symmetric monoidal (∞,1)-category of spectra
A trialgebra is meant to be a structure at least similar to or involving an associative algebra but equipped with two further compatible algebra structures, the first additional one making it a bialgebra, the second one making it then a trialgebra.
In the spirit of Tannaka duality, this is to be such that the category of modules of the associative algebra underlying the trialgebra inherits the structure not just of a monoidal category (as it does for a bialgebra), but even of a Hopf monoidal category.
Notice that next the 2-category of module categories over a Hopf monoidal category inherits the structure of a monoidal 2-category. This finally has a 3-category of module 2-categories.
In the spirit of n-modules by iterated internalization, this 3-category may be thought of as the 4-module (over the groun field), presented by the trialgebra.
Accordingly, trialgebras may be thought of as 4-bases for spaces of quantum states of extended 4d TQFTs . This has been the main motivation for considering them in (Pfeiffer 04) (not quite in the above language), in turn forming a 2-basis for the corresponding Hopf monoidal categories in (Crane-Frenkel 04). A review of related literature is in (Baez-Lauda 09, p. 98)
trialgebra, 3-module
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 |
A proposal for the definition of trialgebras is in
motivated from the definition of Hopf monoidal categories in
A survey of related references is in p. 98 of
Last revised on June 12, 2023 at 18:18:32. See the history of this page for a list of all contributions to it.