symmetric monoidal (∞,1)-category of spectra
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 $R$ be some commutative ring (or E-infinity ring).
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
satisfying the evident associative law and unit law.
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
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).
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 |
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.