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# Quantum Topology

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**Volume 1, Issue 3, 2010, pp. 209–273**

**DOI: 10.4171/QT/6**

Published online: 2010-08-19

Fusion categories and homotopy theory

Pavel Etingof^{[1]}, Dmitri Nikshych

^{[2]}and Victor Ostrik

^{[3]}(1) MIT, Cambridge, USA

(2) University of New Hampshire, Durham, USA

(3) University of Oregon, Eugene, USA

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(__of invertible

**)**`C`**-bimodule categories, called the Brauer–Picard groupoid of**

`C`**, such that equivalence classes of**

`C``G`-extensions of

**are in bijection with homotopy classes of maps from**

`C``BG`to the classifying space of

__BrPic(__. 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.

**)**`C`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(

**) 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**

`C`_{A}, where

`A`is a finite abelian group, then BrPic(

**) is the orthogonal group O(**

`C``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

_{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.

*Keywords: *Fusion categories, module categories, homotopy theory, higher groupoids

Etingof Pavel, Nikshych Dmitri, Ostrik Victor: Fusion categories and homotopy theory. *Quantum Topol.* 1 (2010), 209-273. doi: 10.4171/QT/6