Mutation Pairs in Abelian Categories

Xu, Jinde; Zhou, Panyue; Ouyang, Baiyu

doi:10.1080/00927872.2015.1053900

Cited by 2 publications

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“…Example 3.20. [XZO,Theorem 2.12] Let X ⊆ A be subcategories of an abelian category C . If A is extension closed and (A, A) is an X -mutation pair, then the quotient category A/X is a triangulated category.…”

Section: Examplesmentioning

confidence: 99%

Triangulated quotient categories revisited

2018

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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.

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Section: Examplesmentioning

confidence: 99%