Bialgebras

class sage.categories.bialgebras.Bialgebras(base, name=None)

Bases: sage.categories.category_types.Category_over_base_ring

The category of bialgebras

EXAMPLES:

sage: Bialgebras(ZZ)
Category of bialgebras over Integer Ring
sage: Bialgebras(ZZ).super_categories()
[Category of algebras over Integer Ring, Category of coalgebras over Integer Ring]

class Super(base_category)

Bases: sage.categories.super_modules.SuperModulesCategory

EXAMPLES:

sage: C = Algebras(QQ).Super()
sage: C
Category of super algebras over Rational Field
sage: C.base_category()
Category of algebras over Rational Field
sage: sorted(C.super_categories(), key=str)
[Category of graded algebras over Rational Field,
 Category of super modules over Rational Field]

sage: AlgebrasWithBasis(QQ).Super().base_ring()
Rational Field
sage: HopfAlgebrasWithBasis(QQ).Super().base_ring()
Rational Field

Bialgebras.WithBasis

alias of BialgebrasWithBasis

Bialgebras.additional_structure()

Return None.

Indeed, the category of bialgebras defines no additional structure: a morphism of coalgebras and of algebras between two bialgebras is a bialgebra morphism.

See also

Category.additional_structure()

Todo

This category should be a CategoryWithAxiom.

EXAMPLES:

sage: Bialgebras(QQ).additional_structure()

Bialgebras.super_categories()

EXAMPLES:

sage: Bialgebras(QQ).super_categories()
[Category of algebras over Rational Field, Category of coalgebras over Rational Field]