This article is built on the previous work of the authors [Int. Math. Res. Not. 2011, No. 24, 5644–5705 (2011; Zbl 1250.18005)]. The main idea of this sequence of works is to obtain analogues for tensor categories of the relation between exact sequences of groups, or more generally Hopf algebras, and extensions of these objects. The main result of this paper characterizes the exact sequences of (finite) tensor categories that correspond to equivariantizations, which are an important tool in constructions and classifications of fusion categories.
An exact sequence \( \mathcal{C}' \longrightarrow \mathcal{C} \overset{F}{\longrightarrow} \mathcal{C}'' \) of tensor categories is an equivariantization exact sequence, if it is equivalent to one of the form \(\mathrm{rep}(G) \longrightarrow (\mathcal{C}'')^{G} \longrightarrow \mathcal{C}''\), where \(G\) is a finite group scheme that acts on \(\mathcal{C}''\) and \( (\mathcal{C}'')^{G}\) is the corresponding equivariantization.
By considering the adjoint of the functor \(F\), an exact sequence induces an algebra in \(\mathcal{C}\) that lifts to an algebra in the center \(\mathcal{Z}(\mathcal{C})\) and the sequence is called central if the two categories of modules are equivalent under the forgetful functor \(U: \mathcal{Z}(\mathcal{C}) \longrightarrow \mathcal{C}\). Central exact sequences are shown to be equivalent to an associated normal Hopf monad being exact and cocommutative and to equivariantization exact sequences. The authors give two proofs of this statement and thereby connect to yet another characterization of equivariantization in [P. Etingof et al., Adv. Math. 226, No. 1, 176–205 (2011; Zbl 1210.18009)].
As applications, central exact sequences for Tambara-Yamagami categories are considered and the authors give sufficient criteria for recognizing tensor functors that fit into an equivariantization exact sequence.