Bases: sage.categories.category_types.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]
TESTS:
sage: TestSuite(Coalgebras(ZZ)).run()
Bases: sage.categories.dual.DualObjectsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
sage: class FooBars(CovariantConstructionCategory):
... _functor_category = "FooBars"
sage: Category.FooBars = lambda self: FooBars.category_of(self)
sage: C = FooBars(ModulesWithBasis(ZZ))
sage: C
Category of foo bars of modules with basis over Integer Ring
sage: C.base_category()
Category of modules with basis over Integer Ring
sage: latex(C)
\mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
sage: TestSuite(C).run()
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]
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)])
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)
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)])
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.
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
Bases: sage.categories.realizations.RealizationsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
sage: class FooBars(CovariantConstructionCategory):
... _functor_category = "FooBars"
sage: Category.FooBars = lambda self: FooBars.category_of(self)
sage: C = FooBars(ModulesWithBasis(ZZ))
sage: C
Category of foo bars of modules with basis over Integer Ring
sage: C.base_category()
Category of modules with basis over Integer Ring
sage: latex(C)
\mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
sage: TestSuite(C).run()
alias of Realizations.ParentMethods
Bases: sage.categories.tensor.TensorProductsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
sage: class FooBars(CovariantConstructionCategory):
... _functor_category = "FooBars"
sage: Category.FooBars = lambda self: FooBars.category_of(self)
sage: C = FooBars(ModulesWithBasis(ZZ))
sage: C
Category of foo bars of modules with basis over Integer Ring
sage: C.base_category()
Category of modules with basis over Integer Ring
sage: latex(C)
\mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
sage: TestSuite(C).run()
alias of TensorProducts.ElementMethods
alias of TensorProducts.ParentMethods
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
Bases: sage.categories.with_realizations.WithRealizationsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
sage: class FooBars(CovariantConstructionCategory):
... _functor_category = "FooBars"
sage: Category.FooBars = lambda self: FooBars.category_of(self)
sage: C = FooBars(ModulesWithBasis(ZZ))
sage: C
Category of foo bars of modules with basis over Integer Ring
sage: C.base_category()
Category of modules with basis over Integer Ring
sage: latex(C)
\mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
sage: TestSuite(C).run()
alias of WithRealizations.ParentMethods
EXAMPLES:
sage: Coalgebras(QQ).super_categories()
[Category of vector spaces over Rational Field]