/ /
/
/
,
,
,
,
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
for over /
/ | |
---|---|
$A$ | $Mod_A$ |
$R$- | $Mod_R$- |
= with -preserving | |
strict : with | |
with | |
(correct version) | (without fiber functor) |
with generalized | |
with | |
with | |
() | with |
with | |
() | with and Schur smallness |
form | form |
2-Tannaka duality for over
$A$ | $Mod_A$ |
$R$- | $Mod_R$- |
(with some duality and strictness structure) |
3-Tannaka duality for over
$A$ | $Mod_A$ |
$R$- | $Mod_R$- |
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.