How to make reduced and non-reduced simplicial chain complexes -- simplicial homological constructors

Let $K$ be an abstract simplicial complex on the vertex set $[n] = \{1\dots,n\}$. We represent dimension $i$ faces of $K$ as lexicographically ordered cardinality $i+1$ subsets $\{l_0 < \dots < l_i\}$ of $[n]$. We further let $C_i$ be the free abelian group on the set of $i$-dimensional faces of $K$.

By these conventions the differential from the free abelian group of $K$'s dimension $i$ faces to the free abelian group of $K$'s dimension $i-1$ faces sends each $\{l_0 < \dots < l_i\}$ to the signed sum of all lexicographically ordered sets $\{l_0 < \dots < l_i\} \backslash \{l_j\}$ as $j$ ranges from $0$ to $i$.

Here, the sign in front of $\{l_0 < \dots < l_i\} \backslash \{l_j\}$ is $(-1)^j$.

Using the constructors simplicialChainComplex and reducedSimplicialChainComplex respectively, non-reduced and reduced simplicial chain complexes can be constructed in the following way.

i1 : K = abstractSimplicialComplex({{1,2,3,4}, {2,3,5},{1,5}})

o1 = AbstractSimplicialComplex{-1 => {{}}                                                                   }
                               0 => {{1}, {2}, {3}, {4}, {5}}
                               1 => {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}}
                               2 => {{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {2, 3, 5}}
                               3 => {{1, 2, 3, 4}}

o1 : AbstractSimplicialComplex

i2 : k = simplicialChainComplex K

       5       9       5       1
o2 = ZZ  <-- ZZ  <-- ZZ  <-- ZZ
                              
     0       1       2       3

o2 : Complex

i3 : k.dd

           5                                       9
o3 = 0 : ZZ  <---------------------------------- ZZ  : 1
                | -1 -1 -1 -1 0  0  0  0  0  |
                | 1  0  0  0  -1 -1 -1 0  0  |
                | 0  1  0  0  1  0  0  -1 -1 |
                | 0  0  1  0  0  1  0  1  0  |
                | 0  0  0  1  0  0  1  0  1  |

           9                           5
     1 : ZZ  <---------------------- ZZ  : 2
                | 1  1  0  0  0  |
                | -1 0  1  0  0  |
                | 0  -1 -1 0  0  |
                | 0  0  0  0  0  |
                | 1  0  0  1  1  |
                | 0  1  0  -1 0  |
                | 0  0  0  0  -1 |
                | 0  0  1  1  0  |
                | 0  0  0  0  1  |

           5               1
     2 : ZZ  <---------- ZZ  : 3
                | -1 |
                | 1  |
                | -1 |
                | 1  |
                | 0  |

o3 : ComplexMap

i4 : kRed = reducedSimplicialChainComplex K

       1       5       9       5       1
o4 = ZZ  <-- ZZ  <-- ZZ  <-- ZZ  <-- ZZ
                                      
     -1      0       1       2       3

o4 : Complex

i5 : kRed.dd

            1                      5
o5 = -1 : ZZ  <----------------- ZZ  : 0
                 | 1 1 1 1 1 |

           5                                       9
     0 : ZZ  <---------------------------------- ZZ  : 1
                | -1 -1 -1 -1 0  0  0  0  0  |
                | 1  0  0  0  -1 -1 -1 0  0  |
                | 0  1  0  0  1  0  0  -1 -1 |
                | 0  0  1  0  0  1  0  1  0  |
                | 0  0  0  1  0  0  1  0  1  |

           9                           5
     1 : ZZ  <---------------------- ZZ  : 2
                | 1  1  0  0  0  |
                | -1 0  1  0  0  |
                | 0  -1 -1 0  0  |
                | 0  0  0  0  0  |
                | 1  0  0  1  1  |
                | 0  1  0  -1 0  |
                | 0  0  0  0  -1 |
                | 0  0  1  1  0  |
                | 0  0  0  0  1  |

           5               1
     2 : ZZ  <---------- ZZ  : 3
                | -1 |
                | 1  |
                | -1 |
                | 1  |
                | 0  |

o5 : ComplexMap

The source of this document is in AbstractSimplicialComplexes.m2:526:0.