chain_complex_homspace.py

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
    {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], -1: [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 <[email protected]>
#
#  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
import sage.homology.chain_complex_morphism as chain_complex_morphism

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(17,[[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(17,[[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 chain_complex_morphism.ChainComplexMorphism(f, self.domain(), self.codomain())