simplicial_complex_homset.py

r"""
Homsets between simplicial complexes

AUTHORS:

- Travis Scrimshaw (2012-08-18): Made all simplicial complexes immutable to
  work with the homset cache.

EXAMPLES:

::

    sage: S = simplicial_complexes.Sphere(1)
    sage: T = simplicial_complexes.Sphere(2)
    sage: H = Hom(S,T)
    sage: f = {0:0,1:1,2:3}
    sage: x = H(f)
    sage: x
    Simplicial complex morphism {0: 0, 1: 1, 2: 3} from Simplicial complex with vertex set (0, 1, 2) and facets {(1, 2), (0, 2), (0, 1)} to Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (0, 1, 2), (1, 2, 3), (0, 1, 3)}
    sage: x.is_injective()
    True
    sage: x.is_surjective()
    False
    sage: x.image()
    Simplicial complex with vertex set (0, 1, 3) and facets {(1, 3), (0, 3), (0, 1)}
    sage: from sage.homology.simplicial_complex import Simplex
    sage: s = Simplex([1,2])
    sage: x(s)
    (1, 3)

TESTS::

    sage: S = simplicial_complexes.Sphere(1)
    sage: T = simplicial_complexes.Sphere(2)
    sage: H = Hom(S,T)
    sage: loads(dumps(H))==H
    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.simplicial_complex_morphism as simplicial_complex_morphism

def is_SimplicialComplexHomset(x):
    """
    Return ``True`` if and only if ``x`` is a simplicial complex homspace.

    EXAMPLES::

        sage: S = SimplicialComplex(is_mutable=False)
        sage: T = SimplicialComplex(is_mutable=False)
        sage: H = Hom(S, T)
        sage: H
        Set of Morphisms from Simplicial complex with vertex set () and facets {()} to Simplicial complex with vertex set () and facets {()} in Category of simplicial complexes
        sage: from sage.homology.simplicial_complex_homset import is_SimplicialComplexHomset
        sage: is_SimplicialComplexHomset(H)
        True
    """
    return isinstance(x, SimplicialComplexHomset)

class SimplicialComplexHomset(sage.categories.homset.Homset):
    def __call__(self, f):
        """
        INPUT:

        - ``f`` -- a dictionary with keys exactly the vertices of the domain
           and values vertices of the codomain

        EXAMPLES::

            sage: S = simplicial_complexes.Sphere(3)
            sage: T = simplicial_complexes.Sphere(2)
            sage: f = {0:0,1:1,2:2,3:2,4:2}
            sage: H = Hom(S,T)
            sage: x = H(f)
            sage: x
            Simplicial complex morphism {0: 0, 1: 1, 2: 2, 3: 2, 4: 2} from Simplicial complex with vertex set (0, 1, 2, 3, 4) and 5 facets to Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (0, 1, 2), (1, 2, 3), (0, 1, 3)}
        """
        return simplicial_complex_morphism.SimplicialComplexMorphism(f,self.domain(),self.codomain())

    def diagonal_morphism(self,rename_vertices=True):
        r"""
        Returns the diagonal morphism in `Hom(S, S \times S)`.

        EXAMPLES::

            sage: S = simplicial_complexes.Sphere(2)
            sage: H = Hom(S,S.product(S, is_mutable=False))
            sage: d = H.diagonal_morphism()
            sage: d
            Simplicial complex morphism {0: 'L0R0', 1: 'L1R1', 2: 'L2R2', 3: 'L3R3'} from
            Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (0, 1, 2), (1, 2, 3), (0, 1, 3)}
            to Simplicial complex with 16 vertices and 96 facets

            sage: T = SimplicialComplex([[0], [1]], is_mutable=False)
            sage: U = T.product(T,rename_vertices = False, is_mutable=False)
            sage: G = Hom(T,U)
            sage: e = G.diagonal_morphism(rename_vertices = False)
            sage: e
            Simplicial complex morphism {0: (0, 0), 1: (1, 1)} from
            Simplicial complex with vertex set (0, 1) and facets {(0,), (1,)}
            to Simplicial complex with 4 vertices and facets {((1, 1),), ((1, 0),), ((0, 0),), ((0, 1),)}
        """

        if self._codomain == self._domain.product(self._domain,rename_vertices=rename_vertices):
            X = self._domain.product(self._domain,rename_vertices=rename_vertices)
            f = dict()
            if rename_vertices:
                for i in self._domain.vertices().set():
                    f[i] = "L"+str(i)+"R"+str(i)
            else:
                for i in self._domain.vertices().set():
                    f[i] = (i,i)
            return simplicial_complex_morphism.SimplicialComplexMorphism(f, self._domain,X)
        else:
            raise TypeError, "Diagonal morphism is only defined for Hom(X,XxX)."

    def identity(self):
        """
        Returns the identity morphism of `Hom(S,S)`.

        EXAMPLES::

            sage: S = simplicial_complexes.Sphere(2)
            sage: H = Hom(S,S)
            sage: i = H.identity()
            sage: i.is_identity()
            True

            sage: T = SimplicialComplex([[0,1]], is_mutable=False)
            sage: G = Hom(T,T)
            sage: G.identity()
            Simplicial complex morphism {0: 0, 1: 1} from
            Simplicial complex with vertex set (0, 1) and facets {(0, 1)} to
            Simplicial complex with vertex set (0, 1) and facets {(0, 1)}
        """
        if self.is_endomorphism_set():
            f = dict()
            for i in self._domain.vertices().set():
                f[i]=i
            return simplicial_complex_morphism.SimplicialComplexMorphism(f,self._domain,self._codomain)
        else:
            raise TypeError("Identity map is only defined for endomorphism sets.")

    def an_element(self):
        """
        Returns a (non-random) element of ``self``.

        EXAMPLES::

            sage: S = simplicial_complexes.KleinBottle()
            sage: T = simplicial_complexes.Sphere(5)
            sage: H = Hom(S,T)
            sage: x = H.an_element()
            sage: x
            Simplicial complex morphism {0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: 0, 6: 0, 7: 0} from Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6, 7) and 16 facets to Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6) and 7 facets
        """
        X_vertices = self._domain.vertices().set()
        try:
            i = self._codomain.vertices().set().__iter__().next()
        except StopIteration:
            if len(X_vertices) == 0:
                return dict()
            else:
                raise TypeError, "There are no morphisms from a non-empty simplicial complex to an empty simplicial comples."
        f = dict()
        for x in X_vertices:
            f[x]=i
        return simplicial_complex_morphism.SimplicialComplexMorphism(f,self._domain,self._codomain)