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 classifiying 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 paper is that the 2-truncation of BrPic(C)
is canonically the 2-groupoid of braided autoequivalences 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 autoequivalences 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
rederive (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.
