CW Complexes

class sage.categories.cw_complexes.CWComplexes(s=None)

Bases: sage.categories.category_singleton.Category_singleton

The category of CW complexes.

A CW complex is a Closure-finite cell complex in the Weak topology.

REFERENCES:

Note

The notion of “finite” is that the number of cells is finite.

EXAMPLES:

sage: from sage.categories.cw_complexes import CWComplexes
sage: C = CWComplexes(); C
Category of CW complexes
Compact_extra_super_categories()

Return extraneous super categories for CWComplexes().Compact().

A compact CW complex is finite, see Proposition A.1 in [Hat2002].

Todo

Fix the name of finite CW complexes.

EXAMPLES:

sage: from sage.categories.cw_complexes import CWComplexes
sage: CWComplexes().Compact() # indirect doctest
Category of finite finite dimensional CW complexes
sage: CWComplexes().Compact() is CWComplexes().Finite()
True
class Connected(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom

The category of connected CW complexes.

class ElementMethods
dimension()

Return the dimension of self.

EXAMPLES:

sage: from sage.categories.cw_complexes import CWComplexes
sage: X = CWComplexes().example()
sage: X.an_element().dimension()
2
class Finite(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom

Category of finite CW complexes.

A finite CW complex is a CW complex with a finite number of cells.

class ParentMethods
dimension()

Return the dimension of self.

EXAMPLES:

sage: from sage.categories.cw_complexes import CWComplexes
sage: X = CWComplexes().example()
sage: X.dimension()
2
extra_super_categories()

Return the extra super categories of self.

A finite CW complex is a compact finite-dimensional CW complex.

EXAMPLES:

sage: from sage.categories.cw_complexes import CWComplexes
sage: C = CWComplexes().Finite()
sage: C.extra_super_categories()
[Category of finite dimensional CW complexes,
 Category of compact topological spaces]
class FiniteDimensional(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom

Category of finite dimensional CW complexes.

class ParentMethods
cells()

Return the cells of self.

EXAMPLES:

sage: from sage.categories.cw_complexes import CWComplexes
sage: X = CWComplexes().example()
sage: C = X.cells()
sage: sorted((d, C[d]) for d in C.keys())
[(0, (0-cell v,)),
 (1, (0-cell e1, 0-cell e2)),
 (2, (2-cell f,))]
dimension()

Return the dimension of self.

EXAMPLES:

sage: from sage.categories.cw_complexes import CWComplexes
sage: X = CWComplexes().example()
sage: X.dimension()
2
class SubcategoryMethods
Connected()

Return the full subcategory of the connected objects of self.

EXAMPLES:

sage: from sage.categories.cw_complexes import CWComplexes
sage: CWComplexes().Connected()
Category of connected CW complexes
FiniteDimensional()

Return the full subcategory of the finite dimensional objects of self.

EXAMPLES:

sage: from sage.categories.cw_complexes import CWComplexes
sage: C = CWComplexes().FiniteDimensional(); C
Category of finite dimensional CW complexes
super_categories()

EXAMPLES:

sage: from sage.categories.cw_complexes import CWComplexes
sage: CWComplexes().super_categories()
[Category of topological spaces]