category with duals (list of them)
dualizable object (what they have)
A fusion category is a rigid semisimple linear (Vect-enriched) monoidal category (“tensor category”), with only finitely many isomorphism classes of simple objects, such that the endomorphisms of the unit object form just the ground field .
The name “fusion category” comes from the central examples of structures whose canonical tensor product is called a “fusion product”, notably representations of loop groups and of Hopf algebras and of vertex operator algebras.
Some of the easiest examples are:
Fusion categories were first systematically studied by Etingof, Nikshych and Ostrik in On fusion categories. This paper listed many examples and proved many properties of fusion categories. One of the important conjectures made in that paper was the following:
Every fusion category admits a pivotal structure.
Providing a certain condition is satisfied, a pivotal structure on a fusion category can be shown to correspond to a ‘twisted’ monoidal natural transformation of the identity functor on the category, where the twisting is given by the pivotal symbols.
Given the data of a fusion category one can build a 3d extended TQFT by various means. This is explained by the fact, see below, that fusion categories are (probably precisely) the fully dualizable objects in the 3-category of monoidal categories. By the homotopy hypothesis this explains how they induce 3d TQFTs.
Write for the (infinity,3)-category which has as
This is due to (Douglas & Schommer-Pries & Snyder 13).
Via the cobordism theorem prop. 2 implies that fusion categories encode extended TQFTs on 3-dimensional framed cobordisms, while their -homotopy fixed points encode extended 3d TQFTs on general (not framed) cobordisms.
These 3d TQFTs hence arise from similar algebraic data as the Turaev-Viro model and the Reshetikhin-Turaev construction, however there are various slight differences. See (Douglas & Schommer-Pries & Snyder 13, p. 5).
Here are three things such that it’d be awesome if they were sorted out on this page:
Kuperberg’s theorem saying that abelian semisimple implies linear over some field. Finite, connected, semisimple, rigid tensor categories are linear
Together 1 and 2 let you go between the two different obvious notions of semisimple which seem a bit muddled here when I clicked through the links.
Some classical references include
A review is also in chapter 6 of
Takahiro Hayashi, A canonical Tannaka duality for finite seimisimple tensor categories (arXiv:math/9904073)
and for the case of modular tensor categories in
For some discussion see