An *abstract simplicial complex* on a finite set is a family of subsets that is closed under taking subsets. The elements in the abstract simplicial complex are called its *faces*. The faces having cardinality 1 are its *vertices* and the maximal faces (order by inclusion) are its *facets*. Following the combinatorial conventions, every nonempty abstract simplicial complex has the empty set as a face.

In this package, a simplicial complex is represented by its Stanley–Reisner ideal. The vertices are identified with a subset of the variables in a polynomial ring and each face is identified with the product of the corresponding variables. A nonface is any subset of the variables that does not belong to the simplicial complex and each nonface is again identified with a product of variables. The Stanley-Reisner ideal of a simplicial complex is generated by monomials corresponding to its nonfaces.

- Making an abstract simplicial complex -- information about the basic constructors
- Finding attributes and properties -- information about accessing features of an abstract simplicial complex
- Working with associated chain complexes -- information about the chain complexes and their homogenizations
- Working with simplicial maps -- information about simplicial maps and the induced operations

- bartnetteSphereComplex(PolynomialRing) -- make a non-polytopal 3-sphere with 8 vertices
- barycentricSubdivision(SimplicialComplex,Ring) -- create the barycentric subdivision of a simplicial complex
- bjornerComplex(PolynomialRing) -- make a shellable 2-polyhedron with 6 vertices
- dual(SimplicialComplex) -- make the Alexander dual of an abstract simplicial complex
- dunceHatComplex(PolynomialRing) -- make a realization of the dunce hat with 8 vertices
- elementaryCollapse(SimplicialComplex,RingElement) -- construct the elementary collapse of a free face in a simplicial complex
- grunbaumBallComplex(PolynomialRing) -- make a nonshellable 3-ball with 14 vertices and 29 facets
- image(SimplicialMap) -- construct the image of a simplicial map
- inducedSubcomplex(SimplicialComplex,List) -- make the induced simplicial complex on a subset of vertices
- kleinBottleComplex(PolynomialRing) -- make a minimal triangulation of the Klein bottle
- link(SimplicialComplex,RingElement) -- make the link of a face in an abstract simplicial complex
- nonPiecewiseLinearSphereComplex(PolynomialRing) -- make a non-piecewise-linear 5-sphere with 18 vertices
- poincareSphereComplex(PolynomialRing) -- make a homology 3-sphere with 16 vertices
- prune(SimplicialComplex) -- make a minimal presentation of an abstract simplicial complex
- realProjectiveSpaceComplex(ZZ,PolynomialRing) -- make a small triangulation of real projective space
- rudinBallComplex(PolynomialRing) -- make a nonshellable 3-ball with 14 vertices and 41 facets
- simplexComplex(ZZ,PolynomialRing) -- make the simplex as an abstract simplicial complex
- simplicialComplex(List) -- make an abstract simplicial complex from a list of faces
- simplicialComplex(Matrix) -- see simplicialComplex(List) -- make an abstract simplicial complex from a list of faces
- simplicialComplex(Ideal) -- see simplicialComplex(MonomialIdeal) -- make a simplicial complex from its Stanley–Reisner ideal
- simplicialComplex(MonomialIdeal) -- make a simplicial complex from its Stanley–Reisner ideal
- skeleton(ZZ,SimplicialComplex) -- make a new simplicial complex generated by all faces of a bounded dimension
- smallManifold(ZZ,ZZ,ZZ,PolynomialRing) -- get a small manifold from the Frank Lutz database
- source(SimplicialMap) -- get the source of the map
- substitute(SimplicialComplex,PolynomialRing) -- change the ring of a simplicial complex
- target(SimplicialMap) -- get the target of the map
- wedge(SimplicialComplex,SimplicialComplex,RingElement,RingElement) -- make the wedge sum of two abstract simplicial complexes
- zieglerBallComplex(PolynomialRing) -- make a nonshellable 3-ball with 10 vertices and 21 facets

