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# Morphisms of chain complexes

AUTHORS:

- Benjamin Antieau <d.ben.antieau@gmail.com> (2009.06)
- Travis Scrimshaw (2012-08-18): Made all simplicial complexes immutable to work with the homset cache.

This module implements morphisms of chain complexes. The input is a dictionary whose keys are in the grading group of the chain complex and whose values are matrix morphisms.

EXAMPLES:

    from sage.matrix.constructor import zero_matrix
    sage: S = simplicial_complexes.Sphere(1)
    sage: S
    Simplicial complex with vertex set (0, 1, 2) and facets {(1, 2), (0, 2), (0, 1)}
    sage: C = S.chain_complex()
    sage: C.differential()
    {0: [], 1: [ 1  1  0]
    [ 0 -1 -1]
    [-1  0  1], 2: []}
    sage: f = {0:zero_matrix(ZZ,3,3),1:zero_matrix(ZZ,3,3)}
    sage: G = Hom(C,C)
    sage: x = G(f)
    sage: x
    Chain complex morphism from Chain complex with at most 2 nonzero terms over Integer Ring to Chain complex with at most 2 nonzero terms over Integer Ring
    sage: x._matrix_dictionary
    {0: [0 0 0]
    [0 0 0]
    [0 0 0], 1: [0 0 0]
    [0 0 0]
    [0 0 0]}

class `sage.homology.chain_complex_morphism.``ChainComplexMorphism`(matrices, C, D, check=True)

  Bases: `sage.structure.sage_object.SageObject`

  An element of this class is a morphism of chain complexes.

`sage.homology.chain_complex_morphism.``is_ChainComplexMorphism`(x)

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

  EXAMPLES:

      sage: from sage.homology.chain_complex_morphism import is_ChainComplexMorphism
      sage: S = simplicial_complexes.Sphere(14)
      sage: H = Hom(S,S)
      sage: i = H.identity()  # long time (8s on sage.math, 2011)
      sage: S = simplicial_complexes.Sphere(6)
      sage: H = Hom(S,S)
      sage: i = H.identity()
      sage: x = i.associated_chain_complex_morphism()
      sage: x # indirect doctest
      Chain complex morphism from Chain complex with at most 7 nonzero terms over
      Integer Ring to Chain complex with at most 7 nonzero terms over Integer Ring
      sage: is_ChainComplexMorphism(x)
      True

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