On the existence of Auslander-Reiten $n$-exangles in $n$-exangulated categories

TL;DR

This paper proves that locally finite $n$-exangulated categories always contain Auslander-Reiten $n$-exangles, unifying and extending several known results across different categorical frameworks.

Contribution

It establishes the existence of Auslander-Reiten $n$-exangles in locally finite $n$-exangulated categories, generalizing previous results in related categories.

Findings

Existence of Auslander-Reiten $n$-exangles in locally finite $n$-exangulated categories

Unification of results across triangulated, extriangulated, and $n$-abelian categories

Extension of known theorems to a broader categorical context

Abstract

Let $\mathscr{C}$ be an $n$-exangulated category. In this note, we show that if $\mathscr{C}$ is locally finite, then $\mathscr{C}$ has Auslander-Reiten $n$-exangles. This unifies and extends results of Xiao-Zhu, Zhu-Zhuang, Zhou and Xie-Lu-Wang for triangulated, extriangulated, $(n+2)$-angulated and $n$-abelian categories, respectively.