# Bialgebras¶

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

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]


TESTS:

sage: TestSuite(Bialgebras(ZZ)).run()

class ElementMethods
class Bialgebras.ParentMethods

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.

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]


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Associative algebras

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Bialgebras with basis