Simplicial Complexes

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

Bases: sage.categories.category_singleton.Category_singleton

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)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom

Category of finite simplicial complexes.

class ParentMethods
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
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]