category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A fusion category is a rigid semisimple linear (Vect-enriched) monoidal category, with only finitely many isomorphism classes of simple objects, such that the endomorphisms of the unit object form just the ground field $k$.
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.
Under Tannaka duality, every fusion catgeory $C$ arises as the representation category of a weak Hopf algebra. (Ostrik)
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 $MonCat$ of monoidal categories. By the homotopy hypothesis this explains how they induce 3d TQFTs.
Write $MonCat_{bim}$ for the (infinity,3)-category which has as
objects monoidal categories,
morphism bimodules of these,
and so on.
With its natural tensor product, $MonCat$ is a symmetric monoidal (infinity,3)-category.
A monoidal category which is fusion is fully dualizable in the (infinity,3)-category $MonCat_{bim}$, def. 2.
This is due to (Douglas & Schommer-Pries & Snyder 13).
Via the cobordism theorem prop. 2 implies that fusion categories enocde extended TQFTs on 3-dimensional framed cobordisms, while their $O(3)$-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
Some correct version of the claim that abelian semisimple is the same as idempotent complete and nondegenerate. Math Overflow question
Good notation distinguishing simple versus absolutely simple? (is $End(V) = k$ or just $V$ has no nontrivial proper subobjects).
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.
fusion category
Some classical references include
P. Etingof, D. Nikshych and V. Ostrik, On fusion categories.
P. Etingof and D. Calaque, Lectures on tensor categories.
Pavel Etingof, D. Nikshych and V. Ostrik, Fusion categories and homotopy theory , Quantum Topology, 1(2010), 209-273. (Earlier version available as ArXiv:0909.3140
A review is also in chapter 6 of
The Tannaka duality to weak Hopf algebras is discussed in
Takahiro Hayashi, A canonical Tannaka duality for finite seimisimple tensor categories (arXiv:math/9904073)
Victor Ostrik, Module categories, weak Hopf algebras and modular invariants (arXiv:math/0111139)
The relation to 3d TQFT is clarified via the cobordism hypothesis in
Chris Douglas, Chris Schommer-Pries, Noah Snyder, The Structure of Fusion Categories via 3D TQFTs (talk pdf)
Chris Douglas, Chris Schommer-Pries, Noah Snyder, Dualizable tensor categories (arXiv:1312.7188)
For some discussion see