Abelian quotients of categories of $n$-exangles

Abstract

The notion of $n$-exangulated categories was introduced by Herschend-Liu-Nakaoka, which is a simultaneous generalization of $n$-exact categories in the sense of Jasso and $(n+2)$-angulated categories in the sense of Geiss-Kelier-Oppermann. Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category with enough projectives $\mathcal{P}$ and $\mathcal{M}$ a full subcategory of $\mathscr{C}$ containing $\mathcal{P}$. In the present paper, It is proved that a certian quotient category of $\mathfrak{s}$-def$(\mathcal{M})$ is abelian. We denoted by $S(\mathscr{C})$ the category of $n$-exangles, whose object are given by distinguished $n$-exangles in $\mathscr{C}$. If $\mathcal{M}=\mathscr{C}$, we obtain that a certain ideal quotient category $S(\mathscr{C})/\mathcal{R}_2$ is equivalent to the category of finitely presented modules mod-$(\mathscr{C}/[\mathcal{P}])$. Furthermore, we present the quotient category $S(\mathscr{C})/\mathcal{R}_2$ always has an abelian structure when taking $n$ as an even number. The abelian quotient $S(\mathscr{C})/\mathcal{R}_2$ admits some nice properties. We describe the projective objects in $S(\mathscr{C})/\mathcal{R}_2$ and characterize the simple objects in $S(\mathscr{C})/\mathcal{R}_2$ as Auslander-Reiten $n$-exangle sequences in $\mathscr{C}$.