Pre-$(n+2)$-angulated categories

Abstract

In this article, we introduce the notion of pre-$(n+2)$-angulated categories as higher dimensional analogues of pre-triangulated categories defined by Beligiannis-Reiten. We first show that the idempotent completion of a pre-$(n+2)$-angulated category admits a unique structure of pre-$(n+2)$-angulated category. Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category and $\mathcal{X}$ be a strongly functorially finite subcategory of $\mathcal{C}$. We then show that the quotient category $\mathcal{C}/\mathcal{X}$ is a pre-$(n+2)$-angulated category.These results allow to construct several examples of pre-$(n+2)$-angulated categories. Moreover, we also give a necessary and sufficient condition for the quotient $\mathcal{C}/\mathcal{X}$ to be an $(n+2)$-angulated category.

20 pages. arXiv admin note: text overlap with arXiv:2108.07985, arXiv:2006.02223