Fusion categories over \(\mathbb{C}\) are natural generalisations of the representation category of a finite group. A fusion category is a rigid semi-simple linear monoidal category, having only finitely many isomorphism classes of simple objects and such that the endomorphisms of the unit object form the ground field. The classification of general fusion categories seems out of reach at the moment. This paper makes a step in the classification process, however, by adapting and extending methods from homotopy theory to the classification of \(G\)-extensions of a given fusion category.
The paper introduces a 3-groupoid, the Brauer-Picard groupoid, consisting of fusion categories, invertible bimodule categories between them, equivalences of such as 2-morphisms and with 3-morphisms isomorphisms of such equivalences. It then studies the classifying spaces of various related 2-groupoids, etc. The main results include an explicit description of extensions of fusion categories by a finite group \(G\), in terms of elements related to certain cohomology groups of \(G\).