nLab
fusion category

Definition

Definition

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 $k$.

Examples

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:

• Representations of a finite group or (finite super-group]
• For a given finite group $G$ and a 3-cocycle on $G$ with values in a field $k$ (an element of $H^3(G,k^\times)$), take $G$-graded vector spaces with the cocycle as associator.

Properties

Relation to weak Hopf algebras

Under Tannaka duality, every fusion category $C$ arises as the representation category of a weak Hopf algebra. (Ostrik)

Relation to pivotal and spherical categories

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.

Relation to extended 3d TQFT

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.

Definition

Write $MonCat_{bim}$ for the (infinity,3)-category which has as

• objects monoidal categories,

• morphism bimodules of these,

• and so on.

Proposition

With its natural tensor product, $MonCat$ is a symmetric monoidal (infinity,3)-category.

Proposition

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).

Remark

Via the cobordism theorem prop. 2 implies that fusion categories encode 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).

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 $End(V) = k$ 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.

Related concepts

• fusion category

  fusion ring, Frobenius-Perron dimension

• pivotal category

• spherical category

• Deligne's theorem on tensor categories

• graded fusion category

References

Some classical references include

• P. Etingof, D. Nikshych and V. Ostrik, On fusion categories.

• Pavel Etingof and Damien Calaque, Lectures on tensor categories.

• Michael Müger, Tensor categories: A selective guided tour.

• 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

• Bruce Bartlett, On unitary 2-representations of finite groups and topological quantum field theory.

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)

and for the case of modular tensor categories in

• Bruce Bartlett, Christopher Douglas, Chris Schommer-Pries, Jamie Vicary, Modular categories as representations of the 3-dimensional bordism 2-category (arXiv:1509.06811)

For some discussion see

• Math Overflow, Why are fusion categories interesting? .