Fusion categories and homotopy theory

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 classifying spaces of certain higher groupoids. In particular, to every fusion category $\mathcal{C}$ we attach the 3-groupoid $\underline{\underline{\mathrm{BrPic}}}(\mathcal{C})$ of invertible $\mathcal{C}$-bimodule categories, called the Brauer–Picard groupoid of $\mathcal{C}$, such that equivalence classes of $G$-extensions of $\mathcal{C}$ are in bijection with homotopy classes of maps from $BG$ to the classifying space of $\underline{\underline{\mathrm{BrPic}}}(\mathcal{C})$. 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 article is that the 2-truncation of $\underline{\underline{\mathrm{BrPic}}}(\mathcal{C})$ is canonically equivalent to the 2-groupoid of braided auto-equivalences of the Drinfeld center $\mathcal{Z}(\mathcal{C})$ of $\mathcal{C}$. In particular, this implies that the Brauer–Picard group $\mathrm{BrPic}(\mathcal{C})$ (i.e., the group of equivalence classes of invertible $\mathcal{C}$-bimodule categories) is naturally isomorphic to the group of braided auto-equivalences of $\mathcal{Z}(\mathcal{C})$. Thus, if $\mathcal{C}={\mathrm{Vec}}_A$, where $A$ is a finite abelian group, then $\mathrm{BrPic}(\mathcal{C})$ is the orthogonal group $\mathrm{O}(A\oplus A^{\ast })$. This allows one to obtain a rather explicit classification of extensions in this case; in particular, in the case $G={\mathbb{Z}}_2$, we re-derive (without computations) the classical result of Tambara and Yamagami. Moreover, we explicitly describe the category of all $({\mathrm{Vec}}_{A_1},{\mathrm{Vec}}_{A_2})$-bimodule categories (not necessarily invertible ones) by showing that it is equivalent to the hyperbolic part of the category of Lagrangian correspondences.