nLab
hopfish algebra

Context

Higher algebra

Algebraic theories

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Algebras and modules

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Higher algebras

  • symmetric monoidal (∞,1)-category of spectra

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Model category presentations

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Geometry on formal duals of algebras

Theorems

Contents

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

Definition

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.

Definition

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).

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

for over /

/
AAMod AMod_A
RR-Mod RMod_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

AAMod AMod_A
RR-Mod RMod_R-
(with some duality and strictness structure)

3-Tannaka duality for over

AAMod AMod_A
RR-Mod RMod_R-

References

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.