Triangulated quotient categories revisited

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.