symmetric monoidal (∞,1)-category of spectra
The lift of the notion of Hopf bialgebra from associative algebras to tensor categories.
By the corresponding higher Tannaka duality, the 2-category of module categories over a Hopf monoidal category is a monoidal 2-category with duals.
Related to 4d TQFT as Hopf algebras are related to 3d TQFT.
Just as monoidal categories with fiber functor are the categories of modules of a Hopf algebra, so Hopf monoidal categories are supposed to be the categories of modules of a trialgebra.
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 is due to
A proposal for the corresponding definition of trialgebras is in
A survey of related references is in p. 98 of
Last revised on August 21, 2016 at 04:25:59. See the history of this page for a list of all contributions to it.