Tensor functors between Morita duals of fusion categories

Abstract

Given a fusion category C and an indecomposable C-module category M, the fusion category C * M of C-module endofunctors of M is called the (Morita) dual fusion category of C with respect to M. We describe tensor functors between two arbitrary duals C * M and D * N in terms of data associated to C and D. We apply the results to G-equivariantizations of fusion categories and group-theoretical fusion categories. We describe the orbits of the action of the Brauer-Picard group on the set of module categories and we propose a categorification of the Rosenberg-Zelinsky sequence for fusion categories.2010 Mathematics Subject Classification. 16T05, 18D10.

On the Brauer–Picard groups of fusion categories

“…Namely, two A-module categories belong to the same orbit if and only if the corresponding dual fusion categories are equivalent. A related result is established by Galindo and Plavnik in [GP,Theorem 1.4]. …”

“…The invertible objects of Z(C(G, ω)) are well known, see, e.g., [DPR]. The exact sequence (24) in the next proposition can be found in [GP,Example 6.2]. We include its proof for the sake of completeness.…”

“…The exact sequence (17) can also be obtained by combining two exact sequences from [GP,Theorem 1.3].…”

Fusion categories associated with subfactors with index $3 + sqrt{5}$

“…The outer autmorphism group Out(C) of a fusion category C is the quotient of the group of tensor autoequivalences of C (considered modulo monoidal natural isomorphism) by the subgroup of inner autoequivalences (conjugation by invertible objects). If C M is a module category and D = ( C M) * , then the invertible C-D bimodule categories which extend C M are parametrized by Out(D) (see [GP14]).…”

Hopf cocycle deformations and invariant theory

“…In case the Hopf algebra H is semisimple, Ocneanu rigidity tells us that there are only finitely many 2-cocycles up to equivalence (see [ENO05]). For a large class of Hopf algebras, the group theoretical Hopf algebras, such cocycles can be classified explicitly by group theoretical data (see [ENO05] and [GP17]). The general classification of equivalence classes of 2-cocycles is in general open.…”