : T2 = prune simplicialTensor({K1,K2}) |
|                                          |
|      1      3      2                     |
|o6 = Q  <-- Q  <-- Q                      |
|                                          |
|     0      1      2                      |
|                                          |
|o6 : Complex                              |
+------------------------------------------+
|i7 : T2.dd                                |
|                                          |
|          1                          3    |
|o7 = 0 : Q  <---------------------- Q  : 1|
|               | x_2 x_1 x_1x_2 |         |
|                                          |
|          3                         2     |
|     1 : Q  <--------------------- Q  : 2 |
|               {1} | -x_1 -x_1 |          |
|               {1} | x_2  0    |          |
|               {2} | 0    1    |          |
|                                          |
|o7 : ComplexMap                           |
+------------------------------------------+
|i8 : phi1 = extend(T1,T2,id_(T1_0))       |
|                                          |
|          1             1                 |
|o8 = 0 : Q  <--------- Q  : 0             |
|               | 1 |                      |
|                                          |
|          2                       3       |
|     1 : Q  <------------------- Q  : 1   |
|               {1} | 1 0 0   |            |
|               {1} | 0 1 x_2 |            |
|                                          |
|          1                     2         |
|     2 : Q  <----------------- Q  : 2     |
|               {2} | -1 -1 |              |
|                                          |
|o8 : ComplexMap                           |
+------------------------------------------+
|i9 : phi2 = extend(T2,T1,id_(T1_0))       |
|                                          |
|          1             1                 |
|o9 = 0 : Q  <--------- Q  : 0             |
|               | 1 |                      |
|                                          |
|          3                   2           |
|     1 : Q  <--------------- Q  : 1       |
|               {1} | 1 0 |                |
|               {1} | 0 1 |                |
|               {2} | 0 0 |                |
|                                          |
|          2                  1            |
|     2 : Q  <-------------- Q  : 2        |
|               {2} | -1 |                 |
|               {2} | 0  |                 |
|                                          |
|o9 : ComplexMap                           |
+------------------------------------------+
|i10 : phi1*phi2 == id_T1                  |
|                                          |
|o10 = true                                |
+------------------------------------------+
|i11 : isNullHomotopic(phi2*phi1 - id_T2)  |
|                                          |
|o11 = true                                |
+------------------------------------------+

Here is how to access specific components of the face maps for a tensor product
of simplicial modules:

+-----------------------------------------------------------------------------+
|i12 : S1 = simplicialModule(K1, 4)                                           |
|                                                                             |
|       1      2      3      4      5                                         |
|o12 = Q  <-- Q  <-- Q  <-- Q  <-- Q <-- ...                                  |
|                                                                             |
|      0      1      2      3      4                                          |
|                                                                             |
|o12 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i13 : S2 = simplicialModule(K2, 4)                                           |
|                                                                             |
|       1      2      3      4      5                                         |
|o13 = Q  <-- Q  <-- Q  <-- Q  <-- Q <-- ...                                  |
|                                                                             |
|      0      1      2      3      4                                          |
|                                                                             |
|o13 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i14 : S12 = S1**S2                                                           |
|                                                                             |
|       1      4      9      16      25                                       |
|o14 = Q  <-- Q  <-- Q  <-- Q   <-- Q  <-- ...                                |
|                                                                             |
|      0      1      2      3       4                                         |
|                                                                             |
|o14 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i15 : indices S12_4 --lists the indices of the summands                      |
|                                                                             |
|o15 = {{0, 0}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 0}, {1, 1}, {1, 2}, {1,   |
|      -----------------------------------------------------------------------|
|      3}, {1, 4}, {2, 0}, {2, 1}, {2, 2}, {2, 3}, {2, 4}, {3, 0}, {3, 1}, {3,|
|      -----------------------------------------------------------------------|
|      2}, {3, 3}, {3, 4}, {4, 0}, {4, 1}, {4, 2}, {4, 3}, {4, 4}}            |
|                                                                             |
|o15 : List                                                                   |
+-----------------------------------------------------------------------------+
|i16 : netList flatten for i in indices S12_4 list (                          |
|          for j in indices S12_3 list ( if (dd^S12_(4,0))_[i]^[j]==0 then    |
|              horizontalJoin {net (dd^S12_(4,0))_[i]^[j], " : ", net i, "    |
|                                                                             |
|      +-------------------------------+                                      |
|o16 = || 1 | : {0, 0} --> {0, 0}      |                                      |
|      +-------------------------------+                                      |
|      || x_2 | : {0, 1} --> {0, 0}    |                                      |
|      +-------------------------------+                                      |
|      |{1} | 1 | : {0, 2} --> {0, 1}  |                                      |
|      +-------------------------------+                                      |
|      |{1} | 1 | : {0, 3} --> {0, 2}  |                                      |
|      +-------------------------------+                                      |
|      |{1} | 1 | : {0, 4} --> {0, 3}  |                                      |
|      +-------------------------------+                                      |
|      || x_1 | : {1, 0} --> {0, 0}    |                                      |
|      +-------------------------------+                                      |
|      || x_1x_2 | : {1, 1} --> {0, 0} |                                      |
|      +-------------------------------+                                      |
|      |{1} | x_1 | : {1, 2} --> {0, 1}|                                      |
|      +-------------------------------+                                      |
|      |{1} | x_1 | : {1, 3} --> {0, 2}|                                      |
|      +-------------------------------+                                      |
|      |{1} | x_1 | : {1, 4} --> {0, 3}|                                      |
|      +-------------------------------+                                      |
|      |{1} | 1 | : {2, 0} --> {1, 0}  |                                      |
|      +-------------------------------+                                      |
|      |{1} | x_2 | : {2, 1} --> {1, 0}|                                      |
|      +-------------------------------+                                      |
|      |{2} | 1 | : {2, 2} --> {1, 1}  |                                      |
|      +-------------------------------+                                      |
|      |{2} | 1 | : {2, 3} --> {1, 2}  |                                      |
|      +-------------------------------+                                      |
|      |{2} | 1 | : {2, 4} --> {1, 3}  |                                      |
|      +-------------------------------+                                      |
|      |{1} | 1 | : {3, 0} --> {2, 0}  |                                      |
|      +-------------------------------+                                      |
|      |{1} | x_2 | : {3, 1} --> {2, 0}|                                      |
|      +-------------------------------+                                      |
|      |{2} | 1 | : {3, 2} --> {2, 1}  |                                      |
|      +-------------------------------+                                      |
|      |{2} | 1 | : {3, 3} --> {2, 2}  |                                      |
|      +-------------------------------+                                      |
|      |{2} | 1 | : {3, 4} --> {2, 3}  |                                      |
|      +-------------------------------+                                      |
|      |{1} | 1 | : {4, 0} --> {3, 0}  |                                      |
|      +-------------------------------+                                      |
|      |{1} | x_2 | : {4, 1} --> {3, 0}|                                      |
|      +-------------------------------+                                      |
|      |{2} | 1 | : {4, 2} --> {3, 1}  |                                      |
|      +-------------------------------+                                      |
|      |{2} | 1 | : {4, 3} --> {3, 2}  |                                      |
|      +-------------------------------+                                      |
|      |{2} | 1 | : {4, 4} --> {3, 3}  |                                      |
|      +-------------------------------+                                      |
|-----------------------------------------------------------------------------|
|continue else                                                                |
|--> " , net j} ))                                                            |
+-----------------------------------------------------------------------------+

One reason for using the simplicial tensor product is that its components as a
simplicial module are more canonically built, and thus naturally extend
functorial maps on the category of R-modules to the category of chain
complexes.

We can see actually see this in an example. The exterior power functor admits a
canonical comultiplication map $$\bigwedge^{i+j} \to \bigwedge^i \otimes
\bigwedge^j.$$ This means that for any chain complex $C$ there is a canonical
inclusion of complexes $$\bigwedge^{i+j} C \hookrightarrow \bigwedge^i C
\otimes \bigwedge^j C.$$ Constructing this inclusion directly using the naive
definition of the exterior power functor on complexes would be extremely
unnatural, but filtering through the simplicial category makes this a very easy
task:

+-----------------------------------------------------------------------------+
|i17 : Q = ZZ/101[a,b];                                                       |
+-----------------------------------------------------------------------------+
|i18 : K = koszulComplex vars Q                                               |
|                                                                             |
|       1      2      1                                                       |
|o18 = Q  <-- Q  <-- Q                                                        |
|                                                                             |
|      0      1      2                                                        |
|                                                                             |
|o18 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i19 : SK = simplicialModule(K,6) --want top degree 6 since the resulting     |
|                                                                             |
|       1      3      6      10      15      21      28                       |
|o19 = Q  <-- Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- Q  <-- ...                |
|                                                                             |
|      0      1      2      3       4       5       6                         |
|                                                                             |
|o19 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|complexes should have length 6                                               |
+-----------------------------------------------------------------------------+
|i20 : w21K = extPower(2, SK) ** SK                                           |
|                                                                             |
|             9      90      450      1575      4410      10584               |
|o20 = 0 <-- Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q     <-- ...        |
|                                                                             |
|      0     1      2       3        4         5         6                    |
|                                                                             |
|o20 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i21 : w3K = extPower(3, SK)                                                  |
|                                                                             |
|             1      20      120      455      1330      3276                 |
|o21 = 0 <-- Q  <-- Q   <-- Q    <-- Q    <-- Q     <-- Q    <-- ...          |
|                                                                             |
|      0     1      2       3        4        5         6                     |
|                                                                             |
|o21 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i22 : H = hashTable for i from 0 to 6 list i => dual wedgeProduct(2,1, dual  |
|-----------------------------------------------------------------------------|
|SK_i);                                                                       |
+-----------------------------------------------------------------------------+
|i23 : inclusion = map(w21K, w3K, H);                                         |
+-----------------------------------------------------------------------------+
|i24 : isWellDefined inclusion                                                |
|                                                                             |
|o24 = true                                                                   |
+-----------------------------------------------------------------------------+
|i25 : isCommutative inclusion                                                |
|                                                                             |
|o25 = true                                                                   |
+-----------------------------------------------------------------------------+
|i26 : prune ker inclusion --should be 0 since it is an inclusion             |
|                                                                             |
|o26 = 0 <-- 0 <-- 0 <-- 0 <-- 0 <-- 0 <-- 0<-- ...                           |
|                                                                             |
|      0     1     2     3     4     5     6                                  |
|                                                                             |
|o26 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i27 : S21' = complex { matrix {{a, b,                                        |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|       2      5      6      5      2                                         |
|o27 = Q  <-- Q  <-- Q  <-- Q  <-- Q                                          |
|                                                                             |
|      0      1      2      3      4                                          |
|                                                                             |
|o27 : Complex                                                                |
|-----------------------------------------------------------------------------|
|0, 0, -b}, {0, 0, a, b, a}}, matrix {{-b, b, 0, 0,                           |
| b, 0}, {a, -a, 0, 0, 0, b}, {0, 0, -b, b, -a, 0}, {0, 0, a, -a, 0, -a}, {0, |
|matrix {{a, b, 0, -b, -b}, {a, b, 0, 0, -2*b}, {0, a, b, a, 0}, {0, a, b, 0, |
|  -b, b}, {0, 0, 0, a, -a}}, matrix {{3*b, 0}, {-a, 2*b}, {0, -3*a}, {a, b}, |
|-----------------------------------------------------------------------------|
|0, 0, 0, a, b}},                                                             |
|a}, {0, 0, 0,                                                                |
|{a, b}}}                                                                     |
+-----------------------------------------------------------------------------+
|i28 : S21 = S21'[-1]  --this is the S^(2,1) schur functor; note the ranks    |
|                                                                             |
|       2      5      6      5      2                                         |
|o28 = Q  <-- Q  <-- Q  <-- Q  <-- Q                                          |
|                                                                             |
|      1      2      3      4      5                                          |
|                                                                             |
|o28 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i29 : prune HH S21 --note the homology                                       |
|                                                                             |
|o29 = cokernel | b a 0 0 | <-- cokernel {1} | b a | <-- cokernel {2} | b a 0 |
|               | 0 0 b a |                                       {2} | 0 0 b |
|                               2                                             |
|      1                                                 3                    |
|                                                                             |
|o29 : Complex                                                                |
|-----------------------------------------------------------------------------|
|0 |                                                                          |
|a |                                                                          |
+-----------------------------------------------------------------------------+

Just for sake of illustration, let us see how the above example changes in the
modular setting; note that the only characteristic for which the inclusion
$$\bigwedge^3 \hookrightarrow \bigwedge^2 \otimes \bigwedge^1$$ is not
canonically split is for characteristic 3, so we should expect the Dold-Kan
extension of the Schur functor $\mathbb{S}^{(2,1)} (K)$ to look different in
characteristic 3. (the resulting minimization of $\mathbb{S}^{(2,1)} (K)$
should not have coefficients, for instance).

+-----------------------------------------------------------------------------+
|i30 : Q = ZZ/3[a,b];                                                         |
+-----------------------------------------------------------------------------+
|i31 : K = koszulComplex vars Q                                               |
|                                                                             |
|       1      2      1                                                       |
|o31 = Q  <-- Q  <-- Q                                                        |
|                                                                             |
|      0      1      2                                                        |
|                                                                             |
|o31 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i32 : SK = simplicialModule(K,6) --want top degree 6 since the resulting     |
|                                                                             |
|       1      3      6      10      15      21      28                       |
|o32 = Q  <-- Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- Q  <-- ...                |
|                                                                             |
|      0      1      2      3       4       5       6                         |
|                                                                             |
|o32 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|complexes should have length 6                                               |
+-----------------------------------------------------------------------------+
|i33 : w21K = extPower(2, SK) ** SK                                           |
|                                                                             |
|             9      90      450      1575      4410      10584               |
|o33 = 0 <-- Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q     <-- ...        |
|                                                                             |
|      0     1      2       3        4         5         6                    |
|                                                                             |
|o33 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i34 : w3K = extPower(3, SK)                                                  |
|                                                                             |
|             1      20      120      455      1330      3276                 |
|o34 = 0 <-- Q  <-- Q   <-- Q    <-- Q    <-- Q     <-- Q    <-- ...          |
|                                                                             |
|      0     1      2       3        4        5         6                     |
|                                                                             |
|o34 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i35 : H = hashTable for i from 0 to 6 list i => dual wedgeProduct(2,1, dual  |
|-----------------------------------------------------------------------------|
|SK_i);                                                                       |
+-----------------------------------------------------------------------------+
|i36 : inclusion = map(w21K, w3K, H);                                         |
+-----------------------------------------------------------------------------+

Note that the above method of computing the Schur functor $\mathbb{S}^{(2,1)}
(K)$ is significantly faster than the schurMap command.

