AUTHORS:
Bases: sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring
The category of associative and unital algebras over a given base ring.
An associative and unital algebra over a ring is a module over
which is itself a ring.
Warning
Algebras will be eventually be replaced by magmatic_algebras.MagmaticAlgebras for consistency with e.g. Wikipedia article Algebras which assumes neither associativity nor the existence of a unit (see trac ticket #15043).
Todo
Should be a commutative ring?
EXAMPLES:
sage: Algebras(ZZ)
Category of algebras over Integer Ring
sage: sorted(Algebras(ZZ).super_categories(), key=str)
[Category of associative algebras over Integer Ring,
Category of rings,
Category of unital algebras over Integer Ring]
TESTS:
sage: TestSuite(Algebras(ZZ)).run()
Bases: sage.categories.cartesian_product.CartesianProductsCategory
The category of algebras constructed as cartesian products of algebras
This construction gives the direct product of algebras. See discussion on:
A cartesian product of algebras is endowed with a natural algebra structure.
EXAMPLES:
sage: C = Algebras(QQ).CartesianProducts()
sage: C.extra_super_categories()
[Category of algebras over Rational Field]
sage: sorted(C.super_categories(), key=str)
[Category of Cartesian products of commutative additive groups,
Category of Cartesian products of distributive magmas and additive magmas,
Category of Cartesian products of monoids,
Category of algebras over Rational Field]
alias of CommutativeAlgebras
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 algebras over the Rational Field is dual to the category of coalgebras over the same field:
sage: C = Algebras(QQ)
sage: C.dual()
Category of duals of algebras over Rational Field
sage: C.dual().extra_super_categories()
[Category of coalgebras over Rational Field]
Warning
This is only correct in certain cases (finite dimension, ...). See trac ticket #15647.
alias of GradedAlgebras
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()
EXAMPLES:
sage: Algebras(QQ).TensorProducts().extra_super_categories()
[Category of algebras over Rational Field]
sage: Algebras(QQ).TensorProducts().super_categories()
[Category of algebras over Rational Field]
Meaning: a tensor product of algebras is an algebra
alias of AlgebrasWithBasis