A remark on a theorem by C. Amiot

TL;DR

This paper discusses how connectedness assumptions affect the classification of certain triangulated categories, showing they are determined by their underlying categories and suspension actions.

Contribution

It extends Amiot's classification results by removing the connectedness assumption, highlighting the role of suspension functor actions.

Findings

Connected triangulated categories are determined by their underlying categories.

Dropping connectedness, categories are determined by underlying categories plus suspension actions.

The proof provides insights into the structure of triangulated categories.

Abstract

C. Amiot has classified the connected triangulated k-categories with finitely many isoclasses of indecomposables satisfying suitable hypotheses. We remark that her proof shows that these triangulated categories are determined by their underlying k-linear categories. We observe that if the connectedness assumption is dropped, the triangulated categories are still determined by their underlying k-categories together with the action of the suspension functor on the set of isoclasses of indecomposables.