Exact Sequences of Tensor Categories

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 formula (such that formula is finite) in terms of normal, faithful Hopf monads on formula and also in terms of self-trivializing commutative algebras in the center of formula⁠. More generally, we show that, given any dominant tensor functor formula admitting an exact (right or left) adjoint, there exists a canonical commutative algebra (A,σ) in the center of formula such that F is tensor equivalent to the free module functor formula⁠, where formula denotes the category of A-modules in formula endowed with a monoidal structure defined using σ. 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.