hopfish algebra



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.


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


A sesquiunital sesquialgebra over RR is an associative algebra AA over RR 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 RR-algebra AA equipped with

  • a product A RAA \otimes_R A-AA-bimodule Δ\Delta;

  • a unit RR-AA-bimodule ϵ\epsilon

satisfying the evident associative law and unit law.


A preantipode for a sesquiunital sesquialgebra AA is a left AAA \otimes A-module SS equipped with an isomorphism of right AAA \otimes A-modules

S *Hom A(ϵ,Δ). S^* \simeq Hom_A(\epsilon, \Delta) \,.

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

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

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


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 algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_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
AAMod AMod_A
RR-2-algebraMod RMod_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
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module


The notion was introduced in

Last revised on April 8, 2013 at 17:23:15. See the history of this page for a list of all contributions to it.