- algebraicShifting(SimplicialComplex) -- see algebraicShifting -- the algebraic shifting of a simplicial complex
- boundaryMap(ZZ,SimplicialComplex) -- make a boundary map between the oriented faces of an abstract simplicial complex
- chainComplex(SimplicialComplex) -- create the chain complex associated to a simplicial complex.
- coefficientRing(SimplicialComplex) -- get the coefficient ring of the underlying polynomial ring
- connectedComponents(SimplicialComplex) -- find the connected components of an abstract simplicial complex
- dim(SimplicialComplex) -- find the dimension of an abstract simplicial complex
- faces(SimplicialComplex) -- get the list of faces for an abstract simplicial complex
- faces(ZZ,SimplicialComplex) -- get the $i$-faces of an abstract simplicial complex
- facets(SimplicialComplex) -- get the list of maximal faces
- flagfVector(List,SimplicialComplex) -- compute a flag $f$-number of a colored simplicial complex
- flagfVector(SimplicialComplex) -- compute the flag $f$-vector of an colored simplicial complex
- fVector(SimplicialComplex) -- compute the f-vector of an abstract simplicial complex
- HH^ZZ SimplicialComplex -- see HH^ZZ(SimplicialComplex,Ring) -- compute the reduced cohomology of an abstract simplicial complex
- HH^ZZ(SimplicialComplex,Ring) -- compute the reduced cohomology of an abstract simplicial complex
- HH^ZZ(SimplicialComplex,SimplicialComplex) -- compute the relative homology of two simplicial complexes
- HH_ZZ SimplicialComplex -- see HH_ZZ(SimplicialComplex,Ring) -- compute the reduced homology of an abstract simplicial complex
- HH_ZZ(SimplicialComplex,Ring) -- compute the reduced homology of an abstract simplicial complex
- HH SimplicialComplex -- see homology(SimplicialComplex,Ring) -- compute the reduced homology of an abstract simplicial complex
- homology(Nothing,SimplicialComplex) -- see homology(SimplicialComplex,Ring) -- compute the reduced homology of an abstract simplicial complex
- homology(Nothing,SimplicialComplex,Ring) -- see homology(SimplicialComplex,Ring) -- compute the reduced homology of an abstract simplicial complex
- homology(SimplicialComplex,Ring) -- compute the reduced homology of an abstract simplicial complex
- HH_ZZ(SimplicialComplex,SimplicialComplex) -- see homology(SimplicialComplex,SimplicialComplex) -- compute the relative homology of two simplicial complexes
- homology(Nothing,SimplicialComplex,SimplicialComplex) -- see homology(SimplicialComplex,SimplicialComplex) -- compute the relative homology of two simplicial complexes
- homology(SimplicialComplex,SimplicialComplex) -- compute the relative homology of two simplicial complexes
- ideal(SimplicialComplex) -- get the Stanley–Reisner ideal
- isProper(SimplicialComplex) -- whether an abstract simplicial complex is properly colored
- isPure(SimplicialComplex) -- whether the facets are equidimensional
- isWellDefined(SimplicialComplex) -- whether underlying data is uncontradictory
- map(SimplicialComplex,List) -- see map(SimplicialComplex,SimplicialComplex,Matrix) -- create a simplicial map between simplicial complexes
- map(SimplicialComplex,Matrix) -- see map(SimplicialComplex,SimplicialComplex,Matrix) -- create a simplicial map between simplicial complexes
- map(SimplicialComplex,RingMap) -- see map(SimplicialComplex,SimplicialComplex,Matrix) -- create a simplicial map between simplicial complexes
- map(SimplicialComplex,SimplicialComplex,List) -- see map(SimplicialComplex,SimplicialComplex,Matrix) -- create a simplicial map between simplicial complexes
- map(SimplicialComplex,SimplicialComplex,Matrix) -- create a simplicial map between simplicial complexes
- map(SimplicialComplex,SimplicialComplex,RingMap) -- see map(SimplicialComplex,SimplicialComplex,Matrix) -- create a simplicial map between simplicial complexes
- monomialIdeal(SimplicialComplex) -- get the Stanley–Reisner monomial ideal
- net(SimplicialComplex) -- make a symbolic representation of an abstract simplicial complex
- ring(SimplicialComplex) -- get the polynomial ring of its Stanley–Reisner ideal
- SimplicialComplex * SimplicialComplex -- make the join for two abstract simplicial complexes
- star(SimplicialComplex,RingElement) -- make the star of a face
- vertices(SimplicialComplex) -- get the list of the vertices for an abstract simplicial complex

The object SimplicialComplex is a type, with ancestor classes HashTable < Thing.