This method is also implemented functorially, so it can be applied to
simplicial maps and complex maps.

+-----------------------------------------------------------------------------+
|i37 : F = freeResolution( (ideal vars Q)^2)                                  |
|                                                                             |
|       1      3      2                                                       |
|o37 = Q  <-- Q  <-- Q                                                        |
|                                                                             |
|      0      1      2                                                        |
|                                                                             |
|o37 : Complex                                                                |
+-----------------------------------------------------------------------------+
|i38 : phi = extend(K, F, id_(K_0))                                           |
|                                                                             |
|           1             1                                                   |
|o38 = 0 : Q  <--------- Q  : 0                                               |
|                | 1 |                                                        |
|                                                                             |
|           2                     3                                           |
|      1 : Q  <----------------- Q  : 1                                       |
|                {1} | a b 0 |                                                |
|                {1} | 0 0 b |                                                |
|                                                                             |
|           1                   2                                             |
|      2 : Q  <--------------- Q  : 2                                         |
|                {2} | 0 b |                                                  |
|                                                                             |
|o38 : ComplexMap                                                             |
+-----------------------------------------------------------------------------+
|i39 : prune simplicialTensor(phi, phi)                                       |
|                                                                             |
|           1             1                                                   |
|o39 = 0 : Q  <--------- Q  : 0                                               |
|                | 1 |                                                        |
|                                                                             |
|           8                                                      15         |
|      1 : Q  <-------------------------------------------------- Q   : 1     |
|                {1} | a b 0 0 0 0 0  0  0  0  0  0  0  0  0  |               |
|                {1} | 0 0 b 0 0 0 0  0  0  0  0  0  0  0  0  |               |
|                {1} | 0 0 0 a b 0 0  0  0  0  0  0  0  0  0  |               |
|                {1} | 0 0 0 0 0 b 0  0  0  0  0  0  0  0  0  |               |
|                {2} | 0 0 0 0 0 0 a2 ab 0  ab b2 0  0  0  0  |               |
|                {2} | 0 0 0 0 0 0 0  0  ab 0  0  b2 0  0  0  |               |
|                {2} | 0 0 0 0 0 0 0  0  0  0  0  0  ab b2 0  |               |
|                {2} | 0 0 0 0 0 0 0  0  0  0  0  0  0  0  b2 |               |
|                                                                             |
|           19                                                                |
|      2 : Q   <--------------------------------------------------------------|
|                 {2} | 0 b 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                 {2} | 0 0 0 0 a2 ab 0  0  0  0  ab b2 0  0  0  0  0  0  0  0|
|                 {2} | 0 0 0 0 0  0  ab 0  0  0  0  0  b2 0  0  0  0  0  0  0|
|                 {2} | 0 0 0 0 0  0  0  a2 ab 0  0  0  0  ab b2 0  0  0  0  0|
|                 {2} | 0 0 0 0 0  0  0  0  0  ab 0  0  0  0  0  b2 0  0  0  0|
|                 {2} | 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  ab b2 0  0|
|                 {2} | 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  b2 0|
|                 {2} | 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  a|
|                 {2} | 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                 {2} | 0 0 0 b 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                 {3} | 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                 {3} | 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                 {3} | 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                 {3} | 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                 {3} | 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                 {3} | 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                 {3} | 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                 {3} | 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                 {4} | 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                                                                             |
|           18                                                                |
|      3 : Q   <--------------------------------------------------------------|
|                 {3} | 0 ab 0 b2 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0|
|                 {3} | 0 0  0 0  0 b2 0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0|
|                 {3} | 0 0  0 0  0 0  0 ab 0 0  0 b2 0 0  0 0  0 0  0 0 0 0 0|
|                 {3} | 0 0  0 0  0 0  0 0  0 ab 0 0  0 b2 0 0  0 0  0 0 0 0 0|
|                 {3} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 b2 0 0  0 0 0 0 0|
|                 {3} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 b2 0 0 0 0 0|
|                 {3} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0|
|                 {3} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0|
|                 {3} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0|
|                 {3} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0|
|                 {3} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0|
|                 {3} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0|
|                 {4} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0|
|                 {4} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0|
|                 {4} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0|
|                 {4} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0|
|                 {4} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0|
|                 {4} | 0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0|
|                                                                             |
|           6                                                                 |
|      4 : Q  <---------------------------------------------------------------|
|                {4} | 0 0 0 b2 0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 0  0 0  0 0  ||
|                {4} | 0 0 0 0  0 0 0 0 0 b2 0 0  0 0 0 0 0 0 0 0  0 0  0 0  ||
|                {4} | 0 0 0 0  0 0 0 0 0 0  0 b2 0 0 0 0 0 0 0 0  0 0  0 0  ||
|                {4} | 0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 b2 0 0  0 0  ||
|                {4} | 0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 0  0 b2 0 0  ||
|                {4} | 0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 0  0 0  0 b2 ||
|                                                                             |
|o39 : ComplexMap                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|  0  0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  |
|  0  0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  |
|  0  0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  |
|  0  0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  |
|  0  0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  |
|  0  0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  |
|  0  0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  |
|b b2 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  |
|  0  b2 0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  |
|  0  0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  |
|  0  0  0 ab 0 b2 0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  |
|  0  0  0 0  0 0  0 b2 0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  |
|  0  0  0 0  0 0  0 0  0 ab 0 b2 0 0  0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  |
|  0  0  0 0  0 0  0 0  0 0  0 0  0 b2 0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  |
|  0  0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 ab b2 0  0  0  0  0 0 0 0  |
|  0  0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  b2 0  0  0  0 0 0 0  |
|  0  0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  ab b2 0  0 0 0 0  |
|  0  0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  0  0  b2 0 0 0 0  |
|  0  0  0 0  0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 b2 |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
| 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 ab b2 0  0  0  0  0  0  0  0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  b2 0  0  0  0  0  0  0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  ab b2 0  0  0  0  0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  0  0  b2 0  0  0  0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  0  0  0  ab b2 0  0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  0  0  0  0  0  b2 0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 b2 0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0  0 0 0 0 0 b2 0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0  0 0 0 0 0 0  0 b2 0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
| 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0 0 |
|                                                                             |
|    24                                                                       |
|-- Q   : 4                                                                   |
|-----------------------------------------------------------------------------|
|     50                                                                      |
|--- Q   : 2                                                                  |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
||                                                                            |
|                                                                             |
|                  60                                                         |
|---------------- Q   : 3                                                     |
|0  0 0  0 0  |                                                               |
|0  0 0  0 0  |                                                               |
|0  0 0  0 0  |                                                               |
|0  0 0  0 0  |                                                               |
|0  0 0  0 0  |                                                               |
|0  0 0  0 0  |                                                               |
|0  0 0  0 0  |                                                               |
|0  0 0  0 0  |                                                               |
|0  0 0  0 0  |                                                               |
|0  0 0  0 0  |                                                               |
|0  0 0  0 0  |                                                               |
|0  0 0  0 0  |                                                               |
|0  0 0  0 0  |                                                               |
|0  0 0  0 0  |                                                               |
|0  0 0  0 0  |                                                               |
|b2 0 0  0 0  |                                                               |
|0  0 b2 0 0  |                                                               |
|0  0 0  0 b2 |                                                               |
+-----------------------------------------------------------------------------+

Caveat
======

The user should remember to prune the output upon normalizing a simplicial
tensor product.

See also
========

  * extPower -- compute the Dold-Puppe extension of the
    exterior power functor to simplicial modules
  * symmetricQuotient -- computes the image of the
    surjection from the simplicial tensor product onto the second symmetric
    power of a simplicial module

Ways to use simplicialTensor:
=============================

  * "simplicialTensor(Complex,Complex)"
  * "simplicialTensor(ComplexMap,ComplexMap)"
  * "simplicialTensor(List)"
  * "simplicialTensor(ZZ,Complex)"
  * "simplicialTensor(ZZ,SimplicialModule)"
  * simplicialTensor(SimplicialModule,SimplicialModule) (missing documentation)

For the programmer
==================

The object simplicialTensor, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3737:0.

simplicialTensor(...,Degeneracy=>...) -- whether to compute degeneracy maps
***************************************************************************

  * Usage:
        simplicialTensor(..., Degeneracy => true)

Description
===========

If Degeneracy => true, the degeneracy maps are also computed alongside the face
maps. By default, degeneracy maps are not computed.

See also
========

  * simplicialTensor -- compute the simplicial tensor
    product and cache direct sum indices for easy access

Functions with optional argument named Degeneracy:
==================================================

  * extPower(...,Degeneracy=>...)
    -- whether to compute
    degeneracy maps
  * schurMap(...,Degeneracy=>...)
    -- whether to compute
    degeneracy maps
  * simplicialModule(...,Degeneracy=>...)
    -- whether to
    compute degeneracy maps
  * simplicialTensor(...,Degeneracy=>...)
    -- whether to
    compute degeneracy maps

Further information
===================

  * Default value: false (Macaulay2Doc),
  * Function: simplicialTensor, -- compute the
    simplicial tensor product and cache direct sum indices for easy access
  * Option key: Degeneracy, -- the class of all
    simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:4098:0.

simplicialTensor(...,TopDegree=>...) -- top degree for the construction
***********************************************************************

  * Usage:
        simplicialTensor(..., TopDegree => n)

Description
===========

Specifies the top simplicial degree for the resulting simplicial module.

See also
========

  * simplicialTensor -- compute the simplicial tensor
    product and cache direct sum indices for easy access

Functions with optional argument named TopDegree:
=================================================

  * extPower(...,TopDegree=>...)
    -- top degree for the
    construction
  * schurMap(...,TopDegree=>...)
    -- top degree for the
    construction
  * simplicialTensor(...,TopDegree=>...)
    -- top degree for
    the construction

Further information
===================

  * Default value: null
  * Function: simplicialTensor, -- compute the
    simplicial tensor product and cache direct sum indices for easy access
  * Option key: TopDegree, -- construct the Dold-Puppe
    extension of the Schur functor
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:4112:0.

source(SimplicialModuleMap) -- get the source of a map of simplicial modules
****************************************************************************

  * Function: source (Macaulay2Doc),
  * Usage:
        C = source f
  * Inputs:
      * f, a Map of Simplicial Modules,
  * Outputs:
      * C, a Simplicial Module,

Description
===========

Given a simplicial module map $f : C \to D$ this method returns the simplicial
module $C$.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[a..d]                                                        |
|                                                                             |
|o1 = R                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : I = ideal(a^2, b^2, c^2)                                                |
|                                                                             |
|             2   2   2                                                       |
|o2 = ideal (a , b , c )                                                      |
|                                                                             |
|o2 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i3 : J = I + ideal(a*b*c)                                                    |
|                                                                             |
|             2   2   2                                                       |
|o3 = ideal (a , b , c , a*b*c)                                               |
|                                                                             |
|o3 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i4 : FI = simplicialModule(freeResolution I, Degeneracy => true)             |
|                                                                             |
|      1      4      10      20                                               |
|o4 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o4 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i5 : FJ = simplicialModule(freeResolution J, Degeneracy => true)             |
|                                                                             |
|      1      5      15      34                                               |
|o5 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o5 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i6 : f = randomSimplicialMap(FJ, FI, Cycle=>true)                            |
|                                                                             |
|          1              1                                                   |
|o6 = 0 : R  <---------- R  : 0                                               |
|               | 24 |                                                        |
|                                                                             |
|          5                           4                                      |
|     1 : R  <----------------------- R  : 1                                  |
|               {0} | 24 0  0  0  |                                           |
|               {2} | 0  24 0  0  |                                           |
|               {2} | 0  0  24 0  |                                           |
|               {2} | 0  0  0  24 |                                           |
|               {3} | 0  0  0  0  |                                           |
|                                                                             |
|          15                                             10                  |
|     2 : R   <----------------------------------------- R   : 2              |
|                {0} | 24 0  0  0  0  0  0  0  0  0  |                        |
|                {2} | 0  24 0  0  0  0  0  0  0  0  |                        |
|                {2} | 0  0  24 0  0  0  0  0  0  0  |                        |
|                {2} | 0  0  0  24 0  0  0  0  0  0  |                        |
|                {3} | 0  0  0  0  0  0  0  0  0  0  |                        |
|                {2} | 0  0  0  0  24 0  0  0  0  0  |                        |
|                {2} | 0  0  0  0  0  24 0  0  0  0  |                        |
|                {2} | 0  0  0  0  0  0  24 0  0  0  |                        |
|                {3} | 0  0  0  0  0  0  0  0  0  0  |                        |
|                {4} | 0  0  0  0  0  0  0  24 0  0  |                        |
|                {4} | 0  0  0  0  0  0  0  0  0  0  |                        |
|                {4} | 0  0  0  0  0  0  0  0  0  0  |                        |
|                {4} | 0  0  0  0  0  0  0  0  24 0  |                        |
|                {4} | 0  0  0  0  0  0  0  0  0  0  |                        |
|                {4} | 0  0  0  0  0  0  0  0  0  24 |                        |
|                                                                             |
|          34                                                                 |
|     3 : R   <---------------------------------------------------------------|
|                {0} | 24 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  24 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  24 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  0  24 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  0  0  24 0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  0  0  0  24 0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  0  0  0  0  24 0  0  0  0  0  0  0  0  0  0  0  0|
|                {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  0  0  0  0  0  24 0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  0  0  0  0  0  0  24 0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  0  0  0  0  0  0  0  24 0  0  0  0  0  0  0  0  0|
|                {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  24 0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  24 0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  24 0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  24 0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  24 0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  24 0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  24 0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  24 0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  2|
|                {5} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {5} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {5} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                                                                             |
|o6 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|            20                                                               |
|---------- R   : 3                                                           |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|4 0    |                                                                     |
|  24c  |                                                                     |
|  -24b |                                                                     |
|  24a  |                                                                     |
+-----------------------------------------------------------------------------+
|i7 : source f                                                                |
|                                                                             |
|      1      4      10      20                                               |
|o7 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o7 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i8 : assert isWellDefined f                                                  |
+-----------------------------------------------------------------------------+
|i9 : assert isSimplicialMorphism f                                           |
+-----------------------------------------------------------------------------+
|i10 : assert(source f == FI)                                                 |
+-----------------------------------------------------------------------------+
|i11 : assert(target f == FJ)                                                 |
+-----------------------------------------------------------------------------+

The face/degeneracy map in a simplicial module is considered to have type
SimplicialModuleMap.

