# nLab bialgebra

### Context

#### Algebra

higher algebra

universal algebra

# Bialgebra

## Idea

A bialgebra (or bigebra) is both an algebra and a coalgebra, where the operations of either one are homomorphisms for the other. A bialgebra structure on an associative algebra is precisely such as to make its category of modules into a monoidal category equipped with a fiber functor.

A bialgebra is one of the ingredients in the concept of Hopf algebra.

## Definition

A bialgebra is a monoid in the category of coalgebras. Equivalently, it is a comonoid in the category of algebras. Equivalently, it is a monoid in the category of comonoids in Vect — or equivalently, a comonoid in the category of monoids in Vect.

More generally, a bimonoid in a monoidal category $M$ is a monoid in the category of comonoids in $M$ — or equivalently, a comonoid in the category of monoids in $M$. So, a bialgebra is a bimonoid in $Vect$.

## Properties

### Relation to sesquialgebras

Bialgebras are precisely those sesquialgebras $A$ whose product $A \otimes A$-$A$-bimodule is induced from an algebra homomorphism $A \to A \otimes A$ and whose unit $k$-$A$ bimodule is induced from an algebra homomorphism $A \to k$.

### Tannaka duality and categories of modules

The structure of a bialgebra on an associative algebra equips its category of modules with the structure of a monoidal category and a monoidal fiber functor. In fact that construction is an equivalence. This is the statement of Tannaka duality for bialgebras. For instance (Bakke)

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
$A$$Mod_A$
$R$-algebra$Mod_R$-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
$A$$Mod_A$
$R$-2-algebra$Mod_R$-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category3-category of module 2-categories
$A$$Mod_A$
$R$-3-algebra$Mod_R$-4-module

## Examples

Notions of bialgebra with further structure notably include Hopf algebras and their variants.

Tannaka duality for bialgebras

• Tørris Koløen Bakke, Hopf algebras and monoidal categories (2007) (pdf)