Triangulated quotient categories revisited

TL;DR

This paper revisits triangulated quotient categories within extriangulated categories, establishing new conditions for when such quotients are triangulated and providing a classification of their subcategories.

Contribution

It introduces a mutation-based framework unifying previous constructions of triangulated quotients and extends results to extriangulated categories with applications to exact and triangulated categories.

Findings

Classification of thick triangulated subcategories

Equivalence conditions for triangulated quotients involving mutation and Auslander-Reiten translation

Examples of extriangulated categories that are neither exact nor triangulated

Abstract

Extriangulated categories were introduced by Nakaoka and Palu by extracting the similarities between exact categories and triangulated categories. A notion of mutation of subcategories in an extriangulated category is defined in this article. Let $\cal A$ be an extension closed subcategory of an extriangulated category $\cal C$. Then the quotient category $\cal M:=\cal{A}/\cal{X}$ carries naturally a triangulated structure whenever $(\cal A,\cal A)$ forms an $\cal X$-mutation pair. This result unifies many previous constructions of triangulated quotient categories, and using it gives a classification of thick triangulated subcategories of pretriangulated category $\cal{C}/\cal{X}$, where $\cal X$ is functorially finite in $\cal C$. When $\cal C$ has Auslander-Reiten translation $\tau$, we prove that for a functorially finite subcategory $\cal X$ of $\cal C$ containing projectives and injectives, $\cal{C}/\cal{X}$ is a triangulated category if and only if $(\cal C,\cal C)$ is $\cal X-$mutation if and only if $\tau \underline{\cal X}=\bar{\cal X}.$ This generalizes a result by J{\o}rgensen who proved the equivalence between the first and the third conditions for triangulated categories. Furthermore, we show that for such a subcategory $\cal X$ of the extriangulated category $\cal C$, $\cal C$ admits a new extriangulated structure such that $\cal C$ is a Frobenius extriangulated category. Applications to exact categories and triangulated categories are given. From the applications we present examples that extriangulated categories are neither exact categories nor triangulated categories.