# nLab Hopf monoidal category

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

The lift of the notion of Hopf bialgebra from associative algebras to tensor categories.

By the corresponding higher Tannaka duality, the 2-category of module categories over a Hopf monoidal category is a monoidal 2-category with duals.

Related to 4d TQFT as Hopf algebras are related to 3d TQFT.

## Properties

### Relation to trialgebras

Just as monoidal categories with fiber functor are the categories of modules of a Hopf algebra, so Hopf monoidal categories are supposed to be the categories of modules of a trialgebra.

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

## References

The notion is due to

A proposal for the corresponding definition of trialgebras is in

A survey of related references is in p. 98 of

• John Baez, Aaron Lauda, A prehistory of $n$-categorical physics, in Deep beauty, 13-128, Cambridge Univ. Press, Cambridge, 2011 (arXiv:0908.2469)

Last revised on September 9, 2019 at 22:36:35. See the history of this page for a list of all contributions to it.