# Abelian quotients arising from extriangulated categories via morphism
categories

Research paper by **Zengqiang Lin**

Indexed on: **03 Aug '20**Published on: **30 Jul '20**Published in: **arXiv - Mathematics - Representation Theory**

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#### 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.