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
\mathcal C
we attach the 3-groupoid
\underline{\underline{\mathrm{BrPic}}}(\mathcal C)
of invertible
\mathcal C
-bimodule categories, called the Brauer–Picard groupoid of
\mathcal C
, such that equivalence classes of
G
-extensions of
\mathcal C
are in bijection with homotopy classes of maps from
BG
to the classifying space of
\underline{\underline{\mathrm{BrPic}}}(\mathcal 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
\underline{\underline{\mathrm{BrPic}}}(\mathcal C)
is canonically equivalent to the 2-groupoid of braided auto-equivalences of the Drinfeld center
\mathcal Z(\mathcal C)
of
\mathcal C
. In particular, this implies that the Brauer–Picard group
\mathrm{BrPic}(\mathcal C)
(i.e., the group of equivalence classes of invertible
\mathcal C
-bimodule categories) is naturally isomorphic to the group of braided auto-equivalences of
\mathcal Z(\mathcal C)
. Thus, if
\mathcal C = \mathrm{Vec}_A
, where
A
is a finite abelian group, then
\mathrm{BrPic}(\mathcal C)
is the orthogonal group
\mathrm{O}(A \oplus A^*)
. This allows one to obtain a rather explicit classification of extensions in this case; in particular, in the case
G = \mathbb Z_2
, we re-derive (without computations) the classical result of Tambara and Yamagami. Moreover, we explicitly describe the category of all
(\mathrm{Vec}_{A_1},\mathrm{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.
