nLab sesquialgebra

Idea

A sesquiunital sesquiagebra is an algebra object internal to the monoidal 2-category 2Mod of algebras, bimodules and bimodule intertwiners. This means that it is an algebra equipped with an additional associative product and unit which are both exhibited by bimodules. If these bimodules come from algebra homomorphisms then the sesquialgebra is a bialgebra.

The structure of a sesquialgebra is just so that the category of modules of the underlying algebra is itself a monoidal category. In this sense sesquialgebras are a placeholder for 2-algebras. Moreover, in as far as these 2-algebras themselves are regarded as placeholders for their 2-category of 2-modules a sesquialgebra presents a 3-module/3-vector space.

Sesquialgebras with an extra grouplike-property have been called hopfish algebras.

Definition

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.

Properties

Tannaka duality and 2-rings

Precomposition with the product and unit bimodule makes the category of modules over the underlying associative algebra of a sesquialgebra itself into a monoidal category.

See for instance (Vercruysse, 5.3.3).

In fact, regarding the category of modules $Mod_A$ as a 2-abelian group?, the structure of a sesquialgebra on $A$ is equivalently the structure of a 2-ring (see there) on $Mod_A$.

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

• Joost Vercruysse, Hopf algebras—Variant notions and reconstruction theorems (arXiv:1202.3613)

See also the references at hopfish algebra.