Triangulated quotient categories revisited

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 tau, 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 tau (X) under bar = (X) over bar. This generalizes a result by Jorgensen 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. (C) 2018 Elsevier Inc. All rights reserved.