Exact sequences of tensor categories

ALAIN BRUGUIÈRES Y SONIA NATALE

INTERNATIONAL MATHEMATICS RESEARCH NOTICES

OXFORD UNIV PRESS

Año: 2011 vol. 2011 p. 5644 - 5705

1073-7928

We introduce the notions of normal tensor functor and exact sequence of tensor categories.We show that exact sequences of tensor categories generalize strictly exactsequences of Hopf algebras as defined by Schneider, and in particular, exact sequencesof (finite) groups. We classify exact sequences of tensor categories C→C→C (suchthat C is finite) in terms of normal, faithful Hopf monads on C and also in terms ofself-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, thereexists a canonical commutative algebra (A, σ) in the center of C such that F is tensorequivalent to the free module functor C→modC(A, σ), where modC(A, σ) denotes thecategory of A-modules in C endowed with a monoidal structure defined using σ. Were-interpret equivariantization under a finite group action on a tensor category and, inparticular, the modularization construction, in terms of exact sequences, Hopf monads,and commutative central algebras. As an application, we prove that a braided fusioncategory whose dimension is odd and square-free is equivalent, as a fusion category, tothe representation category of a group.