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

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 $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category and $\mathscr{X}$ be a strongly functorially finite subcategory of $\mathscr{C}$. We then show that the quotient category $\mathscr{C}/\mathscr{X}$ 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 $\mathscr{C}/\mathscr{X}$ to be an $(n+2)$-angulated category.