+-----------------------------------------------------------------+
|i12 : kk = coker vars R                                          |
|                                                                 |
|o12 = cokernel | a b c d |                                       |
|                                                                 |
|                             1                                   |
|o12 : R-module, quotient of R                                    |
+-----------------------------------------------------------------+
|i13 : F = simplicialModule(freeResolution kk, Degeneracy => true)|
|                                                                 |
|       1      5      15      35      70                          |
|o13 = R  <-- R  <-- R   <-- R   <-- R  <-- ...                   |
|                                                                 |
|      0      1      2       3       4                            |
|                                                                 |
|o13 : SimplicialModule                                           |
+-----------------------------------------------------------------+
|i14 : source dd^F == F                                           |
|                                                                 |
|o14 = true                                                       |
+-----------------------------------------------------------------+
|i15 : target dd^F == F                                           |
|                                                                 |
|o15 = true                                                       |
+-----------------------------------------------------------------+
|i16 : degree dd^F == -1                                          |
|                                                                 |
|o16 = true                                                       |
+-----------------------------------------------------------------+

See also
========

  * Making simplicial modules -- information
    about the basic constructors
  * target(SimplicialModuleMap) -- get
    the target of a map of simplicial modules
  * randomSimplicialMap(SimplicialModule,SimplicialModule)
    -- a random
    map of simplicial modules

Ways to use this method:
========================

  * source(SimplicialModuleMap) -- get
    the source of a map of simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1732:0.

symmetricQuotient -- computes the image of the surjection from the simplicial tensor product onto the second symmetric power of a simplicial module
***************************************************************************************************************************************************

  * Usage:
        symmetricQuotient(S)
  * Inputs:
      * S, a Simplicial Module, or Complex (Complexes),
        or Module (Macaulay2Doc), optional
        integer argument specifies top degree
  * Outputs:
      * a Map of Simplicial Modules, The induced
        map representing the image of the surjection from the simplicial tensor
        product onto the second symmetric power of S.

Description
===========

This function computes the induced map on the cokernel of the map constructed
by the exteriorInclusion, method, which is explicitly
giving the map $$S \otimes S \to \operatorname{Sym}^2 (S).$$ Let us see some
examples:

+-----------------------------------------------------------------------------+
|i1 : Q = ZZ/2[a,b];                                                          |
+-----------------------------------------------------------------------------+
|i2 : K = koszulComplex vars Q;                                               |
+-----------------------------------------------------------------------------+
|i3 : phi = prune symmetricQuotient(K,4)                                      |
|                                                                             |
|          1             1                                                    |
|o3 = 0 : Q  <--------- Q  : 0                                                |
|               | 1 |                                                         |
|                                                                             |
|          5                               8                                  |
|     1 : Q  <--------------------------- Q  : 1                              |
|               {1} | 1 0 1 0 0 0 0 0 |                                       |
|               {1} | 0 1 0 1 0 0 0 0 |                                       |
|               {2} | 0 0 0 0 1 0 0 0 |                                       |
|               {2} | 0 0 0 0 0 1 1 0 |                                       |
|               {2} | 0 0 0 0 0 0 0 1 |                                       |
|                                                                             |
|          10                                                     19          |
|     2 : Q   <------------------------------------------------- Q   : 2      |
|                {2} | 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                |
|                {2} | 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 |                |
|                {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                |
|                {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                |
|                {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                |
|                {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 |                |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 |                |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 |                |
|                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 |                |
|                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                |
|                                                                             |
|          9                                                   18             |
|     3 : Q  <----------------------------------------------- Q   : 3         |
|               {3} | 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                   |
|               {3} | 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                   |
|               {3} | 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 |                   |
|               {3} | 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                   |
|               {3} | 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 |                   |
|               {3} | 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 |                   |
|               {4} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 |                   |
|               {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |                   |
|               {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                   |
|                                                                             |
|          3                           6                                      |
|     4 : Q  <----------------------- Q  : 4                                  |
|               {4} | 0 1 0 0 0 1 |                                           |
|               {4} | 1 0 0 0 1 0 |                                           |
|               {4} | 0 0 1 0 0 1 |                                           |
|                                                                             |
|o3 : ComplexMap                                                              |
+-----------------------------------------------------------------------------+
|i4 : isWellDefined phi                                                       |
|                                                                             |
|o4 = true                                                                    |
+-----------------------------------------------------------------------------+
|i5 : isCommutative phi                                                       |
|                                                                             |
|o5 = true                                                                    |
+-----------------------------------------------------------------------------+
|i6 : prune coker phi == 0                                                    |
|                                                                             |
|o6 = true                                                                    |
+-----------------------------------------------------------------------------+
|i7 : prune HH phi                                                            |
|                                                                             |
|o7 = 0 : cokernel | b a | <--------- cokernel | b a | : 0                    |
|                             | 1 |                                           |
|                                                                             |
|o7 : ComplexMap                                                              |
+-----------------------------------------------------------------------------+
|i8 : prune coker oo --the induced map on homology is NOT surjective, in      |
|                                                                             |
|o8 = cokernel {2} | b a |                                                    |
|                                                                             |
|     2                                                                       |
|                                                                             |
|o8 : Complex                                                                 |
|-----------------------------------------------------------------------------|
|contrast to the case when 2 is a unit                                        |
+-----------------------------------------------------------------------------+

See also
========

  * tensorLES -- computes the long exact sequence of homology
    induced by the canonical short exact sequence of complexes
  * exteriorInclusion -- computes the image of the
    2nd exterior power into the tensor product
  * simplicialTensor -- compute the simplicial tensor
    product and cache direct sum indices for easy access

Ways to use symmetricQuotient:
==============================

  * "symmetricQuotient(Complex)"
  * "symmetricQuotient(Complex,ZZ)"
  * "symmetricQuotient(Module)"
  * "symmetricQuotient(SimplicialModule)"

For the programmer
==================

The object symmetricQuotient, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3947:0.

target(SimplicialModuleMap) -- get the target of a map of simplicial modules
****************************************************************************

  * Function: target (Macaulay2Doc),
  * Usage:
        C = target f
  * Inputs:
      * f, a Map of Simplicial Modules,
  * Outputs:
      * C, a Simplicial Module,

Description
===========

Given a simplicial module map $f : C \to D$ this method returns the simplicial
module $D$.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[a..d]                                                        |
|                                                                             |
|o1 = R                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : I = ideal(a^2, b^2, c^2)                                                |
|                                                                             |
|             2   2   2                                                       |
|o2 = ideal (a , b , c )                                                      |
|                                                                             |
|o2 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i3 : J = I + ideal(a*b*c)                                                    |
|                                                                             |
|             2   2   2                                                       |
|o3 = ideal (a , b , c , a*b*c)                                               |
|                                                                             |
|o3 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i4 : FI = simplicialModule(freeResolution I, Degeneracy => true)             |
|                                                                             |
|      1      4      10      20                                               |
|o4 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o4 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i5 : FJ = simplicialModule(freeResolution J, Degeneracy => true)             |
|                                                                             |
|      1      5      15      34                                               |
|o5 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o5 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i6 : f = randomSimplicialMap(FJ, FI, Cycle=>true)                            |
|                                                                             |
|          1              1                                                   |
|o6 = 0 : R  <---------- R  : 0                                               |
|               | 24 |                                                        |
|                                                                             |
|          5                           4                                      |
|     1 : R  <----------------------- R  : 1                                  |
|               {0} | 24 0  0  0  |                                           |
|               {2} | 0  24 0  0  |                                           |
|               {2} | 0  0  24 0  |                                           |
|               {2} | 0  0  0  24 |                                           |
|               {3} | 0  0  0  0  |                                           |
|                                                                             |
|          15                                             10                  |
|     2 : R   <----------------------------------------- R   : 2              |
|                {0} | 24 0  0  0  0  0  0  0  0  0  |                        |
|                {2} | 0  24 0  0  0  0  0  0  0  0  |                        |
|                {2} | 0  0  24 0  0  0  0  0  0  0  |                        |
|                {2} | 0  0  0  24 0  0  0  0  0  0  |                        |
|                {3} | 0  0  0  0  0  0  0  0  0  0  |                        |
|                {2} | 0  0  0  0  24 0  0  0  0  0  |                        |
|                {2} | 0  0  0  0  0  24 0  0  0  0  |                        |
|                {2} | 0  0  0  0  0  0  24 0  0  0  |                        |
|                {3} | 0  0  0  0  0  0  0  0  0  0  |                        |
|                {4} | 0  0  0  0  0  0  0  24 0  0  |                        |
|                {4} | 0  0  0  0  0  0  0  0  0  0  |                        |
|                {4} | 0  0  0  0  0  0  0  0  0  0  |                        |
|                {4} | 0  0  0  0  0  0  0  0  24 0  |                        |
|                {4} | 0  0  0  0  0  0  0  0  0  0  |                        |
|                {4} | 0  0  0  0  0  0  0  0  0  24 |                        |
|                                                                             |
|          34                                                                 |
|     3 : R   <---------------------------------------------------------------|
|                {0} | 24 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  24 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  24 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  0  24 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  0  0  24 0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  0  0  0  24 0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  0  0  0  0  24 0  0  0  0  0  0  0  0  0  0  0  0|
|                {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  0  0  0  0  0  24 0  0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  0  0  0  0  0  0  24 0  0  0  0  0  0  0  0  0  0|
|                {2} | 0  0  0  0  0  0  0  0  0  24 0  0  0  0  0  0  0  0  0|
|                {3} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  24 0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  24 0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  24 0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  24 0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  24 0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  24 0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  24 0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  24 0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {4} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  2|
|                {5} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {5} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                {5} | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
|                                                                             |
|o6 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|            20                                                               |
|---------- R   : 3                                                           |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|  0    |                                                                     |
|4 0    |                                                                     |
|  24c  |                                                                     |
|  -24b |                                                                     |
|  24a  |                                                                     |
+-----------------------------------------------------------------------------+
|i7 : source f                                                                |
|                                                                             |
|      1      4      10      20                                               |
|o7 = R  <-- R  <-- R   <-- R  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o7 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i8 : assert isWellDefined f                                                  |
+-----------------------------------------------------------------------------+
|i9 : assert isSimplicialMorphism f                                           |
+-----------------------------------------------------------------------------+
|i10 : assert(source f == FI)                                                 |
+-----------------------------------------------------------------------------+
|i11 : assert(target f == FJ)                                                 |
+-----------------------------------------------------------------------------+

The face/degeneracy map in a simplicial module is considered to have type
SimplicialModuleMap.

+-----------------------------------------------------------------+
|i12 : kk = coker vars R                                          |
|                                                                 |
|o12 = cokernel | a b c d |                                       |
|                                                                 |
|                             1                                   |
|o12 : R-module, quotient of R                                    |
+-----------------------------------------------------------------+
|i13 : F = simplicialModule(freeResolution kk, Degeneracy => true)|
|                                                                 |
|       1      5      15      35      70                          |
|o13 = R  <-- R  <-- R   <-- R   <-- R  <-- ...                   |
|                                                                 |
|      0      1      2       3       4                            |
|                                                                 |
|o13 : SimplicialModule                                           |
+-----------------------------------------------------------------+
|i14 : source dd^F == F                                           |
|                                                                 |
|o14 = true                                                       |
+-----------------------------------------------------------------+
|i15 : target dd^F == F                                           |
|                                                                 |
|o15 = true                                                       |
+-----------------------------------------------------------------+
|i16 : degree dd^F == -1                                          |
|                                                                 |
|o16 = true                                                       |
+-----------------------------------------------------------------+

See also
========

  * Making simplicial modules -- information
    about the basic constructors
  * source(SimplicialModuleMap) -- get
    the source of a map of simplicial modules
  * freeResolution(Ideal) (Complexes) -- compute a free
    resolution of a module or ideal
  * randomSimplicialMap(SimplicialModule,SimplicialModule)
    -- a random
    map of simplicial modules

Ways to use this method:
========================

  * target(SimplicialModuleMap) -- get
    the target of a map of simplicial modules
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1774:0.

tensorLES -- computes the long exact sequence of homology induced by the canonical short exact sequence of complexes
********************************************************************************************************************

  * Usage:
        tensorLES(C, d)
  * Inputs:
      * C, a complex (Complexes), A complex.
      * d, an integer (Macaulay2Doc), An integer specifying the top
        degree for constructing the simplicial module.
  * Outputs:
      * a complex (Complexes), The long exact sequence of
        homology induced by the canonical short exact sequence \( 0 \to
        \wedge^2 C \to T^2 C \to Sym^2 C \to 0 \).

Description
===========

This function computes the long exact sequence of homology associated with the
canonical short exact sequence of complexes: \( 0 \to \bigwedge^2 C \to C
\otimes C \to \operatorname{Sym}^2 C \to 0 \).

This function first computes the exterior inclusion map on the complex C up to
degree d using the exteriorInclusion, function. Then
it constructs the induced map on the cokernel of this exterior inclusion map,
which is the surjection to the second symmetric power of C. Finally, it
computes the long exact sequence of homology using the longExactSequence (Complexes),
function on the
pruned maps of the induced maps.

