Simplicial Complexes

class sage.categories.simplicial_complexes.SimplicialComplexes

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

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

Python

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

class Connected(base_category)

Bases: CategoryWithAxiom

The category of connected simplicial complexes.

EXAMPLES:

Sage

sage: from sage.categories.simplicial_complexes import SimplicialComplexes
sage: C = SimplicialComplexes().Connected()
sage: TestSuite(C).run()

Python

>>> from sage.all import *
>>> from sage.categories.simplicial_complexes import SimplicialComplexes
>>> C = SimplicialComplexes().Connected()
>>> 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

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

Python

>>> 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

Bases: object

faces()

Return the faces of self.

EXAMPLES:

Sage

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)}}

Python

>>> 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()

Return the facets of self.

EXAMPLES:

Sage

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)]

Python

>>> 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

Bases: object

Connected()

Return the full subcategory of the connected objects of self.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.categories.simplicial_complexes import SimplicialComplexes
>>> SimplicialComplexes().Connected()
Category of connected simplicial complexes

super_categories()

Return the super categories of self.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.categories.simplicial_complexes import SimplicialComplexes
>>> SimplicialComplexes().super_categories()
[Category of sets]