nLab trialgebra

Idea

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)

Related concepts

• algebra, module

  • 2d TQFT

• bialgebra, 2-module

  • quantum group

  • 3d TQFT

• trialgebra, 3-module

  • quantum 2-group

  • 4d TQFT

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

A proposal for the definition of trialgebras is in

• Hendryk Pfeiffer, 2-Groups, trialgebras and their Hopf categories of representations, Adv. Math. 212 No. 1 (2007) 62-108 (arXiv:math/0411468)

motivated from the definition of Hopf monoidal categories in

• Louis Crane, Igor Frenkel, Four dimensional topological quantum field theory, Hopf categories, and the canonical bases, J.Math.Phys. 35 (1994) 5136-5154, (arXiv:hep-th/9405183)

A survey of related references is in p. 98 of

• John Baez, Aaron Lauda, A prehistory of $n$-categorical physics, in Deep beauty, 13-128, Cambridge Univ. Press, Cambridge, 2011 (arXiv:0908.2469)