This long exact sequence is only interesting in characteristic 2, since the
above sequence is split whenever 2 is a unit, so there are no interesting
connecting homomorphisms. Let's see some examples in characteristic 2 of the
connecting homomorphisms:

+-----------------------------------------------------------------------------+
|i1 : Q = ZZ/2[a,b]                                                           |
|                                                                             |
|o1 = Q                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : K = koszulComplex vars Q                                                |
|                                                                             |
|      1      2      1                                                        |
|o2 = Q  <-- Q  <-- Q                                                         |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o2 : Complex                                                                 |
+-----------------------------------------------------------------------------+
|i3 : prune tensorLES(K,4)                                                    |
|                                                                             |
|o3 = cokernel | b a | <-- cokernel | b a | <-- 0 <-- 0 <-- cokernel {1} | b a|
|                                                                    {1} | 0 0|
|     0                    1                    2     3                       |
|                                                           4                 |
|                                                                             |
|o3 : Complex                                                                 |
|-----------------------------------------------------------------------------|
|0 0 | <-- cokernel {1} | a 0 b | <-- cokernel {2} | b a | <-- cokernel {2} | |
|b a |              {1} | 0 b a |                                             |
|                                     6                        7              |
|          5                                                                  |
|-----------------------------------------------------------------------------|
|b a | <-- cokernel {2} | b a |                                               |
|                                                                             |
|          8                                                                  |
+-----------------------------------------------------------------------------+
|i4 : oo.dd_6 --nontrivial connecting homomorphism                            |
|                                                                             |
|o4 = {1} | b |                                                               |
|     {1} | 0 |                                                               |
|                                                                             |
|o4 : Matrix                                                                  |
+-----------------------------------------------------------------------------+
|i5 : F = freeResolution( (ideal vars Q)^3)                                   |
|                                                                             |
|      1      4      3                                                        |
|o5 = Q  <-- Q  <-- Q                                                         |
|                                                                             |
|     0      1      2                                                         |
|                                                                             |
|o5 : Complex                                                                 |
+-----------------------------------------------------------------------------+
|i6 : prune tensorLES(F,4)                                                    |
|                                                                             |
|o6 = cokernel | b3 ab2 a2b a3 | <-- cokernel | b3 ab2 a2b a3 | <-- 0 <-- 0   |
|                                                                             |
|     0                              1                              2     3   |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o6 : Complex                                                                 |
|-----------------------------------------------------------------------------|
|<-- cokernel {3} | b 0 0 b3 ab2 a2b a3 0  0  0  | <-- cokernel {3} | b 0 0   |
|             {3} | a b 0 0  0   0   0  b3 0  0  |              {3} | a b 0   |
|             {3} | 0 a b 0  0   0   0  0  b3 0  |              {3} | 0 a b   |
|             {3} | 0 0 a 0  0   0   0  0  0  b3 |              {3} | 0 0 a   |
|                                                                             |
|    4                                                 5                      |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|ab2 a3 0  0  | <-- cokernel {6} | b a 0 0 0 0 0 0 0 0 0 0 | <-- cokernel {6} |
|0   0  b3 0  |              {6} | 0 0 b a 0 0 0 0 0 0 0 0 |              {6} |
|0   0  0  0  |              {6} | 0 0 0 0 b a 0 0 0 0 0 0 |              {6} |
|0   0  0  b3 |              {6} | 0 0 0 0 0 0 b a 0 0 0 0 |              {6} |
|                            {6} | 0 0 0 0 0 0 0 0 b a 0 0 |              {6} |
|                            {6} | 0 0 0 0 0 0 0 0 0 0 b a |              {6} |
|                                                                         {6} |
|                   6                                                     {6} |
|                                                                         {6} |
|                                                                             |
|                                                                7            |
|-----------------------------------------------------------------------------|
|| b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- cokernel {6} | b a 0 0 0 0 0 0 0 |
|| 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |              {6} | 0 0 b a 0 0 0 0 0 |
|| 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 |              {6} | 0 0 0 0 b a 0 0 0 |
|| 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 |              {6} | 0 0 0 0 0 0 b a 0 |
|| 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 |              {6} | 0 0 0 0 0 0 0 0 b |
|| 0 0 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 |              {6} | 0 0 0 0 0 0 0 0 0 |
|| 0 0 0 0 0 0 0 0 0 0 0 0 b a 0 0 0 0 |                                      |
|| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a 0 0 |     8                                |
|| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a |                                      |
|-----------------------------------------------------------------------------|
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|a 0 0 |                                                                      |
|0 b a |                                                                      |
+-----------------------------------------------------------------------------+
|i7 : L = complex {K.dd_1, map(source K.dd_1,target K.dd_2 ,K.dd_2*K.dd_1),   |
|                                                                             |
|      1      2      2      1                                                 |
|o7 = Q  <-- Q  <-- Q  <-- Q                                                  |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o7 : Complex                                                                 |
|-----------------------------------------------------------------------------|
|K.dd_2}                                                                      |
+-----------------------------------------------------------------------------+
|i8 : hL = elapsedTime prune tensorLES(L,4)                                   |
| -- 2.23785s elapsed                                                         |
|                                                                             |
|                                                                             |
|o8 = cokernel | b a | <-- cokernel | b a | <-- 0 <-- 0 <-- cokernel | a 0 b ||
|                                                                    | 0 b a ||
|     0                    1                    2     3                       |
|                                                           4                 |
|                                                                             |
|                                                                             |
|                                                                             |
|o8 : Complex                                                                 |
|-----------------------------------------------------------------------------|
|                                                                             |
|<-- cokernel | a 0 b | <-- cokernel | b a 0 0 | <-- cokernel | b a 0 0 0 0 0 |
|             | 0 b a |              | 0 0 b a |              | 0 0 b a 0 0 0 |
|                                                             | 0 0 0 0 b a 0 |
|    5                      6                                 | 0 0 0 0 0 0 b |
|                                                                             |
|                                                    7                        |
|-----------------------------------------------------------------------------|
|                                                                             |
|0 | <-- cokernel | 0   0   b a 0 0 0  0  | <-- cokernel | ab2 a2b b2 a2 | <--|
|0 |              | 0   0   0 0 b a 0  0  |                                   |
|0 |              | ab2 a2b 0 0 0 0 b2 a2 |     9                             |
|a |                                                                          |
|        8                                                                    |
|-----------------------------------------------------------------------------|
|                                                          25      50      25 |
|cokernel | a2 b2 ab | <-- cokernel | ab2 a2b b2 a2 | <-- Q   <-- Q   <-- Q   |
|                                                                             |
|10                        11                             12      13      14  |
+-----------------------------------------------------------------------------+
|i9 : netList {hL.dd_3, hL.dd_6, hL.dd_9, hL.dd_12, hL.dd_15} --two nontrivial|
|                                                                             |
|     +------------------------------------------------------+                |
|o9 = |0                                                     |                |
|     +------------------------------------------------------+                |
|     |0                                                     |                |
|     +------------------------------------------------------+                |
|     || 0 |                                                 |                |
|     || 0 |                                                 |                |
|     || 1 |                                                 |                |
|     +------------------------------------------------------+                |
|     || 0 0 0 0 0 0 0 0 ab 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||                |
|     +------------------------------------------------------+                |
|     |0                                                     |                |
|     +------------------------------------------------------+                |
|-----------------------------------------------------------------------------|
|connecting homs                                                              |
+-----------------------------------------------------------------------------+

See also
========

  * exteriorInclusion -- computes the image of the
    2nd exterior power into the tensor product
  * simplicialTensor -- compute the simplicial tensor
    product and cache direct sum indices for easy access

Ways to use tensorLES:
======================

  * "tensorLES(Complex,ZZ)"

For the programmer
==================

The object tensorLES, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3993:0.

tensorwithComponents -- compute the tensor product of direct summands, caching the components for easy access
*************************************************************************************************************

  * Usage:
        tensorwithComponents(M, N)
  * Inputs:
      * M, a module (Macaulay2Doc),
      * N, a module (Macaulay2Doc), Two modules whose tensor
        product of direct summands is to be computed.
  * Outputs:
      * a module (Macaulay2Doc), The tensor product of M and N,
        with cached components for easier access

Description
===========

This function computes the tensor product of two modules M and N, but
additionally caches the components based on the component indices of the
original modules. This caching is particularly useful for easily accessing
induced maps on components of tensor products of direct sums.

If M and N have cached index components, then this function will use those
indices. This function is mostly used for extracting components of the
simplicial tensor product, since the full face/degeneracy maps are typically
much too large to be displayed on their own.

+---------------------------------------------------------------------+
|i1 : Q = ZZ/101[a..b]                                                |
|                                                                     |
|o1 = Q                                                               |
|                                                                     |
|o1 : PolynomialRing                                                  |
+---------------------------------------------------------------------+
|i2 : M = Q^2++Q^3                                                    |
|                                                                     |
|      5                                                              |
|o2 = Q                                                               |
|                                                                     |
|o2 : Q-module, free                                                  |
+---------------------------------------------------------------------+
|i3 : N = Q^3++Q^4                                                    |
|                                                                     |
|      7                                                              |
|o3 = Q                                                               |
|                                                                     |
|o3 : Q-module, free                                                  |
+---------------------------------------------------------------------+
|i4 : components(M**N) --all components of M and N have been forgotten|
|                                                                     |
|       35                                                            |
|o4 = {Q  }                                                           |
|                                                                     |
|o4 : List                                                            |
+---------------------------------------------------------------------+
|i5 : T = tensorwithComponents(M,N)                                   |
|                                                                     |
|      35                                                             |
|o5 = Q                                                               |
|                                                                     |
|o5 : Q-module, free                                                  |
+---------------------------------------------------------------------+
|i6 : components T                                                    |
|                                                                     |
|       6   8   9   12                                                |
|o6 = {Q , Q , Q , Q  }                                               |
|                                                                     |
|o6 : List                                                            |
+---------------------------------------------------------------------+
|i7 : indices T                                                       |
|                                                                     |
|o7 = {{0, 0}, {0, 1}, {1, 0}, {1, 1}}                                |
|                                                                     |
|o7 : List                                                            |
+---------------------------------------------------------------------+

This method also works for lists of modules, for purposes of iterating:

+-----------------------------------------------------------------------------+
|i8 : tensorwithComponents {M,M,M}                                            |
|                                                                             |
|      125                                                                    |
|o8 = Q                                                                       |
|                                                                             |
|o8 : Q-module, free                                                          |
+-----------------------------------------------------------------------------+
|i9 : indices oo                                                              |
|                                                                             |
|o9 = {{0, 0, 0}, {0, 0, 1}, {0, 1, 0}, {0, 1, 1}, {1, 0, 0}, {1, 0, 1}, {1,  |
|     ------------------------------------------------------------------------|
|     1, 0}, {1, 1, 1}}                                                       |
|                                                                             |
|o9 : List                                                                    |
+-----------------------------------------------------------------------------+

This method is also functorial, which makes it easy to restrict maps to tensor
summands:

+-----------------------------------------------------------------------------+
|i10 : L = directSum {Q^2, Q^3, Q^4}                                          |
|                                                                             |
|       9                                                                     |
|o10 = Q                                                                      |
|                                                                             |
|o10 : Q-module, free                                                         |
+-----------------------------------------------------------------------------+
|i11 : f = L_[0,2]                                                            |
|                                                                             |
|o11 = | 1 0 0 0 0 0 |                                                        |
|      | 0 1 0 0 0 0 |                                                        |
|      | 0 0 0 0 0 0 |                                                        |
|      | 0 0 0 0 0 0 |                                                        |
|      | 0 0 0 0 0 0 |                                                        |
|      | 0 0 1 0 0 0 |                                                        |
|      | 0 0 0 1 0 0 |                                                        |
|      | 0 0 0 0 1 0 |                                                        |
|      | 0 0 0 0 0 1 |                                                        |
|                                                                             |
|              9      6                                                       |
|o11 : Matrix Q  <-- Q                                                        |
+-----------------------------------------------------------------------------+
|i12 : phi = tensorwithComponents(f, f)                                       |
|                                                                             |
|o12 = | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|      -----------------------------------------------------------------------|
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|                                                                             |
|              81      36                                                     |
|o12 : Matrix Q   <-- Q                                                       |
+-----------------------------------------------------------------------------+
|i13 : phi^[{0,2}]_[{0,1}]                                                    |
|                                                                             |
|o13 = | 0 0 0 1 0 0 0 0 |                                                    |
|      | 0 0 0 0 0 0 0 0 |                                                    |
|      | 0 0 0 0 0 0 0 0 |                                                    |
|      | 0 0 0 0 0 0 0 0 |                                                    |
|      | 0 0 0 0 1 0 0 0 |                                                    |
|      | 0 0 0 0 0 1 0 0 |                                                    |
|      | 0 0 0 0 0 0 1 0 |                                                    |
|      | 0 0 0 0 0 0 0 1 |                                                    |
|                                                                             |
|              8      8                                                       |
|o13 : Matrix Q  <-- Q                                                        |
+-----------------------------------------------------------------------------+

Ways to use tensorwithComponents:
=================================

  * "tensorwithComponents(List)"
  * "tensorwithComponents(Matrix,Matrix)"
  * "tensorwithComponents(Module,Module)"

For the programmer
==================

The object tensorwithComponents, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:3397:0.

