Exact sequences of tensor categories

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.Comment: 39 page

On the center of fusion categories

M\"uger proved in 2003 that the center of a spherical fusion category C of non-zero dimension over an algebraically closed field is a modular fusion category whose dimension is the square of that of C. We generalize this theorem to a pivotal fusion category C over an arbitrary commutative ring K, without any condition on the dimension of the category. (In this generalized setting, modularity is understood as 2-modularity in the sense of Lyubashenko.) Our proof is based on an explicit description of the Hopf algebra structure of the coend of the center of C. Moreover we show that the dimension of C is invertible in K if and only if any object of the center of C is a retract of a `free' half-braiding. As a consequence, if K is a field, then the center of C is semisimple (as an abelian category) if and only if the dimension of C is non-zero. If in addition K is algebraically closed, then this condition implies that the center is a fusion category, so that we recover M\"uger's result

Hopf monads on monoidal categories

We define Hopf monads on an arbitrary monoidal category, extending the definition given previously for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode. Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a category of bimodules admitting a right adjoint). We show that any finite tensor category is the category of finite-dimensional modules over a Hopf algebroid. Any Hopf algebra in the center of a monoidal category C gives rise to a Hopf monad on C. The Hopf monads so obtained are exactly the augmented Hopf monads. More generally if a Hopf monad T is a retract of a Hopf monad P, then P is a cross product of T by a Hopf algebra of the center of the category of T-modules (generalizing the Radford-Majid bosonization of Hopf algebras). We show that the comonoidal comonad of a Hopf adjunction is canonically represented by a cocommutative central coalgebra. As a corollary, we obtain an extension of Sweedler's Hopf module decomposition theorem to Hopf monads (in fact to the weaker notion of pre-Hopf monad).Comment: 45 page