nLab
fusion category

Definition

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 $k$.

Remarks

The name “fusion category” comes from the central examples of structures whose canonical tensor product is called a “fusion product”, notably representations of loop group?s and of Hopf algebras and of vertex operator algebras.

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:

Conjecture (Etingof, Nikshcych and Ostrik)

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.

Suggestions

Here are three things such that it’d be awesome if they were sorted out on this page:

1. Kuperberg’s theorem saying that abelian semisimple implies linear over some field. Finite, connected, semisimple, rigid tensor categories are linear

2. Some correct version of the claim that abelian semisimple is the same as idempotent complete and nondegenerate. Math Overflow question

3. Good notation distinguishing simple versus absolutely simple? (is $\mathrm{End}(V)=k$ 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.

References

• Math Overflow, Why are fusion categories interesting?.
• P. Etingof, D. Nikshych and V. Ostrik, On fusion categories.
• P. Etingof and D. Calaque, Lectures on tensor categories.
• M. Müger, Tensor categories: A selective guided tour.
• B. Bartlett, On unitary 2-representations of finite groups and topological quantum field theory (chapter 6).