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 A be an extension closed subcategory of an extriangulated category C . Then the quotient category M := A/X carries naturally a triangulated structure whenever (A, A) forms an 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 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, C /X is a triangulated category if and only if (C , C ) is X −mutation, and if and only if τ X = 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 examples that extriangulated categories are neither exact categories nor triangulated categories.

Phantom Ideals and Cotorsion Pairs in Extriangulated Categories

“…To overcome the difficulty, Nakaoka and Palu [15] introduced the notion of externally triangulated categories (extriangulated categories for short) by a careful looking what is necessary in the definition of cotorsion pairs in exact and triangulated cases. Under this notion, exact categories and extension-closed subcategories of triangulated categories both are externally triangulated, and hence, in some levels, it becomes easy to give uniform statements and proofs for the exact and triangulated settings [15,20].…”

Cluster Subalgebras and Cotorsion Pairs in Frobenius Extriangulated Categories

“…We recall the notion of mutation pairs in extriangulated categories from [ZhZ,Definition 3.2]. (1) For any A ∈ A, there exists an E-triangle…”

“…Theorem 1.8. [ZhZ,Theorem 3.15] Let C be an extriangulated category and let X ⊆ A be two additive subcategories of C . We assume that the following two conditions concerning A and X :…”

Proper classes and Gorensteinness in extriangulated categories

“…The notion of extriangulated categories was introduced by Nakaoka and Palu in [5] as a simultaneous generalization of exact categories and triangulated categories. Exact categories and extension closed subcategories of an extriangulated category are extriangulated categories, while there exist some other examples of extriangulated categories which are neither exact nor triangulated, see [5,12,2]. Hence many results hold on exact categories and triangulated categories can be unified in the same framework.…”