On equivariant triangulated categories

TL;DR

This paper studies the existence and construction of triangulated structures on categories of G-equivariant objects under finite group actions, providing conditions and DG-enhancements for such structures.

Contribution

It establishes the existence of triangulated structures on G-equivariant categories and constructs DG-enhancements under certain conditions, advancing understanding of equivariant triangulated categories.

Findings

Triangulated structures exist on G-equivariant categories under technical conditions.

DG-enhancements of G-equivariant categories can be constructed from DG-actions.

The relation of being an equivariant category is symmetric for finite abelian groups.

Abstract

Consider a finite group $G$ acting on a triangulated category $\mathcal T$. In this paper we investigate triangulated structure on the category $\mathcal T^G$ of $G$-equivariant objects in $\mathcal T$. We prove (under some technical conditions) that such structure exists. Supposed that an action on $\mathcal T$ is induced by a DG-action on some DG-enhancement of $\mathcal T$, we construct a DG-enhancement of $\mathcal T^G$. Also, we show that the relation "to be an equivariant category with respect to a finite abelian group action" is symmetric on idempotent complete additive categories.