Exact sequences of tensor categories

By Alain Bruguières, Sonia Natale. 2010

106 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/048c2b8f06a8172a653a89b8e8fe23c53d5e4a27

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In Algebra and Number Theory

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In Mathematical Physics

Citations in the corpus

Drinfeld Center and Representation Theory for Monoidal Categories (2015, background): “lgebra. Morally, it corresponds to the algebra object representing the forgetful functor on the Drinfeld center which appeared in the work of Brugui´eres–Virelizier [BV07,BV13] and Brugui´eres–Natale [BN11] in the framework of fusion categories. The representations deﬁned by the half-braidings lead ”
Categorified trace for module tensor categories over braided tensor categories (2015, background): “4deals with the left column, and Section4.5deals with the middle column. 19 4.1 The attaching map Adjoints of tensor functors have been studied systematically [Kel74] and arise in many contexts (e.g. [BN11], [DMNO13]). We will reprove several known properties as a way of introducing our graphical no”