Exact sequences of tensor categories

TL;DR

This paper introduces the concept of exact sequences in tensor categories, generalizing classical algebraic structures, and provides classification results and applications to fusion categories and group representations.

Contribution

It defines normal tensor functors and exact sequences of tensor categories, linking them to Hopf monads and commutative algebras, extending classical algebraic notions.

Findings

Classifies exact sequences of tensor categories via Hopf monads and commutative algebras.

Shows that certain dominant tensor functors are equivalent to module categories over commutative algebras.

Proves that odd square-free braided fusion categories are group-theoretical.

Abstract

We introduce the notions of normal tensor functor and exact sequence of tensor categories. We show that exact sequences of tensor categories generalize strictly exact sequences of Hopf algebras as defined by Schneider, and in particular, exact sequences of (finite) groups. We classify exact sequences of tensor categories C' -> C -> C'' (such that C' is finite) in terms of normal faithful Hopf monads on C'' and also, in terms of self-trivializing commutative algebras in the center of C. More generally, we show that, given any dominant tensor functor C -> D admitting an exact (right or left) adjoint there exists a canonical commutative algebra A in the center of C such that F is tensor equivalent to the free module functor C -> mod_C A, where mod_C A denotes the category of A-modules in C endowed with a monoidal structure defined using the half-braiding of A. We re-interpret equivariantization under a finite group action on a tensor category and, in particular, the modularization construction, in terms of exact sequences, Hopf monads and commutative central algebras. As an application, we prove that a braided fusion category whose dimension is odd and square-free is equivalent, as a fusion category, to the representation category of a group.