Abelian Quotients Arising from Extriangulated Categories via Morphism Categories

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 a certain quotient category of s-def(M), the category of s-deflations integral : M-1 -> M-2 with M-1, M-2 is an element of M, is abelian. Our main theorem has two applications. If M = C, we obtain that a 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] congruent to mod-(M/[P]) and M-L/[M] congruent to (mod-(M/[P])(op))(op), where M-L (resp. Omega M) is the full subcategory of C of objects X admitting an s-triangle X -> M-1 -> M-2 -> (resp. X -> M ) with M-1, M-2 is an element of M (resp. M is an element of M and P is an element of P). In particular, we have C/[M] congruent to mod-(M/[P]) and C/[Omega M] congruent to (mod-(M/[P])(op))(op) provided thatMis a cluster-tilting subcategory.