# Group actions on categories and Elagin's Theorem Revisited

## TL;DR

This paper revisits Elagin's theorem on semiorthogonal decompositions under finite group actions on categories, providing a concise proof and establishing a coherence result for group actions.

## Contribution

It offers a simplified proof of Elagin's theorem and demonstrates that any group action can be made strict, extending coherence concepts to categorical group actions.

## Key findings

- Concise proof of Elagin's theorem on semiorthogonal decompositions
- Any G-action on a category is weakly equivalent to a strict G-action
- Establishment of a coherence theorem for G-actions on categories

## 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 $\mathcal{C} = \langle \mathcal{A}, \mathcal{B} \rangle$ 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 $\mathcal{C}^G = \langle \mathcal{A}^G, \mathcal{B}^G \rangle$. 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.