Fusion categories and homotopy theory
Pavel Etingof, Dmitri Nikshych, Victor Ostrik, with an appendix by Ehud Meir
TL;DR
The paper develops a homotopy‑theoretic framework for classifying G‑extensions of fusion categories by mapping BG into classifying spaces of higher groupoids attached to the input category. A central result is that the Drinfeld center functor Z provides a fully faithful embedding BrPic(C) → EqBr(Z(C)), yielding BrPic(C) ≅ EqBr(Z(C)) and enabling explicit computations, e.g. BrPic(Vec_A) ≅ O(A ⊕ A*) where A is finite abelian. Extensions are described by topological data (c,M,α) with obstructions O3 ∈ H^3(G, π2) and O4 ∈ H^4(G, k×) that must vanish, and these results have algebraic analogues via decategorification and cohomological obstructions. The framework recovers and generalizes Tambara–Yamagami classifications, provides a detailed description of the tensor products of module categories, and connects to Lagrangian correspondences for Vec_A. The appendix further elucidates the relationship between group extensions and G‑graded fusion categories through the Lyndon–Hochschild–Serre spectral sequence, tying the categorical obstructions to classical group extension theory and enabling a principled, unified treatment of extensions and actions in both topological and algebraic languages.
Abstract
We apply the yoga of classical homotopy theory to classification problems of G-extensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to classification (up to homotopy) of maps from BG to classifiying spaces of certain higher groupoids. In particular, to every fusion category C we attach the 3-groupoid BrPic(C) of invertible C-bimodule categories, called the Brauer-Picard groupoid of C, such that equivalence classes of G-extensions of C are in bijection with homotopy classes of maps from BG to the classifying space of BrPic(C). This gives rise to an explicit description of both the obstructions to existence of extensions and the data parametrizing them; we work these out both topologically and algebraically. One of the central results of the paper is that the 2-truncation of BrPic(C) is canonically the 2-groupoid of braided autoequivalences of the Drinfeld center Z(C) of C. In particular, this implies that the Brauer-Picard group BrPic(C) (i.e., the group of equivalence classes of invertible C-bimodule categories) is naturally isomorphic to the group of braided autoequivalences of Z(C). Thus, if C=Vec(A), where A is a finite abelian group, then BrPic(C) is the orthogonal group O(A+A^*). This allows one to obtain a rather explicit classification of extensions in this case; in particular, in the case G=Z/2, we rederive (without computations) the classical result of Tambara and Yamagami. Moreover, we explicitly describe the category of all (Vec(A1),Vec(A2))-bimodule categories (not necessarily invertible ones) by showing that it is equivalent to the hyperbolic part of the category of Lagrangian correspondences.