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 and triangulated categories. Let $(\mathcal {C},\mathbb {E},\mathfrak {s})$ be an category with enough projectives $\mathcal {P}$ ${\mathscr{M}}$ full subcategory of {C}$ containing . show that certain quotient $\mathfrak {s}\textup {-def}({\mathscr{M}})$ , the {s}$ -deflations $f {:} M_{1} {\rightarrow } M_{2}$ $M_{1},M_{2} {\in {\mathscr{M}}$ abelian. Our main theorem has two applications. If ${\mathscr{M}} {=} \mathcal we obtain ideal {-tri}(\mathcal {C})/\mathcal {R}_{2}$ equivalent to finitely presented modules $\textup {mod-}(\mathcal {C}/[\mathcal {P}])$ where -tri {C})$ all -triangles. rigid subcategory, ${\mathscr{M}}_{L}/[{\mathscr{M}}]\cong \textup {mod-}({\mathscr{M}}/[\mathcal ${\mathscr{M}}_{L}/[{\Omega }{\mathscr{M}}]\cong (\textup {P}])^{\textup {op}})^{\textup {op}}$ ${\mathscr{M}}_{L}$ (resp. ${\Omega }{\mathscr{M}}$ ) objects X admitting -triangle $M_{1}, M_{2}\in $M\in $P\in ). In particular, have {C}/[{\mathscr{M}}]\cong {C}/[{\Omega provided cluster-tilting subcategory.