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

TL;DR

This paper demonstrates that quotient categories derived from $n$-exangulated categories with cluster tilting subcategories are $n$-abelian, extending previous results and revealing new phenomena in higher homological algebra.

Contribution

It establishes that the quotient of an $n$-exangulated category by a cluster tilting subcategory is an $n$-abelian category, generalizing earlier work and exploring new applications.

Findings

Quotients of $n$-exangulated categories are $n$-abelian.

Extension of Zhou-Zhu's result to $n$-exangulated contexts.

New phenomena observed in $n$-exact categories.

Abstract

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 $\mathscr C$ be an $n$-exangulated category with enough projectives and enough injectives, and $\mathscr X$ a cluster tilting subcategory of $\mathscr C$. In this article, we show that the quotient category $\mathscr C/\mathscr X$ 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.