nLab trialgebra




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)

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_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.