Coverage for local/lib/python2.7/site-packages/sage/categories/cw_complexes.py : 100%

r""" 

CW Complexes 

""" 

#***************************************************************************** 

# Copyright (C) 2015 Travis Scrimshaw <tscrim at ucdavis.edu> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#****************************************************************************** 

from sage.misc.abstract_method import abstract_method 

from sage.misc.cachefunc import cached_method 

from sage.categories.category_singleton import Category_singleton 

from sage.categories.category_with_axiom import CategoryWithAxiom 

from sage.categories.sets_cat import Sets 

class CWComplexes(Category_singleton): 

r""" 

The category of CW complexes. 

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

REFERENCES: 

- :wikipedia:`CW_complex` 

.. 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 

TESTS:: 

sage: TestSuite(C).run() 

""" 

@cached_method 

def super_categories(self): 

""" 

EXAMPLES:: 

sage: from sage.categories.cw_complexes import CWComplexes 

sage: CWComplexes().super_categories() 

[Category of topological spaces] 

""" 

return [Sets().Topological()] 

def _repr_object_names(self): 

""" 

EXAMPLES:: 

sage: from sage.categories.cw_complexes import CWComplexes 

sage: CWComplexes() # indirect doctest 

Category of CW complexes 

""" 

return "CW complexes" 

class SubcategoryMethods: 

@cached_method 

def Connected(self): 

""" 

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 

TESTS:: 

sage: TestSuite(CWComplexes().Connected()).run() 

sage: CWComplexes().Connected.__module__ 

'sage.categories.cw_complexes' 

""" 

return self._with_axiom('Connected') 

@cached_method 

def FiniteDimensional(self): 

""" 

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 

TESTS:: 

sage: from sage.categories.cw_complexes import CWComplexes 

sage: C = CWComplexes().FiniteDimensional() 

sage: TestSuite(C).run() 

sage: CWComplexes().Connected().FiniteDimensional.__module__ 

'sage.categories.cw_complexes' 

""" 

return self._with_axiom('FiniteDimensional') 

class Connected(CategoryWithAxiom): 

""" 

The category of connected CW complexes. 

""" 

class FiniteDimensional(CategoryWithAxiom): 

""" 

Category of finite dimensional CW complexes. 

""" 

class Finite(CategoryWithAxiom): 

""" 

Category of finite CW complexes. 

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

""" 

def extra_super_categories(self): 

""" 

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] 

""" 

return [CWComplexes().FiniteDimensional(), Sets().Topological().Compact()] 

class ParentMethods: 

@cached_method 

def dimension(self): 

""" 

Return the dimension of ``self``. 

EXAMPLES:: 

sage: from sage.categories.cw_complexes import CWComplexes 

sage: X = CWComplexes().example() 

sage: X.dimension() 

2 

""" 

C = self.cells() 

return max(c.dimension() for d in C.keys() for c in C[d]) 

def Compact_extra_super_categories(self): 

""" 

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 

""" 

return (Sets().Finite(),) 

class ElementMethods: 

@abstract_method 

def dimension(self): 

""" 

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 ParentMethods: 

@abstract_method 

def dimension(self): 

""" 

Return the dimension of ``self``. 

EXAMPLES:: 

sage: from sage.categories.cw_complexes import CWComplexes 

sage: X = CWComplexes().example() 

sage: X.dimension() 

2 

""" 

@abstract_method(optional=True) 

def cells(self): 

""" 

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,))] 

"""