From $n$-exangulated categories to $n$-abelian categories

Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of $n$-exact categories in the sense of Jasso and $(n+2)$-angulated in the sense of Geiss-Keller-Oppermann. Let $\mathcal{C}$ be an $n$-exangulated category with enough projectives and enough injectives, and $\mathcal{X}$ a cluster tilting subcategory of $\mathcal{C}$. In this article, we show that the quotient category $\mathcal{C}\mathrm{/}\mathcal{X}$ is an $n$-abelian category. This extends a result of Zhou-Zhu for $(n+2)$-angulated categories. Moreover, it highlights new phenomena when it is applied to $n$-exact categories.