coalgebras.py

r"""
Coalgebras
"""
#*****************************************************************************
#  Copyright (C) 2008 Teresa Gomez-Diaz (CNRS) <[email protected]>
#  Copyright (C) 2008-2009 Nicolas M. Thiery <nthiery at users.sf.net>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#                  http://www.gnu.org/licenses/
#******************************************************************************

from category_types import Category_over_base_ring
from sage.categories.all import Modules, Algebras
from sage.categories.tensor import TensorProductsCategory, tensor
from sage.categories.dual import DualObjectsCategory
from sage.misc.abstract_method import abstract_method
from sage.misc.cachefunc import cached_method

class Coalgebras(Category_over_base_ring):
    """
    The category of coalgebras

    EXAMPLES::

        sage: Coalgebras(QQ)
        Category of coalgebras over Rational Field
        sage: Coalgebras(QQ).super_categories()
        [Category of vector spaces over Rational Field]
        sage: Coalgebras(QQ).all_super_categories()
        [Category of coalgebras over Rational Field,
         Category of vector spaces over Rational Field,
         Category of modules over Rational Field,
         Category of bimodules over Rational Field on the left and Rational Field on the right,
         Category of left modules over Rational Field,
         Category of right modules over Rational Field,
         Category of commutative additive groups,
         Category of commutative additive monoids,
         Category of commutative additive semigroups,
         Category of additive magmas,
         Category of sets,
         Category of sets with partial maps,
         Category of objects]

    TESTS::

        sage: TestSuite(Coalgebras(ZZ)).run()
    """
    def super_categories(self):
        """
        EXAMPLES::

            sage: Coalgebras(QQ).super_categories()
            [Category of vector spaces over Rational Field]
        """
        return [Modules(self.base_ring())]

    class ParentMethods:
        #def __init_add__(self): # The analogue of initDomainAdd
        #    # Will declare the coproduct of self to the coercion mechanism when it exists
        #    pass

        @cached_method
        def tensor_square(self):
            """
            Returns the tensor square of ``self``

            EXAMPLES::

                sage: A = HopfAlgebrasWithBasis(QQ).example()
                sage: A.tensor_square()
                An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field # An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
            """
            return tensor([self, self])

        @abstract_method
        def counit(self, x):
            """
            Returns the counit of x.

            Eventually, there will be a default implementation,
            delegating to the overloading mechanism and forcing the
            conversion back

            EXAMPLES::

                sage: A = HopfAlgebrasWithBasis(QQ).example(); A
                An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
                sage: [a,b] = A.algebra_generators()
                sage: a, A.counit(a)
                (B[(1,2,3)], 1)
                sage: b, A.counit(b)
                (B[(1,3)], 1)

            TODO: implement some tests of the axioms of coalgebras, bialgebras
            and Hopf algebras using the counit.
            """
            

        @abstract_method
        def coproduct(self, x):
            """
            Returns the coproduct of x.

            Eventually, there will be a default implementation,
            delegating to the overloading mechanism and forcing the
            conversion back

            EXAMPLES::

                sage: A = HopfAlgebrasWithBasis(QQ).example(); A
                An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
                sage: [a,b] = A.algebra_generators()
                sage: a, A.coproduct(a)
                (B[(1,2,3)], B[(1,2,3)] # B[(1,2,3)])
                sage: b, A.coproduct(b)
                (B[(1,3)], B[(1,3)] # B[(1,3)])
            """
            #return self.tensor_square()(overloaded_coproduct(x))

    class ElementMethods:
        def coproduct(self):
            """
            Returns the coproduct of ``self``

            EXAMPLES::

                sage: A = HopfAlgebrasWithBasis(QQ).example(); A
                An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
                sage: [a,b] = A.algebra_generators()
                sage: a, a.coproduct()
                (B[(1,2,3)], B[(1,2,3)] # B[(1,2,3)])
                sage: b, b.coproduct()
                (B[(1,3)], B[(1,3)] # B[(1,3)])
            """
            return self.parent().coproduct(self)

        def counit(self):
            """
            Returns the counit of ``self``

            EXAMPLES::

                sage: A = HopfAlgebrasWithBasis(QQ).example(); A
                An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
                sage: [a,b] = A.algebra_generators()
                sage: a, a.counit()
                (B[(1,2,3)], 1)
                sage: b, b.counit()
                (B[(1,3)], 1)
            """
            return self.parent().counit(self)

    class TensorProducts(TensorProductsCategory):

        @cached_method
        def extra_super_categories(self):
            """
            EXAMPLES::

                sage: Coalgebras(QQ).TensorProducts().extra_super_categories()
                [Category of coalgebras over Rational Field]
                sage: Coalgebras(QQ).TensorProducts().super_categories()
                [Category of coalgebras over Rational Field]

            Meaning: a tensor product of coalgebras is a coalgebra
            """
            return [self.base_category()]

        class ParentMethods:
            # TODO: provide this default implementation of one if one_basis is not implemented
            #def one(self):
            #    return tensor(module.one() for module in self.modules)
            pass

        class ElementMethods:
            pass

    class DualObjects(DualObjectsCategory):

        def extra_super_categories(self):
            r"""
            Returns the dual category

            EXAMPLES:

            The category of coalgebras over the Rational Field is dual
            to the category of algebras over the same field::

                sage: C = Coalgebras(QQ)
                sage: C.dual()
                Category of duals of coalgebras over Rational Field
                sage: C.dual().super_categories() # indirect doctest
                [Category of algebras over Rational Field, Category of duals of vector spaces over Rational Field]

            """
            from sage.categories.algebras import Algebras
            return [Algebras(self.base_category().base_ring())]