The Dold-Kan Correspondence in Macaulay2 -- compute the image of a non-negatively graded complex under the Dold-Kan correspondence
**********************************************************************************************************************************

  * Usage:
        simplicialModule(C,d)
        simplicialModule(C)
  * Inputs:
      * C, a complex (Complexes),
      * d, an integer (Macaulay2Doc), an integer specifying the top
        degree of the output SimplicialModule (if d is not specified, the top
        degree is the length of the complex C)

Given any non-negatively graded chain complex, the Dold-Kan correspondence is
an equivalence of categories: $$\text{Ch}_{\geq 0} (R) \leftrightarrow
\text{Simplicial R-modules}.$$ If the simplicial module is obtained as a
Dold-Kan image of some complex $C$, then for each $i$ there is a decomposition
$$S_i = C_0 \oplus C_1^{\binom{i}{1}} \oplus \cdots \oplus C_j^{\binom{i}{j}}
\oplus \cdots.$$ To be technically correct, each of the direct summands $C_j$
of $S_i$ should be thought of as being parametrized by an order preserving
surjection $f : [i] \to [j]$ (where $[n] := \{ 0 ,\dots  , n \}$). To deduce
which surjection corresponds to a given summand, first notice that order
preserving surjections $f : [i] \to [j]$ are in bijection with compositions of
$i+1$ into $j+1$ parts by just listing the sizes of the fibers of the map $f$.
For instance, the composition $(2,2,1)$ corresponds to the surjection $f : [4]
\to [2]$ defined via $$f(0) = f(1) = 0, \quad f(2) = f(3) = 1, \quad f(4) =
2.$$ With this in mind, one can deduce the surjection corresponding to a
summand as follows:

+----------------------------------------------------------------+
|i1 : R = ZZ/101[a..c]                                           |
|                                                                |
|o1 = R                                                          |
|                                                                |
|o1 : PolynomialRing                                             |
+----------------------------------------------------------------+
|i2 : C = koszulComplex vars R                                   |
|                                                                |
|      1      3      3      1                                    |
|o2 = R  <-- R  <-- R  <-- R                                     |
|                                                                |
|     0      1      2      3                                     |
|                                                                |
|o2 : Complex                                                    |
+----------------------------------------------------------------+
|i3 : S = simplicialModule(C, 4, Degeneracy => true)             |
|                                                                |
|      1      4      10      20      35                          |
|o3 = R  <-- R  <-- R   <-- R   <-- R  <-- ...                   |
|                                                                |
|     0      1      2       3       4                            |
|                                                                |
|o3 : SimplicialModule                                           |
+----------------------------------------------------------------+
|i4 : S_2                                                        |
|                                                                |
|      10                                                        |
|o4 = R                                                          |
|                                                                |
|o4 : R-module, free, degrees {0..1, 5:1, 3:2}                   |
+----------------------------------------------------------------+
|i5 : components S_2 --these are all the modules showing up      |
|                                                                |
|       1   3   3   3                                            |
|o5 = {R , R , R , R }                                           |
|                                                                |
|o5 : List                                                       |
+----------------------------------------------------------------+
|i6 : S_(2,1)                                                    |
|                                                                |
|      6                                                         |
|o6 = R                                                          |
|                                                                |
|o6 : R-module, free, degrees {6:1}                              |
+----------------------------------------------------------------+
|i7 : components S_(2,1) --these are only the components of C_1^2|
|                                                                |
|       3   3                                                    |
|o7 = {R , R }                                                   |
|                                                                |
|o7 : List                                                       |
+----------------------------------------------------------------+
|i8 : sort(select(compositions(2,3), i -> all(i, j -> j>0)))     |
|                                                                |
|o8 = {{1, 2}, {2, 1}}                                           |
|                                                                |
|o8 : List                                                       |
+----------------------------------------------------------------+

The ordering of the surjections corresponding to a summand agrees with the
lexicographic ordering of compositions via the bijection mentioned above. Thus
the first summand of $C_1$ appearing in $S_2$ corresponds to the surjection $f
: [2] \to [1]$ given by $f(0) = 0$ and $f(1) = f(2) = 1$, and the second
summand corresponds to the surjection $f: [2] \to [1]$ with $f(0) = f(1) = 0$
and $f(2) = 1$. These surjections may be accessed more directly using the
summandSurjection command:

+----------------------------------------------------------------+
|i9 : summandSurjection(2,0) --the surjection parametrizing C_0  |
|                                                                |
|o9 = {HashTable{0 => 0}}                                        |
|                1 => 0                                          |
|                2 => 0                                          |
|                                                                |
|o9 : List                                                       |
+----------------------------------------------------------------+
|i10 : summandSurjection(2,1) --the surjections parametrizing C_1|
|                                                                |
|o10 = {HashTable{0 => 0}, HashTable{0 => 0}}                    |
|                 1 => 1             1 => 0                      |
|                 2 => 1             2 => 1                      |
|                                                                |
|o10 : List                                                      |
+----------------------------------------------------------------+
|i11 : summandSurjection(2,2) --the surjection parametrizing C_2 |
|                                                                |
|o11 = {HashTable{0 => 0}}                                       |
|                 1 => 1                                         |
|                 2 => 2                                         |
|                                                                |
|o11 : List                                                      |
+----------------------------------------------------------------+

Notice that the individual terms $C_j^{\binom{i}{j}}$ may be accessed as the
$(i,j)$-component of the simplicial module. There are moreover explicit
formulas for the face and degeneracy maps in terms of the differentials of $C$.
These can also be accessed:

+-----------------------------------------------------------------------------+
|i12 : S.dd --face maps                                                       |
|                                                                             |
|                1                   4                                        |
|o12 = (0, 0) : R  <--------------- R  : (1, 0)                               |
|                     | 1 a b c |                                             |
|                                                                             |
|                1                   4                                        |
|      (0, 1) : R  <--------------- R  : (1, 1)                               |
|                     | 1 0 0 0 |                                             |
|                                                                             |
|                4                                      10                    |
|      (1, 0) : R  <---------------------------------- R   : (2, 0)           |
|                     {0} | 1 a b c 0 0 0 0  0  0  |                          |
|                     {1} | 0 0 0 0 1 0 0 -b -c 0  |                          |
|                     {1} | 0 0 0 0 0 1 0 a  0  -c |                          |
|                     {1} | 0 0 0 0 0 0 1 0  a  b  |                          |
|                                                                             |
|                4                                   10                       |
|      (1, 1) : R  <------------------------------- R   : (2, 1)              |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 |                             |
|                     {1} | 0 1 0 0 1 0 0 0 0 0 |                             |
|                     {1} | 0 0 1 0 0 1 0 0 0 0 |                             |
|                     {1} | 0 0 0 1 0 0 1 0 0 0 |                             |
|                                                                             |
|                4                                   10                       |
|      (1, 2) : R  <------------------------------- R   : (2, 2)              |
|                     {0} | 1 0 0 0 0 0 0 0 0 0 |                             |
|                     {1} | 0 1 0 0 0 0 0 0 0 0 |                             |
|                     {1} | 0 0 1 0 0 0 0 0 0 0 |                             |
|                     {1} | 0 0 0 1 0 0 0 0 0 0 |                             |
|                                                                             |
|                10                                                           |
|      (2, 0) : R   <---------------------------------------------------------|
|                      {0} | 1 a b c 0 0 0 0 0 0 0  0  0  0  0  0  0 0 0 0  | |
|                      {1} | 0 0 0 0 1 0 0 0 0 0 -b -c 0  0  0  0  0 0 0 0  | |
|                      {1} | 0 0 0 0 0 1 0 0 0 0 a  0  -c 0  0  0  0 0 0 0  | |
|                      {1} | 0 0 0 0 0 0 1 0 0 0 0  a  b  0  0  0  0 0 0 0  | |
|                      {1} | 0 0 0 0 0 0 0 1 0 0 0  0  0  -b -c 0  0 0 0 0  | |
|                      {1} | 0 0 0 0 0 0 0 0 1 0 0  0  0  a  0  -c 0 0 0 0  | |
|                      {1} | 0 0 0 0 0 0 0 0 0 1 0  0  0  0  a  b  0 0 0 0  | |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  1 0 0 c  | |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 1 0 -b | |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 0 1 a  | |
|                                                                             |
|                10                                                       20  |
|      (2, 1) : R   <--------------------------------------------------- R   :|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 |        |
|                                                                             |
|                10                                                       20  |
|      (2, 2) : R   <--------------------------------------------------- R   :|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 |        |
|                                                                             |
|                10                                                       20  |
|      (2, 3) : R   <--------------------------------------------------- R   :|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |        |
|                                                                             |
|                20                                                           |
|      (3, 0) : R   <---------------------------------------------------------|
|                      {0} | 1 a b c 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0 |
|                      {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 -b -c 0  0  0  0  0  0 |
|                      {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 a  0  -c 0  0  0  0  0 |
|                      {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0  a  b  0  0  0  0  0 |
|                      {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0  0  0  -b -c 0  0  0 |
|                      {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0  0  0  a  0  -c 0  0 |
|                      {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0  0  0  0  a  b  0  0 |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0  0  0  0  0  0  -b -c|
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0  0  0  0  0  0  a  0 |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0  0  0  0  0  0  0  a |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0 |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0 |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0 |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0 |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0 |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0 |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0 |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0 |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0 |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0 |
|                                                                             |
|                20                                                           |
|      (3, 1) : R   <---------------------------------------------------------|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|                20                                                           |
|      (3, 2) : R   <---------------------------------------------------------|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|                20                                                           |
|      (3, 3) : R   <---------------------------------------------------------|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|                20                                                           |
|      (3, 4) : R   <---------------------------------------------------------|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|                                                                             |
|o12 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|   20                                                                        |
|- R   : (3, 0)                                                               |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
| (3, 1)                                                                      |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
| (3, 2)                                                                      |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
| (3, 3)                                                                      |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                      35                                     |
|------------------------------------ R   : (4, 0)                            |
| 0  0 0 0 0 0 0 0 0 0 0  0  0  0 |                                           |
| 0  0 0 0 0 0 0 0 0 0 0  0  0  0 |                                           |
| 0  0 0 0 0 0 0 0 0 0 0  0  0  0 |                                           |
| 0  0 0 0 0 0 0 0 0 0 0  0  0  0 |                                           |
| 0  0 0 0 0 0 0 0 0 0 0  0  0  0 |                                           |
| 0  0 0 0 0 0 0 0 0 0 0  0  0  0 |                                           |
| 0  0 0 0 0 0 0 0 0 0 0  0  0  0 |                                           |
| 0  0 0 0 0 0 0 0 0 0 0  0  0  0 |                                           |
| -c 0 0 0 0 0 0 0 0 0 0  0  0  0 |                                           |
| b  0 0 0 0 0 0 0 0 0 0  0  0  0 |                                           |
| 0  1 0 0 0 0 0 0 0 0 c  0  0  0 |                                           |
| 0  0 1 0 0 0 0 0 0 0 -b 0  0  0 |                                           |
| 0  0 0 1 0 0 0 0 0 0 a  0  0  0 |                                           |
| 0  0 0 0 1 0 0 0 0 0 0  c  0  0 |                                           |
| 0  0 0 0 0 1 0 0 0 0 0  -b 0  0 |                                           |
| 0  0 0 0 0 0 1 0 0 0 0  a  0  0 |                                           |
| 0  0 0 0 0 0 0 1 0 0 0  0  c  0 |                                           |
| 0  0 0 0 0 0 0 0 1 0 0  0  -b 0 |                                           |
| 0  0 0 0 0 0 0 0 0 1 0  0  a  0 |                                           |
| 0  0 0 0 0 0 0 0 0 0 0  0  0  1 |                                           |
|                                                                             |
|                          35                                                 |
|------------------------ R   : (4, 1)                                        |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 1 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 1 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 1 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 1 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 1 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 1 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 1 1 |                                                       |
|                                                                             |
|                          35                                                 |
|------------------------ R   : (4, 2)                                        |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 1 0 0 1 0 0 0 0 0 0 |                                                       |
| 0 1 0 0 1 0 0 0 0 0 |                                                       |
| 0 0 1 0 0 1 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 1 1 0 |                                                       |
|                                                                             |
|                          35                                                 |
|------------------------ R   : (4, 3)                                        |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 1 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 1 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 1 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 1 1 0 0 |                                                       |
|                                                                             |
|                          35                                                 |
|------------------------ R   : (4, 4)                                        |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 0 0 0 0 |                                                       |
| 0 0 0 0 0 0 1 0 0 0 |                                                       |
+-----------------------------------------------------------------------------+
|i13 : S.ss --degeneracy maps                                                 |
|                                                                             |
|                4                 1                                          |
|o13 = (1, 0) : R  <------------- R  : (0, 0)                                 |
|                     {0} | 1 |                                               |
|                     {1} | 0 |                                               |
|                     {1} | 0 |                                               |
|                     {1} | 0 |                                               |
|                                                                             |
|                10                       4                                   |
|      (2, 0) : R   <------------------- R  : (1, 0)                          |
|                      {0} | 1 0 0 0 |                                        |
|                      {1} | 0 0 0 0 |                                        |
|                      {1} | 0 0 0 0 |                                        |
|                      {1} | 0 0 0 0 |                                        |
|                      {1} | 0 1 0 0 |                                        |
|                      {1} | 0 0 1 0 |                                        |
|                      {1} | 0 0 0 1 |                                        |
|                      {2} | 0 0 0 0 |                                        |
|                      {2} | 0 0 0 0 |                                        |
|                      {2} | 0 0 0 0 |                                        |
|                                                                             |
|                10                       4                                   |
|      (2, 1) : R   <------------------- R  : (1, 1)                          |
|                      {0} | 1 0 0 0 |                                        |
|                      {1} | 0 1 0 0 |                                        |
|                      {1} | 0 0 1 0 |                                        |
|                      {1} | 0 0 0 1 |                                        |
|                      {1} | 0 0 0 0 |                                        |
|                      {1} | 0 0 0 0 |                                        |
|                      {1} | 0 0 0 0 |                                        |
|                      {2} | 0 0 0 0 |                                        |
|                      {2} | 0 0 0 0 |                                        |
|                      {2} | 0 0 0 0 |                                        |
|                                                                             |
|                20                                   10                      |
|      (3, 0) : R   <------------------------------- R   : (2, 0)             |
|                      {0} | 1 0 0 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 1 0 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 1 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 1 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 1 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 1 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 0 1 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 1 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 1 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 1 |                            |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                                                                             |
|                20                                   10                      |
|      (3, 1) : R   <------------------------------- R   : (2, 1)             |
|                      {0} | 1 0 0 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 1 0 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 1 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 1 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 1 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 1 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 0 1 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 1 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 1 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 1 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                                                                             |
|                20                                   10                      |
|      (3, 2) : R   <------------------------------- R   : (2, 2)             |
|                      {0} | 1 0 0 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 1 0 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 1 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 1 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 1 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 1 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 0 1 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 1 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 1 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 1 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 |                            |
|                                                                             |
|                35                                                       20  |
|      (4, 0) : R   <--------------------------------------------------- R   :|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |        |
|                                                                             |
|                35                                                       20  |
|      (4, 1) : R   <--------------------------------------------------- R   :|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                                                                             |
|                35                                                       20  |
|      (4, 2) : R   <--------------------------------------------------- R   :|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                                                                             |
|                35                                                       20  |
|      (4, 3) : R   <--------------------------------------------------- R   :|
|                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |        |
|                                                                             |
|o13 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
|(3, 0)                                                                       |
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|(3, 1)                                                                       |
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|(3, 2)                                                                       |
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|(3, 3)                                                                       |
+-----------------------------------------------------------------------------+
|i14 : S.dd_(2,0)                                                             |
|                                                                             |
|o14 = {0} | 1 a b c 0 0 0 0  0  0  |                                         |
|      {1} | 0 0 0 0 1 0 0 -b -c 0  |                                         |
|      {1} | 0 0 0 0 0 1 0 a  0  -c |                                         |
|      {1} | 0 0 0 0 0 0 1 0  a  b  |                                         |
|                                                                             |
|              4      10                                                      |
|o14 : Matrix R  <-- R                                                        |
+-----------------------------------------------------------------------------+

