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

r""" 

Homspaces between chain complexes 

Note that some significant functionality is lacking. Namely, the homspaces 

are not actually modules over the base ring. It will be necessary to 

enrich some of the structure of chain complexes for this to be naturally 

available. On other hand, there are various overloaded operators. ``__mul__`` 

acts as composition. One can ``__add__``, and one can ``__mul__`` with a ring 

element on the right. 

EXAMPLES:: 

sage: S = simplicial_complexes.Sphere(2) 

sage: T = simplicial_complexes.Torus() 

sage: C = S.chain_complex(augmented=True,cochain=True) 

sage: D = T.chain_complex(augmented=True,cochain=True) 

sage: G = Hom(C,D) 

sage: G 

Set of Morphisms from Chain complex with at most 4 nonzero terms over Integer Ring to Chain complex with at most 4 nonzero terms over Integer Ring in Category of chain complexes over Integer Ring 

sage: S = simplicial_complexes.ChessboardComplex(3,3) 

sage: H = Hom(S,S) 

sage: i = H.identity() 

sage: x = i.associated_chain_complex_morphism(augmented=True) 

sage: x 

Chain complex morphism: 

From: Chain complex with at most 4 nonzero terms over Integer Ring 

To: Chain complex with at most 4 nonzero terms over Integer Ring 

sage: x._matrix_dictionary 

{-1: [1], 0: [1 0 0 0 0 0 0 0 0] 

[0 1 0 0 0 0 0 0 0] 

[0 0 1 0 0 0 0 0 0] 

[0 0 0 1 0 0 0 0 0] 

[0 0 0 0 1 0 0 0 0] 

[0 0 0 0 0 1 0 0 0] 

[0 0 0 0 0 0 1 0 0] 

[0 0 0 0 0 0 0 1 0] 

[0 0 0 0 0 0 0 0 1], 1: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] 

[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] 

[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] 

[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0] 

[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0] 

[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0] 

[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0] 

[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0] 

[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0] 

[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0] 

[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0] 

[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0] 

[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0] 

[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0] 

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0] 

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0] 

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0] 

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1], 2: [1 0 0 0 0 0] 

[0 1 0 0 0 0] 

[0 0 1 0 0 0] 

[0 0 0 1 0 0] 

[0 0 0 0 1 0] 

[0 0 0 0 0 1]} 

sage: S = simplicial_complexes.Sphere(2) 

sage: A = Hom(S,S) 

sage: i = A.identity() 

sage: x = i.associated_chain_complex_morphism() 

sage: x 

Chain complex morphism: 

From: Chain complex with at most 3 nonzero terms over Integer Ring 

To: Chain complex with at most 3 nonzero terms over Integer Ring 

sage: y = x*4 

sage: z = y*y 

sage: (y+z) 

Chain complex morphism: 

From: Chain complex with at most 3 nonzero terms over Integer Ring 

To: Chain complex with at most 3 nonzero terms over Integer Ring 

sage: f = x._matrix_dictionary 

sage: C = S.chain_complex() 

sage: G = Hom(C,C) 

sage: w = G(f) 

sage: w == x 

True 

""" 

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

# Copyright (C) 2009 D. Benjamin Antieau <d.ben.antieau@gmail.com> 

# 

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

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty 

# of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. 

# 

# See the GNU General Public License for more details; the full text 

# is available at: 

# 

# http://www.gnu.org/licenses/ 

# 

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

import sage.categories.homset 

from sage.homology.chain_complex_morphism import ChainComplexMorphism 

def is_ChainComplexHomspace(x): 

""" 

Returns ``True`` if and only if ``x`` is a morphism of chain complexes. 

EXAMPLES:: 

sage: from sage.homology.chain_complex_homspace import is_ChainComplexHomspace 

sage: T = SimplicialComplex([[1,2,3,4],[7,8,9]]) 

sage: C = T.chain_complex(augmented=True, cochain=True) 

sage: G = Hom(C,C) 

sage: is_ChainComplexHomspace(G) 

True 

""" 

return isinstance(x,ChainComplexHomspace) 

class ChainComplexHomspace(sage.categories.homset.Homset): 

""" 

Class of homspaces of chain complex morphisms. 

EXAMPLES:: 

sage: T = SimplicialComplex([[1,2,3,4],[7,8,9]]) 

sage: C = T.chain_complex(augmented=True, cochain=True) 

sage: G = Hom(C,C) 

sage: G 

Set of Morphisms from Chain complex with at most 5 nonzero terms over Integer Ring to Chain complex with at most 5 nonzero terms over Integer Ring in Category of chain complexes over Integer Ring 

""" 

def __call__(self, f): 

""" 

`f` is a dictionary of matrices in the basis of the chain complex. 

EXAMPLES:: 

sage: S = simplicial_complexes.Sphere(5) 

sage: H = Hom(S,S) 

sage: i = H.identity() 

sage: C = S.chain_complex() 

sage: G = Hom(C,C) 

sage: x = i.associated_chain_complex_morphism() 

sage: f = x._matrix_dictionary 

sage: y = G(f) 

sage: x == y 

True 

""" 

return ChainComplexMorphism(f, self.domain(), self.codomain())