Simplicial Complexes#

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

Bases: 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 Connected(base_category)#

Bases: CategoryWithAxiom

The category of connected simplicial complexes.

EXAMPLES:

sage: from sage.categories.simplicial_complexes import SimplicialComplexes
sage: C = SimplicialComplexes().Connected()
sage: TestSuite(C).run()
class Finite(base_category)#

Bases: CategoryWithAxiom

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]])           # optional - sage.graphs
sage: S.dimension()                                                 # optional - sage.graphs
2
class ParentMethods#

Bases: object

faces()#

Return the faces of self.

EXAMPLES:

sage: S = SimplicialComplex([[1,3,4], [1,2],[2,5],[4,5]])               # optional - sage.graphs
sage: S.faces()                                                         # optional - sage.graphs
{-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]])               # optional - sage.graphs
sage: sorted(S.facets())                                                # optional - sage.graphs
[(1, 2), (1, 3, 4), (2, 5), (4, 5)]
class SubcategoryMethods#

Bases: object

Connected()#

Return the full subcategory of the connected objects of self.

EXAMPLES:

sage: from sage.categories.simplicial_complexes import SimplicialComplexes
sage: SimplicialComplexes().Connected()
Category of connected simplicial complexes
super_categories()#

Return the super categories of self.

EXAMPLES:

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