If you want to restrict/project a face/degeneracy map to a particular summand
group, this can be done using the array notation. The $k$-th summand group of
$S_i$ corresponds to $C_k^{\binom{i}{k}}$:

+-----------------------------------------------------------------------------+
|i15 : (S.dd_(2,0))_[1]^[0]  --restrict to C_1 summand group of S_2, project  |
|                                                                             |
|o15 = | a b c 0 0 0 |                                                        |
|                                                                             |
|              1      6                                                       |
|o15 : Matrix R  <-- R                                                        |
|-----------------------------------------------------------------------------|
|onto C_0 summand of S_1                                                      |
+-----------------------------------------------------------------------------+
|i16 : (S.dd_(2,0))_[1]^[1]  --restrict to C_1 summand group of S_2, project  |
|                                                                             |
|o16 = {1} | 0 0 0 1 0 0 |                                                    |
|      {1} | 0 0 0 0 1 0 |                                                    |
|      {1} | 0 0 0 0 0 1 |                                                    |
|                                                                             |
|              3      6                                                       |
|o16 : Matrix R  <-- R                                                        |
|-----------------------------------------------------------------------------|
|onto C_1 summand of S_1                                                      |
+-----------------------------------------------------------------------------+
|i17 : (S.dd_(2,0))_[2]^[1]  --restrict to C_2 summand of S_2, project onto   |
|                                                                             |
|o17 = {1} | -b -c 0  |                                                       |
|      {1} | a  0  -c |                                                       |
|      {1} | 0  a  b  |                                                       |
|                                                                             |
|              3      3                                                       |
|o17 : Matrix R  <-- R                                                        |
|-----------------------------------------------------------------------------|
|C_1 summand of S_1                                                           |
+-----------------------------------------------------------------------------+

The first computation tells us that the component of the face map $d_{2,0}$
mapping $$C_1^{\oplus 2} \to  C_0$$ is given by the differential of the
original complex $C$ applied to the first copy of $C_1$ (we use the
compositions corresponding to the surjections to label the free modules). The
second computation shows the component $C_1^{\oplus 2} \to C_1$ is simply the
identity map on the second copy of $C_1$. The third computation shows the
component $$C_2 \to C_1$$ is also given by the differential of $C$.

See also
========

  * Making simplicial modules -- information
    about the basic constructors
  * Making maps between simplicial modules
    -- information about the basic constructors
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:242:0.

topDegree -- top degree of a simplicial module
**********************************************

  * Usage:
        topDegree C
  * Inputs:
      * C, a Simplicial Module,
  * Outputs:
      * d, an integer (Macaulay2Doc), an integer specifying the top
        degree to which the face/degeneracy maps of a simplicial module have
        been computed

Description
===========

In this package, each simplicial module necessarily has a top degree which is a
finite bound up to which Macaulay2 will compute the face and/or degeneracy
maps. This function is mainly used in programming, to loop over all non-zero
modules or maps in the simplicial module.

+-----------------------------------------------------------------------------+
|i1 : S = ZZ/101[a..c];                                                       |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution coker vars S, 8)                    |
|                                                                             |
|      1      4      10      20      35      56      84      120      165     |
|o2 = S  <-- S  <-- S   <-- S   <-- S   <-- S   <-- S   <-- S    <-- S   <--  |
|                                                                             |
|     0      1      2       3       4       5       6       7        8        |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|...                                                                          |
+-----------------------------------------------------------------------------+
|i3 : topDegree C                                                             |
|                                                                             |
|o3 = 8                                                                       |
+-----------------------------------------------------------------------------+
|i4 : C' = simplicialModule(freeResolution coker vars S, 6)                   |
|                                                                             |
|      1      4      10      20      35      56      84                       |
|o4 = S  <-- S  <-- S   <-- S   <-- S   <-- S   <-- S  <-- ...                |
|                                                                             |
|     0      1      2       3       4       5       6                         |
|                                                                             |
|o4 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i5 : assert(topDegree C' == 6)                                               |
+-----------------------------------------------------------------------------+

Indices that are outside of the top degree automatically return the zero
object.

+-------------+
|i6 : C_-1    |
|             |
|o6 = 0       |
|             |
|o6 : S-module|
+-------------+
|i7 : C_9     |
|             |
|o7 = 0       |
|             |
|o7 : S-module|
+-------------+
|i8 : C'_7    |
|             |
|o8 = 0       |
|             |
|o8 : S-module|
+-------------+

The function topDegree does no computation, and just accesses a cached value.

See also
========

  * Basic invariants and properties, --
    information about accessing basic features
  * SimplicialModule _ ZZ -- access individual
    objects of a simplicial module

Ways to use topDegree:
======================

  * "topDegree(SimplicialModule)"
  * "topDegree(SimplicialModuleMap)"

For the programmer
==================

The object topDegree, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:406:0.

truncate(List,SimplicialModule) -- truncation of a simplicial module at a specified degree or set of degrees
************************************************************************************************************

  * Function: truncate (Macaulay2Doc),
  * Usage:
        truncate(d, C)
  * Inputs:
      * d, a list (Macaulay2Doc), or ZZ (Macaulay2Doc),
        if the underlying ring $R$ is singly graded.
      * C, a Simplicial Module, that is homogeneous
        over $R$
  * Outputs:
      * a Simplicial Module, a simplicial module whose
        terms consist of all elements of component-wise degree at least d.

Description
===========

Truncation of homogeneous (graded) modules induces a natural operation on
simplicial modules.

In the singly graded case, the truncation of a homogeneous module $M$ at degree
$d$ is generated by all homogeneous elements of degree at least $d$ in $M$.
This method applies this operation to each term in a simplicial module.

+-----------------------------------------------------------------------------+
|i1 : R = QQ[a,b,c];                                                          |
+-----------------------------------------------------------------------------+
|i2 : I = ideal(a*b, a*c, b*c)                                                |
|                                                                             |
|o2 = ideal (a*b, a*c, b*c)                                                   |
|                                                                             |
|o2 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i3 : C = simplicialModule(freeResolution I, 3, Degeneracy => true)           |
|                                                                             |
|      1      4      9      16                                                |
|o3 = R  <-- R  <-- R  <-- R  <-- ...                                         |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o3 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i4 : D = truncate(3,C)                                                       |
|                                                                             |
|o4 = image | c3 bc2 ac2 b2c abc a2c b3 ab2 a2b a3 | <-- image {0} | c3 bc2   |
|                                                              {2} | 0  0     |
|     0                                                        {2} | 0  0     |
|                                                              {2} | 0  0     |
|                                                                             |
|                                                        1                    |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o4 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|ac2 b2c abc a2c b3 ab2 a2b a3 0 0 0 0 0 0 0 0 0 | <-- image {0} | c3 bc2 ac2 |
|0   0   0   0   0  0   0   0  c b a 0 0 0 0 0 0 |           {2} | 0  0   0   |
|0   0   0   0   0  0   0   0  0 0 0 c b a 0 0 0 |           {2} | 0  0   0   |
|0   0   0   0   0  0   0   0  0 0 0 0 0 0 c b a |           {2} | 0  0   0   |
|                                                            {2} | 0  0   0   |
|                                                            {2} | 0  0   0   |
|                                                            {2} | 0  0   0   |
|                                                            {3} | 0  0   0   |
|                                                            {3} | 0  0   0   |
|                                                                             |
|                                                      2                      |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|b2c abc a2c b3 ab2 a2b a3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image|
|0   0   0   0  0   0   0  c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |          |
|0   0   0   0  0   0   0  0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |          |
|0   0   0   0  0   0   0  0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 |          |
|0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 |          |
|0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 |          |
|0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 |          |
|0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |          |
|0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                        3    |
|-----------------------------------------------------------------------------|
|{0} | c3 bc2 ac2 b2c abc a2c b3 ab2 a2b a3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|{2} | 0  0   0   0   0   0   0  0   0   0  c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|{2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 |
|{2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 |
|{2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 |
|{2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 |
|{2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b |
|{2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|{2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|{2} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|{3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|{3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|{3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|{3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|{3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|{3} | 0  0   0   0   0   0   0  0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |<-- ...                                     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                                            |
+-----------------------------------------------------------------------------+
|i5 : assert isWellDefined D                                                  |
+-----------------------------------------------------------------------------+

Truncating at a degree less than the minimal generators is the identity
operation.

+--------------------------------+
|i6 : assert(C == truncate(0, C))|
+--------------------------------+

In the multi-graded case, the truncation of a homogeneous module at a list of
degrees is generated by all homogeneous elements of degree that are
component-wise greater than or equal to at least one of the degrees.

