Abelian quotients arising from extriangulated categories via morphism categories

TL;DR

This paper studies how certain quotients of extriangulated categories can be made abelian, unifying results for exact and triangulated categories, and explores their relations to module categories and cluster-tilting subcategories.

Contribution

It establishes conditions under which quotients of extriangulated categories are abelian and connects these to module categories, extending known results to a broader setting.

Findings

Certain quotient categories are abelian.

Equivalence of ideal quotient categories with module categories.

Descriptions of subcategories related to cluster-tilting subcategories.

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 $\mathcal{M}$ be a full subcategory of $\mathcal{C}$ containing $\mathcal{P}$. We show that certain quotient category of $\mathfrak{s}\textup{-def}(\mathcal{M})$, the category of $\mathfrak{s}$-deflations $f:M_{1}\rightarrow M_2$ with $M_1,M_2\in\mathcal{M}$, is abelian. Our main theorem has two applications. If $\mathcal{M}=\mathcal{C}$, we obtain that 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 $\mathcal{M}$ is a rigid subcategory, we show that $\mathcal{M}_{L}/[\mathcal{M}]\cong\textup{mod-}(\mathcal{M}/[\mathcal{P}])$ and $\mathcal{M}_{L}/[\Omega\mathcal{M}]\cong(\textup{mod-}(\mathcal{M}/[\mathcal{P}])^{\textup{op}})^{\textup{op}}$, where $\mathcal{M}_L$ (resp. $\Omega\mathcal{M}$) is the full subcategory of $\mathcal{C}$ of objects $X$ admitting an $\mathfrak{s}$-triangle $\xymatrixrowsep{0.1pc}\xymatrix{X\ar[r]&M_1\ar[r] & M_2\ar@{-->}[r]&} (\textup{resp.} \xymatrixrowsep{0.1pc}\xymatrix{X\ar[r]&P\ar[r] & M\ar@{-->}[r]&})$ with $M_1, M_2\in\mathcal{M}$ (resp. $M\in\mathcal{M}$ and $P\in\mathcal{P}$). In particular, we have $\mathcal{C}/[\mathcal{M}]\cong\textup{mod-}(\mathcal{M}/[\mathcal{P}])$ and $\mathcal{C}/[\Omega\mathcal{M}]\cong(\textup{mod-}(\mathcal{M}/[\mathcal{P}])^{\textup{op}})^{\textup{op}}$ provided that $\mathcal{M}$ is a cluster-tilting subcategory.