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 paper. Let A be an extension closed subcategory of an extriangulated category C. Then the additive quotient category M:=A/[X] carries naturally a triangulated structure whenever (A,A) forms an X-mutation pair. This result generalizes many results of the same type for triangulated categories. It is used to give a classification of thick triangulated subcategories of pre-triangulated category C/[X], where X is functorially finite in C. When C has Auslander–Reiten translation τ, we prove that for a functorially finite subcategory X of C containing projectives and injectives, the quotient C/[X] is a triangulated category if and only if (C,C) is X-mutation, and if and only if τX_=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 X of the extriangulated category C, C admits a new extriangulated structure such that C is a Frobenius extriangulated category. Applications to exact categories and triangulated categories are given. From the applications we present extriangulated categories which are neither exact categories nor triangulated categories.