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Central Exact Sequences of Tensor Categories, Equivariantization and Applications

We define equivariantization of tensor categories under tensor group scheme actions and give necessary and sufficient conditions for an exact sequence of tensor categories to be an equivariantization under a finite group or finite group scheme action. We introduce the notion of central exact sequence of tensor categories and use it in order to present an alternative formulation of some known characterizations of equivariantizations for fusion categories, and to extend these characterizations to equivariantizations of finite tensor categories under finite group scheme actions. In particular, we obtain a simple characterization of equivariantizations under actions of finite abelian groups. As an application, we show that if $\C$ is a fusion category and $F: \C \to \D$ is a dominant tensor functor of Frobenius-Perron index $p$, then $F$ is an equivariantization if $p=2$, or if $\C$ is weakly integral and $p$ is the smallest prime factor of $\FPdim \C$.