nLab Hopf monoidal category

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.

Related concepts

• Hopf monoid, Hopf algebra

• Hopf monad

• Hopf V-category?

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebra	category of modules
$A$	$Mod_A$
$R$-algebra	$Mod_R$-2-module
sesquialgebra	2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebra	strict 2-ring: monoidal category with fiber functor
Hopf algebra	rigid monoidal category with fiber functor
hopfish algebra (correct version)	rigid monoidal category (without fiber functor)
weak Hopf algebra	fusion category with generalized fiber functor
quasitriangular bialgebra	braided monoidal category with fiber functor
triangular bialgebra	symmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)	rigid braided monoidal category with fiber functor
triangular Hopf algebra	rigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)	rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld double	form Drinfeld center
trialgebra	Hopf monoidal category

2-Tannaka duality for module categories over monoidal categories

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

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

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

References

The notion is due to

• Louis Crane, Igor Frenkel, Four dimensional topological quantum field theory, Hopf categories, and the canonical bases, J.Math.Phys. 35 (1994) 5136-5154, (arXiv:hep-th/9405183)

A proposal for the corresponding definition of trialgebras is in

• Hendryk Pfeiffer, 2-Groups, trialgebras and their Hopf categories of representations (arXiv:math/0411468)

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)

On its relation to 4d TQFT:

• Hank Chen, Hopf 2-Algebras: Homotopy Higher Symmetries in Physics, PhD thesis (2024) [pdf]