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 $\mathcal{A}$ be an extension closed subcategory of an extriangulated category $\mathcal{C}$. Then the quotient category $\mathcal{M}:=\mathcal{A}/\mathcal{X}$ carries naturally a triangulated structure whenever $(\mathcal{A},\mathcal{A})$ forms an $\mathcal{X}$-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 $\mathcal{C}/\mathcal{X}$, where $\mathcal{X}$ is functorially finite in $\mathcal{C}$. When $\mathcal{C}$ has Auslander-Reiten translation $\tau$, we prove that for a functorially finite subcategory $\mathcal{X}$ of $\mathcal{C}$ containing projectives and injectives, $\mathcal{C}/\mathcal{X}$ is a triangulated category if and only if $(\mathcal{C},\mathcal{C})$ is $\mathcal{X}-$mutation if and only if $\tau \underline{\mathcal{X}}=\bar{\mathcal{X}}.$ This generalizes a result by Jørgensen who proved the equivalence between the first and the third conditions for triangulated categories. Furthermore, we show that for such a subcategory $\mathcal{X}$ of the extriangulated category $\mathcal{C}$, $\mathcal{C}$ admits a new extriangulated structure such that $\mathcal{C}$ 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.

arXiv admin note: text overlap with arXiv:1605.05607 by other authors; add a word to the title

References in corpus (1)

• A homotopy theory of additive categories with suspensions