symmetric monoidal (∞,1)-category of spectra
A sesquiunital sesquiagebra is an algebra object internal to the 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.
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.
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 |
See also the references at hopfish algebra.
Last revised on February 11, 2013 at 14:35:58. See the history of this page for a list of all contributions to it.