Group actions on categories and Elagin’s theorem revisited

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 C= ⟨ A, B⟩ is a semiorthogonal decomposition of a triangulated category which is preserved by the action of G, and CG is triangulated, then there is a semiorthogonal decomposition CG= ⟨ AG, BG⟩. We also prove that any G-action on C is weakly equivalent to a strict G-action which is the analog of the Coherence theorem for monoidal categories. © 2017, Springer International Publishing AG.