# Simplicial Complexes¶

class sage.categories.simplicial_complexes.SimplicialComplexes(s=None)

The category of abstract simplicial complexes.

An abstract simplicial complex $$A$$ is a collection of sets $$X$$ such that:

• $$\emptyset \in A$$,
• if $$X \subset Y \in A$$, then $$X \in A$$.

Todo

Implement the category of simplicial complexes considered as CW complexes and rename this to the category of AbstractSimplicialComplexes with appropriate functors.

EXAMPLES:

sage: from sage.categories.simplicial_complexes import SimplicialComplexes
sage: C = SimplicialComplexes(); C
Category of simplicial complexes

class Finite(base_category)

Category of finite simplicial complexes.

class ParentMethods

Bases: object

dimension()

Return the dimension of self.

EXAMPLES:

sage: S = SimplicialComplex([[1,3,4], [1,2],[2,5],[4,5]])
sage: S.dimension()
2

class ParentMethods

Bases: object

faces()

Return the faces of self.

EXAMPLES:

sage: S = SimplicialComplex([[1,3,4], [1,2],[2,5],[4,5]])
sage: S.faces()
{-1: {()},
0: {(1,), (2,), (3,), (4,), (5,)},
1: {(1, 2), (1, 3), (1, 4), (2, 5), (3, 4), (4, 5)},
2: {(1, 3, 4)}}

facets()

Return the facets of self.

EXAMPLES:

sage: S = SimplicialComplex([[1,3,4], [1,2],[2,5],[4,5]])
sage: sorted(S.facets())
[(1, 2), (1, 3, 4), (2, 5), (4, 5)]

super_categories()

EXAMPLES:

sage: from sage.categories.simplicial_complexes import SimplicialComplexes
sage: SimplicialComplexes().super_categories()
[Category of sets]