Group actions on categories and Elagin's Theorem Revisited

Abstract

After recalling basic definitions and constructions for a finite group $G$ action on a $k$-linear category we give a concise proof of the following theorem of Elagin: if $\mathcal{C}=⟨\mathcal{A},\mathcal{B}⟩$ is a semiorthogonal decomposition of a triangulated category which is preserved by the action of $G$, and ${\mathcal{C}}^G$ is triangulated, then there is a semiorthogonal decomposition ${\mathcal{C}}^G=⟨{\mathcal{A}}^G,{\mathcal{B}}^G⟩$. We also prove that any $G$-action on $\mathcal{C}$ is weakly equivalent to a strict $G$-action which is the analog of the Coherence Theorem for monoidal categories.

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