A note on equivariantization of additive categories and triangulated categories

TL;DR

This paper explores how to reconstruct additive categories from their equivariant objects under finite group actions, especially when the group is solvable, and examines conditions for triangulated structures on these categories.

Contribution

It demonstrates that for solvable groups, the original additive category can be reconstructed from the equivariant category through a finite sequence of equivariantizations and discusses conditions for triangulated structures.

Findings

Reconstruction of additive categories from equivariant categories for solvable groups.

Conditions under which equivariant categories inherit triangulated structures.

Examples of canonical triangulated structures on equivariant categories.

Abstract

In this article, we investigate the category $\mathcal{A}^G$ of equivariant objects of an additive category $\mathcal{A}$ with respect to an action of a finite group $G$. We show that if $G$ is solvable then we can reconstruct $\mathcal{A}$ from $\mathcal{A}^G$ via a finite sequence of equivariantization. We also consider the possibility of a triangulated structure on $\mathcal{A}^G$ canonical in certain sense when $\mathcal{A}$ is triangulated and give several instances in which $\mathcal{A}^G$ is indeed canonically triangulated.