+-----------------------------------------------------------------------------+
|i7 : A = ZZ/101[x_0, x_1, y_0, y_1, y_2, Degrees => {2:{1,0}, 3:{0,1}}];     |
+-----------------------------------------------------------------------------+
|i8 : I = intersect(ideal(x_0, x_1), ideal(y_0, y_1, y_2))                    |
|                                                                             |
|o8 = ideal (x y , x y , x y , x y , x y , x y )                              |
|             1 2   0 2   1 1   0 1   1 0   0 0                               |
|                                                                             |
|o8 : Ideal of A                                                              |
+-----------------------------------------------------------------------------+
|i9 : C = simplicialModule(freeResolution I, 3, Degeneracy => true)           |
|                                                                             |
|      1      7      22      51                                               |
|o9 = A  <-- A  <-- A   <-- A  <-- ...                                        |
|                                                                             |
|     0      1      2       3                                                 |
|                                                                             |
|o9 : SimplicialModule                                                        |
+-----------------------------------------------------------------------------+
|i10 : D1 = prune truncate({{1,1}}, C)                                        |
|                                                                             |
|o10 = cokernel {1, 1} | x_0  y_1  0    0    y_0  0    0    0    0    | <--   |
|               {1, 1} | -x_1 0    y_1  0    0    0    y_0  0    0    |       |
|               {1, 1} | 0    -y_2 0    x_0  0    y_0  0    0    0    |       |
|               {1, 1} | 0    0    -y_2 -x_1 0    0    0    y_0  0    |       |
|               {1, 1} | 0    0    0    0    -y_2 -y_1 0    0    x_0  |       |
|               {1, 1} | 0    0    0    0    0    0    -y_2 -y_1 -x_1 |       |
|                                                                             |
|      0                                                                      |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|o10 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|cokernel {1, 1} | x_0  y_1  0    0    y_0  0    0    0    0    | <-- cokernel|
|         {1, 1} | -x_1 0    y_1  0    0    0    y_0  0    0    |             |
|         {1, 1} | 0    -y_2 0    x_0  0    y_0  0    0    0    |             |
|         {1, 1} | 0    0    -y_2 -x_1 0    0    0    y_0  0    |             |
|         {1, 1} | 0    0    0    0    -y_2 -y_1 0    0    x_0  |             |
|         {1, 1} | 0    0    0    0    0    0    -y_2 -y_1 -x_1 |             |
|         {1, 1} | 0    0    0    0    0    0    0    0    0    |             |
|         {1, 1} | 0    0    0    0    0    0    0    0    0    |             |
|         {1, 1} | 0    0    0    0    0    0    0    0    0    |             |
|         {1, 1} | 0    0    0    0    0    0    0    0    0    |             |
|         {1, 1} | 0    0    0    0    0    0    0    0    0    |             |
|         {1, 1} | 0    0    0    0    0    0    0    0    0    |             |
|                                                                             |
|1                                                                            |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                     2       |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|{1, 1} | x_0  y_1  0    0    y_0  0    0    0    0    | <-- cokernel {1, 1} ||
|{1, 1} | -x_1 0    y_1  0    0    0    y_0  0    0    |              {1, 1} ||
|{1, 1} | 0    -y_2 0    x_0  0    y_0  0    0    0    |              {1, 1} ||
|{1, 1} | 0    0    -y_2 -x_1 0    0    0    y_0  0    |              {1, 1} ||
|{1, 1} | 0    0    0    0    -y_2 -y_1 0    0    x_0  |              {1, 1} ||
|{1, 1} | 0    0    0    0    0    0    -y_2 -y_1 -x_1 |              {1, 1} ||
|{1, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{2, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{2, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 2} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 2} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{2, 1} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 2} | 0    0    0    0    0    0    0    0    0    |              {1, 1} ||
|{1, 2} | 0    0    0    0    0    0    0    0    0    |              {2, 1} ||
|{1, 2} | 0    0    0    0    0    0    0    0    0    |              {2, 1} ||
|{1, 2} | 0    0    0    0    0    0    0    0    0    |              {1, 2} ||
|                                                                     {1, 2} ||
|                                                                     {2, 1} ||
|                                                                     {1, 2} ||
|                                                                     {1, 2} ||
|                                                                     {1, 2} ||
|                                                                     {1, 2} ||
|                                                                     {2, 1} ||
|                                                                     {2, 1} ||
|                                                                     {1, 2} ||
|                                                                     {1, 2} ||
|                                                                     {2, 1} ||
|                                                                     {1, 2} ||
|                                                                     {1, 2} ||
|                                                                     {1, 2} ||
|                                                                     {1, 2} ||
|                                                                     {2, 1} ||
|                                                                     {2, 1} ||
|                                                                     {1, 2} ||
|                                                                     {1, 2} ||
|                                                                     {2, 1} ||
|                                                                     {1, 2} ||
|                                                                     {1, 2} ||
|                                                                     {1, 2} ||
|                                                                     {1, 2} ||
|                                                                     {2, 2} ||
|                                                                     {2, 2} ||
|                                                                     {2, 2} ||
|                                                                     {1, 3} ||
|                                                                     {1, 3} ||
|                                                                             |
|                                                            3                |
|-----------------------------------------------------------------------------|
|x_0  y_1  0    0    y_0  0    0    0    0    |<-- ...                        |
|-x_1 0    y_1  0    0    0    y_0  0    0    |                               |
|0    -y_2 0    x_0  0    y_0  0    0    0    |                               |
|0    0    -y_2 -x_1 0    0    0    y_0  0    |                               |
|0    0    0    0    -y_2 -y_1 0    0    x_0  |                               |
|0    0    0    0    0    0    -y_2 -y_1 -x_1 |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
|0    0    0    0    0    0    0    0    0    |                               |
+-----------------------------------------------------------------------------+
|i11 : D2 = truncate({{1,0}}, C)                                              |
|                                                                             |
|o11 = image | x_1 x_0 | <-- image {0, 0} | x_1 x_0 0 0 0 0 0 0 | <-- image   |
|                                  {1, 1} | 0   0   1 0 0 0 0 0 |             |
|      0                           {1, 1} | 0   0   0 1 0 0 0 0 |             |
|                                  {1, 1} | 0   0   0 0 1 0 0 0 |             |
|                                  {1, 1} | 0   0   0 0 0 1 0 0 |             |
|                                  {1, 1} | 0   0   0 0 0 0 1 0 |             |
|                                  {1, 1} | 0   0   0 0 0 0 0 1 |             |
|                                                                             |
|                            1                                                |
|                                                                             |
|                                                                             |
|                                                                             |
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|o11 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|{0, 0} | x_1 x_0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {0, 0}|
|{1, 1} | 0   0   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1, 1}|
|{1, 1} | 0   0   0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1, 1}|
|{1, 1} | 0   0   0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1, 1}|
|{1, 1} | 0   0   0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1, 1}|
|{1, 1} | 0   0   0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1, 1}|
|{1, 1} | 0   0   0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1, 1}|
|{1, 1} | 0   0   0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1, 1}|
|{1, 1} | 0   0   0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |           {1, 1}|
|{1, 1} | 0   0   0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |           {1, 1}|
|{1, 1} | 0   0   0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |           {1, 1}|
|{1, 1} | 0   0   0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |           {1, 1}|
|{1, 1} | 0   0   0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |           {1, 1}|
|{2, 1} | 0   0   0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |           {1, 1}|
|{2, 1} | 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |           {1, 1}|
|{1, 2} | 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |           {1, 1}|
|{1, 2} | 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |           {1, 1}|
|{2, 1} | 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |           {1, 1}|
|{1, 2} | 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |           {1, 1}|
|{1, 2} | 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |           {2, 1}|
|{1, 2} | 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |           {2, 1}|
|{1, 2} | 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |           {1, 2}|
|                                                                       {1, 2}|
|                                                                       {2, 1}|
|                                                                       {1, 2}|
|                                                                       {1, 2}|
|                                                                       {1, 2}|
|                                                                       {1, 2}|
|                                                                       {2, 1}|
|                                                                       {2, 1}|
|                                                                       {1, 2}|
|                                                                       {1, 2}|
|                                                                       {2, 1}|
|                                                                       {1, 2}|
|                                                                       {1, 2}|
|                                                                       {1, 2}|
|                                                                       {1, 2}|
|                                                                       {2, 1}|
|                                                                       {2, 1}|
|                                                                       {1, 2}|
|                                                                       {1, 2}|
|                                                                       {2, 1}|
|                                                                       {1, 2}|
|                                                                       {1, 2}|
|                                                                       {1, 2}|
|                                                                       {1, 2}|
|                                                                       {2, 2}|
|                                                                       {2, 2}|
|                                                                       {2, 2}|
|                                                                       {1, 3}|
|                                                                       {1, 3}|
|                                                                             |
|                                                                 3           |
|-----------------------------------------------------------------------------|
|| x_1 x_0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|| 0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |<-- ...                                     |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                                            |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                                            |
+-----------------------------------------------------------------------------+
|i12 : D3 = truncate({{0,1}}, C)                                              |
|                                                                             |
|o12 = image | y_2 y_1 y_0 | <-- image {0, 0} | y_2 y_1 y_0 0 0 0 0 0 0 | <-- |
|                                      {1, 1} | 0   0   0   1 0 0 0 0 0 |     |
|      0                               {1, 1} | 0   0   0   0 1 0 0 0 0 |     |
|                                      {1, 1} | 0   0   0   0 0 1 0 0 0 |     |
|                                      {1, 1} | 0   0   0   0 0 0 1 0 0 |     |
|                                      {1, 1} | 0   0   0   0 0 0 0 1 0 |     |
|                                      {1, 1} | 0   0   0   0 0 0 0 0 1 |     |
|                                                                             |
|                                1                                            |
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|                                                                             |
|o12 : SimplicialModule                                                       |
|-----------------------------------------------------------------------------|
|image {0, 0} | y_2 y_1 y_0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <--   |
|      {1, 1} | 0   0   0   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|      {1, 1} | 0   0   0   0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|      {1, 1} | 0   0   0   0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|      {1, 1} | 0   0   0   0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|      {1, 1} | 0   0   0   0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|      {1, 1} | 0   0   0   0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |       |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |       |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |       |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |       |
|      {2, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |       |
|      {2, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |       |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |       |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |       |
|      {2, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |       |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |       |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |       |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |       |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |       |
|                                                                             |
|2                                                                            |
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|-----------------------------------------------------------------------------|
|image {0, 0} | y_2 y_1 y_0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
|      {1, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
|      {2, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
|      {2, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
|      {2, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2, 1} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {2, 2} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 3} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|      {1, 3} | 0   0   0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|                                                                             |
|3                                                                            |
|-----------------------------------------------------------------------------|
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |<-- ...                   |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |                          |
|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |                          |
+-----------------------------------------------------------------------------+
|i13 : D4 = truncate({{1,0},{0,1}}, C);                                       |
+-----------------------------------------------------------------------------+
|i14 : D5 = truncate({{2,2}}, C);                                             |
+-----------------------------------------------------------------------------+
|i15 : assert all({D1,D2,D3,D4,D5}, isWellDefined)                            |
+-----------------------------------------------------------------------------+

See also
========

  * Making simplicial modules -- information
    about the basic constructors
  * truncate(List,Module) (Truncations), --
    truncation of the graded ring, ideal or module at a specified degree or set
    of degrees
  * truncate(List,SimplicialModuleMap)
    -- truncation of a simplicial
    module map at a specified degree or set of degrees

Ways to use this method:
========================

  * truncate(List,SimplicialModule)
    -- truncation of a simplicial module
    at a specified degree or set of degrees
  * "truncate(ZZ,SimplicialModule)"
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:1035:0.

truncate(List,SimplicialModuleMap) -- truncation of a simplicial module map at a specified degree or set of degrees
*******************************************************************************************************************

  * Function: truncate (Macaulay2Doc),
  * Usage:
        truncate(d, f)
  * Inputs:
      * d, a list (Macaulay2Doc), or ZZ (Macaulay2Doc),
        if the underlying ring $R$ is singly graded.
      * f, a Map of Simplicial Modules, that is
        homogeneous over $R$
  * Outputs:
      * a Map of Simplicial Modules, a simplicial
        module map over $R$ whose terms in the source and target consist of all
        elements of component-wise degree at least d.

Description
===========

Truncation of homogeneous (graded) maps induces a natural operation on maps of
simplicial modules.

In the singly graded case, the truncation of a homogeneous module $M$ at degree
$d$ is generated by all homogeneous elements of degree at least $d$ in $M$.
The truncation of a map between homogeneous modules is the induced map between
the truncation of the source and the truncation of the target.  This method
applies this operation to each term in a map of simplicial modules.

+-----------------------------------------------------------------------------+
|i1 : R = QQ[a,b,c];                                                          |
+-----------------------------------------------------------------------------+
|i2 : C = simplicialModule(freeResolution ideal(a*b, a*c, b*c), 3, Degeneracy |
|                                                                             |
|      1      4      9      16                                                |
|o2 = R  <-- R  <-- R  <-- R  <-- ...                                         |
|                                                                             |
|     0      1      2      3                                                  |
|                                                                             |
|o2 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|=> true)                                                                     |
+-----------------------------------------------------------------------------+
|i3 : D = simplicialModule((freeResolution ideal(a*b, a*c, b*c, a^2-b^2))[-1],|
|                                                                             |
|            1      6      19                                                 |
|o3 = 0 <-- R  <-- R  <-- R  <-- ...                                          |
|                                                                             |
|     0     1      2      3                                                   |
|                                                                             |
|o3 : SimplicialModule                                                        |
|-----------------------------------------------------------------------------|
|3, Degeneracy => true)                                                       |
+-----------------------------------------------------------------------------+
|i4 : f = randomSimplicialMap(D,C, Cycle => true)                             |
|                                                                             |
|                   1                                                         |
|o4 = 0 : 0 <----- R  : 0                                                     |
|              0                                                              |
|                                                                             |
|          1                                                                  |
|     1 : R  <----------------------------------------------------------------|
|               | 0 9/2a2+4ab+7/9b2+3/4ac+7/10bc 3/7a2+6/7ab+5/6b2+45/7ac+5bc+|
|                                                                             |
|          6                                                                  |
|     2 : R  <----------------------------------------------------------------|
|               {0} | 0 9/2a2+4ab+7/9b2+3/4ac+7/10bc 3/7a2+6/7ab+5/6b2+45/7ac+|
|               {0} | 0 0                            0                        |
|               {2} | 0 0                            0                        |
|               {2} | 0 0                            0                        |
|               {2} | 0 0                            0                        |
|               {2} | 0 0                            0                        |
|                                                                             |
|          19                                                                 |
|     3 : R   <---------------------------------------------------------------|
|                {0} | 0 9/2a2+4ab+7/9b2+3/4ac+7/10bc 3/7a2+6/7ab+5/6b2+45/7ac|
|                {0} | 0 0                            0                       |
|                {0} | 0 0                            0                       |
|                {2} | 0 0                            0                       |
|                {2} | 0 0                            0                       |
|                {2} | 0 0                            0                       |
|                {2} | 0 0                            0                       |
|                {2} | 0 0                            0                       |
|                {2} | 0 0                            0                       |
|                {2} | 0 0                            0                       |
|                {2} | 0 0                            0                       |
|                {2} | 0 0                            0                       |
|                {2} | 0 0                            0                       |
|                {2} | 0 0                            0                       |
|                {2} | 0 0                            0                       |
|                {3} | 0 0                            0                       |
|                {3} | 0 0                            0                       |
|                {3} | 0 0                            0                       |
|                {4} | 0 0                            0                       |
|                                                                             |
|o4 : SimplicialModuleMap                                                     |
|-----------------------------------------------------------------------------|
|                                                   4                         |
|------------------------------------------------- R  : 1                     |
|10/9c2 7/8a2+7/10ab+7/3b2+5/4ac+65/9bc+3/10c2 |                              |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|5bc+10/9c2 7/8a2+7/10ab+7/3b2+5/4ac+65/9bc+3/10c2 0                          |
|           0                                      9/2a2+4ab+7/9b2+3/4ac+7/10b|
|           0                                      0                          |
|           0                                      0                          |
|           0                                      0                          |
|           0                                      0                          |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|+5bc+10/9c2 7/8a2+7/10ab+7/3b2+5/4ac+65/9bc+3/10c2 0                         |
|            0                                      9/2a2+4ab+7/9b2+3/4ac+7/10|
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|            0                                      0                         |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|  0                                   0                                      |
|c 3/7a2+6/7ab+5/6b2+45/7ac+5bc+10/9c2 7/8a2+7/10ab+7/3b2+5/4ac+65/9bc+3/10c2 |
|  0                                   0                                      |
|  0                                   0                                      |
|  0                                   0                                      |
|  0                                   0                                      |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|   0                                   0                                     |
|bc 3/7a2+6/7ab+5/6b2+45/7ac+5bc+10/9c2 7/8a2+7/10ab+7/3b2+5/4ac+65/9bc+3/10c2|
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|   0                                   0                                     |
|-----------------------------------------------------------------------------|
|                                              9                              |
|-------------------------------------------- R  : 2                          |
|0                     0                  |                                   |
|0                     0                  |                                   |
|5/6b-3/2c             -7/8a-5/6b-5c      |                                   |
|-53/42a-6/7b-16c      59/105a-395/168b-c |                                   |
|6a+49/4b+3/4c         15/4a-2/9b-3/10c   |                                   |
|37/28a-103/18b-37/90c 3/7a+10/9c         |                                   |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
| 0                            0                                   0          |
| 0                            0                                   0          |
| 9/2a2+4ab+7/9b2+3/4ac+7/10bc 3/7a2+6/7ab+5/6b2+45/7ac+5bc+10/9c2 7/8a2+7/10a|
| 0                            0                                   0          |
| 0                            0                                   0          |
| 0                            0                                   0          |
| 0                            0                                   0          |
| 0                            0                                   0          |
| 0                            0                                   0          |
| 0                            0                                   0          |
| 0                            0                                   0          |
| 0                            0                                   0          |
| 0                            0                                   0          |
| 0                            0                                   0          |
| 0                            0                                   0          |
| 0                            0                                   0          |
| 0                            0                                   0          |
| 0                            0                                   0          |
| 0                            0                                   0          |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|                            0                     0                  0       |
|                            0                     0                  0       |
|b+7/3b2+5/4ac+65/9bc+3/10c2 0                     0                  0       |
|                            5/6b-3/2c             -7/8a-5/6b-5c      0       |
|                            -53/42a-6/7b-16c      59/105a-395/168b-c 0       |
|                            6a+49/4b+3/4c         15/4a-2/9b-3/10c   0       |
|                            37/28a-103/18b-37/90c 3/7a+10/9c         0       |
|                            0                     0                  5/6b-3/2|
|                            0                     0                  -53/42a-|
|                            0                     0                  6a+49/4b|
|                            0                     0                  37/28a-1|
|                            0                     0                  0       |
|                            0                     0                  0       |
|                            0                     0                  0       |
|                            0                     0                  0       |
|                            0                     0                  0       |
|                            0                     0                  0       |
|                            0                     0                  0       |
|                            0                     0                  0       |
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|              0                  0                     0                  |  |
|              0                  0                     0                  |  |
|              0                  0                     0                  |  |
|              0                  0                     0                  |  |
|              0                  0                     0                  |  |
|              0                  0                     0                  |  |
|              0                  0                     0                  |  |
|c             -7/8a-5/6b-5c      0                     0                  |  |
|6/7b-16c      59/105a-395/168b-c 0                     0                  |  |
|+3/4c         15/4a-2/9b-3/10c   0                     0                  |  |
|03/18b-37/90c 3/7a+10/9c         0                     0                  |  |
|              0                  5/6b-3/2c             -7/8a-5/6b-5c      |  |
|              0                  -53/42a-6/7b-16c      59/105a-395/168b-c |  |
|              0                  6a+49/4b+3/4c         15/4a-2/9b-3/10c   |  |
|              0                  37/28a-103/18b-37/90c 3/7a+10/9c         |  |
|              0                  0                     0                  |  |
|              0                  0                     0                  |  |
|              0                  0                     0                  |  |
|              0                  0                     0                  |  |
|-----------------------------------------------------------------------------|
| 16                                                                          |
|R   : 3                                                                      |
+-----------------------------------------------------------------------------+
|i5 : g = truncate(3,f);                                                      |
+-----------------------------------------------------------------------------+
|i6 : assert isWellDefined g                                                  |
+-----------------------------------------------------------------------------+
|i7 : assert (source g == truncate(3, source f))                              |
+-----------------------------------------------------------------------------+
|i8 : assert (target g == truncate(3, target f))                              |
+-----------------------------------------------------------------------------+

Truncating at a degree less than the minimal generators is the identity
operation.

