# 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** = VecA, 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 = ℤ2, we re-derive (without computations) the classical result of Tambara and Yamagami. Moreover, we explicitly describe the category of all (VecA1,VecA2)-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