# Fusion categories and homotopy theory

### Pavel Etingof

MIT, Cambridge, USA### Dmitri Nikshych

University of New Hampshire, Durham, USA### Victor Ostrik

University of Oregon, Eugene, USA

## 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 $C$ 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 article is that the 2-truncation of $BrPic (C)$ is canonically equivalent to the 2-groupoid of braided auto-equivalences 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 auto-equivalences 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 re-derive (without computations) the classical result of Tambara and Yamagami. Moreover, we explicitly describe the category of all $(Vec_{A_{1}},Vec_{A_{2}})$-bimodule categories (not necessarily invertible ones) by showing that it is equivalent to the hyperbolic part of the category of Lagrangian correspondences.

## Cite this article

Pavel Etingof, Dmitri Nikshych, Victor Ostrik, Fusion categories and homotopy theory. Quantum Topol. 1 (2010), no. 3, pp. 209–273

DOI 10.4171/QT/6