+--------------------------------+
|i9 : assert(f == truncate(0, f))|
+--------------------------------+

In the multi-graded case, the truncation of a homogeneous module at a list of
degrees is generated by all homogeneous elements of degree that are
component-wise greater than or equal to at least one of the degrees.  As in the
singly graded case, this induces a map between the truncations of the source
and target.

+-----------------------------------------------------------------------------+
|i10 : A = ZZ/101[x_0, x_1, y_0, y_1, y_2, Degrees => {2:{1,0}, 3:{0,1}}];    |
+-----------------------------------------------------------------------------+
|i11 : I = intersect(ideal(x_0, x_1), ideal(y_0, y_1, y_2))                   |
|                                                                             |
|o11 = ideal (x y , x y , x y , x y , x y , x y )                             |
|              1 2   0 2   1 1   0 1   1 0   0 0                              |
|                                                                             |
|o11 : Ideal of A                                                             |
+-----------------------------------------------------------------------------+
|i12 : C = simplicialModule(freeResolution I, 4, Degeneracy => true)          |
|                                                                             |
|       1      7      22      51      100                                     |
|o12 = A  <-- A  <-- A   <-- A   <-- A   <-- ...                              |
|                                                                             |
|      0      1      2       3       4                                        |
|                                                                             |
|o12 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i13 : J = intersect(ideal(x_0^2, x_1^2), ideal(y_0^2, y_1^2, y_2^2))         |
|                                                                             |
|              2 2   2 2   2 2   2 2   2 2   2 2                              |
|o13 = ideal (x y , x y , x y , x y , x y , x y )                             |
|              1 2   0 2   1 1   0 1   1 0   0 0                              |
|                                                                             |
|o13 : Ideal of A                                                             |
+-----------------------------------------------------------------------------+
|i14 : D = simplicialModule(freeResolution J, 4, Degeneracy => true)          |
|                                                                             |
|       1      7      22      51      100                                     |
|o14 = A  <-- A  <-- A   <-- A   <-- A   <-- ...                              |
|                                                                             |
|      0      1      2       3       4                                        |
|                                                                             |
|o14 : SimplicialModule                                                       |
+-----------------------------------------------------------------------------+
|i15 : f = simplicialModule(extend(C.complex, D.complex, id_(A^1)), 2)        |
|                                                                             |
|           1             1                                                   |
|o15 = 0 : A  <--------- A  : 0                                               |
|                | 1 |                                                        |
|                                                                             |
|           7                                                              7  |
|      1 : A  <---------------------------------------------------------- A  :|
|                {0, 0} | 1 0      0      0      0      0      0      |       |
|                {1, 1} | 0 x_0y_0 0      0      0      0      0      |       |
|                {1, 1} | 0 0      x_1y_0 0      0      0      0      |       |
|                {1, 1} | 0 0      0      x_0y_1 0      0      0      |       |
|                {1, 1} | 0 0      0      0      x_1y_1 0      0      |       |
|                {1, 1} | 0 0      0      0      0      x_0y_2 0      |       |
|                {1, 1} | 0 0      0      0      0      0      x_1y_2 |       |
|                                                                             |
|           22                                                                |
|      2 : A   <--------------------------------------------------------------|
|                 {0, 0} | 1 0      0      0      0      0      0      0      |
|                 {1, 1} | 0 x_0y_0 0      0      0      0      0      0      |
|                 {1, 1} | 0 0      x_1y_0 0      0      0      0      0      |
|                 {1, 1} | 0 0      0      x_0y_1 0      0      0      0      |
|                 {1, 1} | 0 0      0      0      x_1y_1 0      0      0      |
|                 {1, 1} | 0 0      0      0      0      x_0y_2 0      0      |
|                 {1, 1} | 0 0      0      0      0      0      x_1y_2 0      |
|                 {1, 1} | 0 0      0      0      0      0      0      x_0y_0 |
|                 {1, 1} | 0 0      0      0      0      0      0      0      |
|                 {1, 1} | 0 0      0      0      0      0      0      0      |
|                 {1, 1} | 0 0      0      0      0      0      0      0      |
|                 {1, 1} | 0 0      0      0      0      0      0      0      |
|                 {1, 1} | 0 0      0      0      0      0      0      0      |
|                 {2, 1} | 0 0      0      0      0      0      0      0      |
|                 {2, 1} | 0 0      0      0      0      0      0      0      |
|                 {1, 2} | 0 0      0      0      0      0      0      0      |
|                 {1, 2} | 0 0      0      0      0      0      0      0      |
|                 {2, 1} | 0 0      0      0      0      0      0      0      |
|                 {1, 2} | 0 0      0      0      0      0      0      0      |
|                 {1, 2} | 0 0      0      0      0      0      0      0      |
|                 {1, 2} | 0 0      0      0      0      0      0      0      |
|                 {1, 2} | 0 0      0      0      0      0      0      0      |
|                                                                             |
|o15 : SimplicialModuleMap                                                    |
|-----------------------------------------------------------------------------|
| 1                                                                           |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|0      0      0      0      0      0         0         0         0         0 |
|0      0      0      0      0      0         0         0         0         0 |
|0      0      0      0      0      0         0         0         0         0 |
|0      0      0      0      0      0         0         0         0         0 |
|0      0      0      0      0      0         0         0         0         0 |
|0      0      0      0      0      0         0         0         0         0 |
|0      0      0      0      0      0         0         0         0         0 |
|0      0      0      0      0      0         0         0         0         0 |
|x_1y_0 0      0      0      0      0         0         0         0         0 |
|0      x_0y_1 0      0      0      0         0         0         0         0 |
|0      0      x_1y_1 0      0      0         0         0         0         0 |
|0      0      0      x_0y_2 0      0         0         0         0         0 |
|0      0      0      0      x_1y_2 0         0         0         0         0 |
|0      0      0      0      0      x_0x_1y_0 0         0         0         0 |
|0      0      0      0      0      0         x_0x_1y_1 0         0         0 |
|0      0      0      0      0      0         0         x_0y_0y_1 0         0 |
|0      0      0      0      0      0         0         0         x_1y_0y_1 0 |
|0      0      0      0      0      0         0         0         0         x_|
|0      0      0      0      0      0         0         0         0         0 |
|0      0      0      0      0      0         0         0         0         0 |
|0      0      0      0      0      0         0         0         0         0 |
|0      0      0      0      0      0         0         0         0         0 |
|-----------------------------------------------------------------------------|
|                                                     22                      |
|--------------------------------------------------- A   : 2                  |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|        0         0         0         0         |                            |
|0x_1y_2 0         0         0         0         |                            |
|        x_0y_0y_2 0         0         0         |                            |
|        0         x_1y_0y_2 0         0         |                            |
|        0         0         x_0y_1y_2 0         |                            |
|        0         0         0         x_1y_1y_2 |                            |
+-----------------------------------------------------------------------------+
|i16 : g1 = prune truncate({{1,1}}, f);                                       |
+-----------------------------------------------------------------------------+
|i17 : g1_0                                                                   |
|                                                                             |
|o17 = {1, 1} | 1 0 0 0 0 0 |                                                 |
|      {1, 1} | 0 1 0 0 0 0 |                                                 |
|      {1, 1} | 0 0 1 0 0 0 |                                                 |
|      {1, 1} | 0 0 0 1 0 0 |                                                 |
|      {1, 1} | 0 0 0 0 1 0 |                                                 |
|      {1, 1} | 0 0 0 0 0 1 |                                                 |
|                                                                             |
|o17 : Matrix                                                                 |
+-----------------------------------------------------------------------------+
|i18 : g1_1                                                                   |
|                                                                             |
|o18 = {1, 1} | 1 0 0 0 0 0 0      0      0      0      0      0      |       |
|      {1, 1} | 0 1 0 0 0 0 0      0      0      0      0      0      |       |
|      {1, 1} | 0 0 1 0 0 0 0      0      0      0      0      0      |       |
|      {1, 1} | 0 0 0 1 0 0 0      0      0      0      0      0      |       |
|      {1, 1} | 0 0 0 0 1 0 0      0      0      0      0      0      |       |
|      {1, 1} | 0 0 0 0 0 1 0      0      0      0      0      0      |       |
|      {1, 1} | 0 0 0 0 0 0 x_0y_0 0      0      0      0      0      |       |
|      {1, 1} | 0 0 0 0 0 0 0      x_1y_0 0      0      0      0      |       |
|      {1, 1} | 0 0 0 0 0 0 0      0      x_0y_1 0      0      0      |       |
|      {1, 1} | 0 0 0 0 0 0 0      0      0      x_1y_1 0      0      |       |
|      {1, 1} | 0 0 0 0 0 0 0      0      0      0      x_0y_2 0      |       |
|      {1, 1} | 0 0 0 0 0 0 0      0      0      0      0      x_1y_2 |       |
|                                                                             |
|o18 : Matrix                                                                 |
+-----------------------------------------------------------------------------+
|i19 : g2 = truncate({{1,0}}, f);                                             |
+-----------------------------------------------------------------------------+
|i20 : g2_1                                                                   |
|                                                                             |
|o20 = {1, 0} | 1 0 0      0      0      0      0      0      |               |
|      {1, 0} | 0 1 0      0      0      0      0      0      |               |
|      {1, 1} | 0 0 x_0y_0 0      0      0      0      0      |               |
|      {1, 1} | 0 0 0      x_1y_0 0      0      0      0      |               |
|      {1, 1} | 0 0 0      0      x_0y_1 0      0      0      |               |
|      {1, 1} | 0 0 0      0      0      x_1y_1 0      0      |               |
|      {1, 1} | 0 0 0      0      0      0      x_0y_2 0      |               |
|      {1, 1} | 0 0 0      0      0      0      0      x_1y_2 |               |
|                                                                             |
|o20 : Matrix                                                                 |
+-----------------------------------------------------------------------------+
|i21 : g3 = truncate({{0,1}}, f);                                             |
+-----------------------------------------------------------------------------+
|i22 : g4 = truncate({{1,0},{0,1}}, f);                                       |
+-----------------------------------------------------------------------------+
|i23 : g4_1                                                                   |
|                                                                             |
|o23 = {0, 1} | 1 0 0 0 0 0      0      0      0      0      0      |         |
|      {0, 1} | 0 1 0 0 0 0      0      0      0      0      0      |         |
|      {0, 1} | 0 0 1 0 0 0      0      0      0      0      0      |         |
|      {1, 0} | 0 0 0 1 0 0      0      0      0      0      0      |         |
|      {1, 0} | 0 0 0 0 1 0      0      0      0      0      0      |         |
|      {1, 1} | 0 0 0 0 0 x_0y_0 0      0      0      0      0      |         |
|      {1, 1} | 0 0 0 0 0 0      x_1y_0 0      0      0      0      |         |
|      {1, 1} | 0 0 0 0 0 0      0      x_0y_1 0      0      0      |         |
|      {1, 1} | 0 0 0 0 0 0      0      0      x_1y_1 0      0      |         |
|      {1, 1} | 0 0 0 0 0 0      0      0      0      x_0y_2 0      |         |
|      {1, 1} | 0 0 0 0 0 0      0      0      0      0      x_1y_2 |         |
|                                                                             |
|o23 : Matrix                                                                 |
+-----------------------------------------------------------------------------+
|i24 : g5 = truncate({{2,2}}, f);                                             |
+-----------------------------------------------------------------------------+
|i25 : assert all({g1,g2,g3,g4,g5}, isWellDefined)                            |
+-----------------------------------------------------------------------------+

See also
========

  * Making maps between simplicial modules
    -- information about the basic constructors
  * truncate(List,Matrix) (Truncations), --
    truncation of a map of free modules
  * truncate(List,SimplicialModule)
    -- truncation of a simplicial module
    at a specified degree or set of degrees

Ways to use this method:
========================

  * truncate(List,SimplicialModuleMap)
    -- truncation of a simplicial
    module map at a specified degree or set of degrees
  * "truncate(ZZ,SimplicialModuleMap)"
-------------------------------------------------------------------------------

The source of this document is in
SimplicialModules/SimplicialModuleDOC.m2:2363:0.