nLab hopfish algebra

Idea

The notion of hopfish algebra is a generalization of that of Hopf algebra designed to behave better with respect to Morita equivalence of algebras. It is defined to be a sesquialgebra (hence a 2-algebra/3-module) which is grouplike in a suitable sense.

The notion subsumes Hopf algebras and weak Hopf algebras.

Definition

Let $R$ be some commutative ring (or E-infinity ring).

Definition

A sesquiunital sesquialgebra over $R$ is an associative algebra $A$ over $R$ equipped with the structure of an algebra object internal to the 2-category 2Mod of associative algebras, bimodules and bimodule intertwiners.

This means that it is an $R$-algebra $A$ equipped with

• a product $A \otimes_R A$-$A$-bimodule $\Delta$;

• a unit $R$-$A$-bimodule $\epsilon$

satisfying the evident associative law and unit law.

Definition

A preantipode for a sesquiunital sesquialgebra $A$ is a left $A \otimes A$-module $S$ equipped with an isomorphism of right $A \otimes A$-modules

$$S^* \simeq Hom_A(\epsilon, \Delta) \,.$$

An preantipode is an antipode if it is a free module over $A$ of rank 1 when regarded as an $A$-$A^{op}$-bimodule.

A sesquiunital sesquialgebra equipped with such an antipode is a hopfish algebra.

This is (TWZ, def. 3.1, def. 3.2).

Properties

Module categories and Tannaka duality

The notion of sesquialgebra generalizes that of bialgebra such that under Tannaka duality sesquialgebras corespondond to monoidal categories generally, while the strictness of bialgebras means that there their monoidal category of modules is equipped with a fiber functor.

Since moreover Hopf algebras correspond to rigid monoidal categories with fiber functor under Tannaka duality, the correct sesqui-algebra generalization of Hopf algebras should have exactly the rigid monoidal categories as module categories, up to equivalence, without necessarily a fiber functor. This is expressed by the following table

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 was introduced in

• Xiang Tang, Alan Weinstein, Chenchang Zhu, Hopfish algebras, Pacific J. Math. 231 (2007), no. 1, 193–216. (arXiv:math/0510421)