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