Abelian quotients arising from extriangulated categories via morphism categories

Abstract

We investigate abelian quotients arising from extriangulated categories via morphism categories, which is a unified treatment for both exact categories and triangulated categories. Let (C, E, s) be an extriangulated category with enough projectives P and M be a full subcategory of C containing P. We show that certain quotient category of s-def(M), the category of s-deflationsis abelian. Our main theorem has two applications. If M = C, we obtain that certain ideal quotient category s-tri(C)/R 2 is equivalent to the category of finitely presented modules mod-C/[P], where s-tri(C) is the category of all s-triangles. If M is a rigid subcategory, we show that M L /[M] ∼ = mod-(M/[P]) and M L /[ΩM] ∼ = (mod-(M/[P]) op ) op , where M L (resp. ΩM) is the full subcategory of C of objects X admitting an striangle X ) with M 1 , M 2 ∈ M (resp. M ∈ M and P ∈ P). In particular, we have C/[M] ∼ = mod-(M/[P]) and C/[ΩM] ∼ = (mod-(M/[P]) op ) op provided that M is a cluster-tilting subcategory. 2010 Mathematics Subject Classification. 18E30. Key words and phrases. extriangulated categories; abelian categories.