Two new classes of n-exangulated categories

TL;DR

This paper introduces two new classes of $n$-exangulated categories, expanding the framework beyond $n$-exact and $(n+2)$-angulated categories, with conditions for quotient categories and new structures via $n$-proper classes.

Contribution

It provides necessary and sufficient conditions for quotient categories to be $n$-exangulated and introduces $n$-proper classes to construct new $n$-exangulated categories beyond existing classes.

Findings

Characterization of when quotient categories are $n$-exangulated.

Introduction of $n$-proper classes in $n$-exangulated categories.

Construction of new $n$-exangulated categories outside $n$-exact and $(n+2)$-angulated classes.

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 and $(n+2)$-angulated categories. Let $\mathcal C$ be an $n$-exangulated category and $\mathcal X$ a full subcategory of $\mathcal C$. If $\mathcal X$ satisfies $\mathcal X\subseteq\mathcal P\cap\mathcal I$, then we give a necessary and sufficient condition for the ideal quotient $\mathcal C/\mathcal X$ to be an $n$-exangulated category, where $\mathcal P$ (resp. $\mathcal I$) is the full subcategory of projective (resp. injective) objects in $\mathcal C$. In addition, we define the notion of $n$-proper class in $\mathcal C$. If $\xi$ is an $n$-proper class in $\mathcal C$, then we prove that $\mathcal C$ admits a new $n$-exangulated structure. These two ways give $n$-exangulated categories which are neither $n$-exact nor $(n+2)$-angulated in general.