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

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); 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); 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(); 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

Python

>>> from sage.all import *
>>> S = simplicial_complexes.Sphere(Integer(2))
>>> T = simplicial_complexes.Torus()
>>> C = S.chain_complex(augmented=True, cochain=True)
>>> D = T.chain_complex(augmented=True, cochain=True)
>>> G = Hom(C, D); 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

>>> S = simplicial_complexes.ChessboardComplex(Integer(3), Integer(3))
>>> H = Hom(S,S)
>>> i = H.identity()
>>> x = i.associated_chain_complex_morphism(augmented=True); 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
>>> 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]}

>>> S = simplicial_complexes.Sphere(Integer(2))
>>> A = Hom(S, S)
>>> i = A.identity()
>>> x = i.associated_chain_complex_morphism(); 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
>>> y = x*Integer(4)
>>> z = y*y
>>> 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
>>> f = x._matrix_dictionary
>>> C = S.chain_complex()
>>> G = Hom(C, C)
>>> w = G(f)
>>> w == x
True

class sage.homology.chain_complex_homspace.ChainComplexHomspace(X, Y, category=None, base=None, check=True)[source]#

Bases: Homset

Class of homspaces of chain complex morphisms.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> T = SimplicialComplex([[Integer(1),Integer(2),Integer(3),Integer(4)],[Integer(7),Integer(8),Integer(9)]])
>>> C = T.chain_complex(augmented=True, cochain=True)
>>> G = Hom(C, C)
>>> 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

sage.homology.chain_complex_homspace.is_ChainComplexHomspace(x)[source]#

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

EXAMPLES:

Sage

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)
doctest:warning...
DeprecationWarning: The function is_ChainComplexHomspace is deprecated;
use 'isinstance(..., ChainComplexHomspace)' instead.
See https://github.com/sagemath/sage/issues/38184 for details.
True

Python

>>> from sage.all import *
>>> from sage.homology.chain_complex_homspace import is_ChainComplexHomspace
>>> T = SimplicialComplex([[Integer(1),Integer(2),Integer(3),Integer(4)],[Integer(7),Integer(8),Integer(9)]])
>>> C = T.chain_complex(augmented=True, cochain=True)
>>> G = Hom(C, C)
>>> is_ChainComplexHomspace(G)
doctest:warning...
DeprecationWarning: The function is_ChainComplexHomspace is deprecated;
use 'isinstance(..., ChainComplexHomspace)' instead.
See https://github.com/sagemath/sage/issues/38184 for details.
True