Existence of n-Auslander–Reiten sequences via a finiteness condition

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

“…Namely, Zhou proved that if an (n+2)-angulated category C is locally finite, then C has Auslander-Reiten (n + 2)-angules. Subsequently, Xie-Lu-Wang [20] proved a similar result to Zhou. More precisely, they showed that if an n-abelian category C is locally finite, then C has n-Auslander-Reiten sequences.…”

“…(1) If C is an n-abelian category, then Definition 3.1 coincides with the definition of n-Auslander-Reiten sequence of n-abelian category (cf. [20]).…”

“…Based on the definition of locally finite (n + 2)-angulated categories and locally finite nabelian categories, [20,24], we define the notion of locally finite n-exangulated categories.…”

“…More precisely, they showed that if an n-abelian category C is locally finite, then C has n-Auslander-Reiten sequences. Based on this idea, we have a natural question of whether their results of Zhou [24] and Xie-Lu-Wang [20] can be unified under the framework of n-exangulated categories or whether the result of Zhu-Zhuang [25] has a higher counterpart. In this article, we give an affirmative answer.…”