We apply the yoga of classical homotopy theory to classification problems of -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 to classifying spaces of certain higher groupoids. In particular, to every fusion category we attach the 3-groupoid of invertible -bimodule categories, called the Brauer–Picard groupoid of , such that equivalence classes of -extensions of are in bijection with homotopy classes of maps from to the classifying space of . 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 is canonically equivalent to the 2-groupoid of braided auto-equivalences of the Drinfeld center of . In particular, this implies that the Brauer–Picard group (i.e., the group of equivalence classes of invertible -bimodule categories) is naturally isomorphic to the group of braided auto-equivalences of . Thus, if , where is a finite abelian group, then is the orthogonal group . This allows one to obtain a rather explicit classification of extensions in this case; in particular, in the case , we re-derive (without computations) the classical result of Tambara and Yamagami. Moreover, we explicitly describe the category of all -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. 209–273DOI 10.4171/QT/6