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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
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