Morita equivalence methods in classification of fusion categories

Abstract

We describe an approach to classification of fusion categories in terms of Morita equivalence. This is usually achieved by analyzing Drinfeld centers of fusion categories and finding Tannakian subcategories therein.Definition 2.2. Let A 1 , A 2 be fusion categories. A tensor functor between A 1 and A 2 is a functor F : A 1 → A 2 along with natural isomorphisms

Modular categories are not determined by their modular data

“…The group-theoretical fusion categories C(G, H, ω, µ) are exactly the categories that are categorically (or weakly) Morita equivalent to the pointed fusion category Vect ω G , see [Ost03], and [Nik13] for an overview. Two fusion categories are categorically Morita equivalent if and only if their Drinfeld centers are equivalent braided monoidal categories [ENO05].…”

Computing higher Frobenius–Schur indicators in fusion categories constructed from inclusions of finite groups

“…By [24], the Drinfeld center of such a bimodule category is equivalent to the Drinfeld center of the "ambient" category. In different language this means that grouptheoretical fusion categories are Morita equivalent to the category of graded vector spaces with twisted associativity; see the survey [19]. We will treat the case of a group-theoretical fusion category defined without cocycles.…”

A Higher Frobenius–Schur Indicator Formula for Group-Theoretical Fusion Categories

“…the module category of a Drinfeld double. This can be expressed as saying that a group-theoretical category C(G, H, ω, ψ is Morita equivalent to Vect ω G (see the survey [16] for the notion of Morita equivalence).…”