Simplicial Complexes¶
- class sage.categories.simplicial_complexes.SimplicialComplexes[source]¶
- 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 complexesand rename this to the category of- AbstractSimplicialComplexeswith appropriate functors.- EXAMPLES: - sage: from sage.categories.simplicial_complexes import SimplicialComplexes sage: C = SimplicialComplexes(); C Category of simplicial complexes - >>> from sage.all import * >>> from sage.categories.simplicial_complexes import SimplicialComplexes >>> C = SimplicialComplexes(); C Category of simplicial complexes - class Connected(base_category)[source]¶
- 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() - >>> from sage.all import * >>> from sage.categories.simplicial_complexes import SimplicialComplexes >>> C = SimplicialComplexes().Connected() >>> TestSuite(C).run() 
 - class Finite(base_category)[source]¶
- Bases: - CategoryWithAxiom- Category of finite simplicial complexes. - class ParentMethods[source]¶
- Bases: - object- dimension()[source]¶
- Return the dimension of - self.- EXAMPLES: - sage: S = SimplicialComplex([[1,3,4], [1,2],[2,5],[4,5]]) # needs sage.graphs sage: S.dimension() # needs sage.graphs 2 - >>> from sage.all import * >>> S = SimplicialComplex([[Integer(1),Integer(3),Integer(4)], [Integer(1),Integer(2)],[Integer(2),Integer(5)],[Integer(4),Integer(5)]]) # needs sage.graphs >>> S.dimension() # needs sage.graphs 2 
 
 
 - class ParentMethods[source]¶
- Bases: - object- faces()[source]¶
- Return the faces of - self.- EXAMPLES: - sage: S = SimplicialComplex([[1,3,4], [1,2],[2,5],[4,5]]) # needs sage.graphs sage: S.faces() # needs 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)}} - >>> from sage.all import * >>> S = SimplicialComplex([[Integer(1),Integer(3),Integer(4)], [Integer(1),Integer(2)],[Integer(2),Integer(5)],[Integer(4),Integer(5)]]) # needs sage.graphs >>> S.faces() # needs 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()[source]¶
- Return the facets of - self.- EXAMPLES: - sage: S = SimplicialComplex([[1,3,4], [1,2],[2,5],[4,5]]) # needs sage.graphs sage: sorted(S.facets()) # needs sage.graphs [(1, 2), (1, 3, 4), (2, 5), (4, 5)] - >>> from sage.all import * >>> S = SimplicialComplex([[Integer(1),Integer(3),Integer(4)], [Integer(1),Integer(2)],[Integer(2),Integer(5)],[Integer(4),Integer(5)]]) # needs sage.graphs >>> sorted(S.facets()) # needs sage.graphs [(1, 2), (1, 3, 4), (2, 5), (4, 5)] 
 
 - class SubcategoryMethods[source]¶
- Bases: - object- Connected()[source]¶
- 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 - >>> from sage.all import * >>> from sage.categories.simplicial_complexes import SimplicialComplexes >>> SimplicialComplexes().Connected() Category of connected simplicial complexes 
 
 - super_categories()[source]¶
- Return the super categories of - self.- EXAMPLES: - sage: from sage.categories.simplicial_complexes import SimplicialComplexes sage: SimplicialComplexes().super_categories() [Category of sets] - >>> from sage.all import * >>> from sage.categories.simplicial_complexes import SimplicialComplexes >>> SimplicialComplexes().super_categories() [Category of sets]