Quotient categories with exact structure from $(n+2)$-rigid subcategories in extriangulated categories

TL;DR

This paper introduces higher extension groups in extriangulated categories and studies quotient categories derived from $(n+2)$-rigid subcategories, establishing their exact structures and conditions for abelianity.

Contribution

It defines higher $ extbf{E}$-extension groups and characterizes quotient categories with exact structures from $(n+2)$-rigid subcategories in extriangulated categories.

Findings

Quotients are equivalent to subcategories of functor categories.

Exact structures can be induced on these quotients.

Under certain conditions, quotients are abelian categories.

Abstract

In this work we introduce the notion of higher $\mathbb{E}$-extension groups for an extriangulated category $\mathcal{C}$ and study the quotients $\mathcal{X}_{n+1}^{\vee}/[\mathcal{X}]$ and $\mathcal{X}_{n+1}^{\wedge}/[\mathcal{X}]$ when $\mathcal{X}$ is an $(n+2)$-rigid subcategory of $\mathcal{C}$. We also prove (under mild conditions) that each one is equivalent to a suitable subcategory of the category of functors of the stable category of $\mathcal{X}_{n}^{\vee}$ and the co-stable category of $\mathcal{X}_{n}^{\wedge}$, respectively. Moreover, it can be induced an exact structure through these equivalences and we analyze when such quotients are weakly idempotent complete, Krull-Schmidt or abelian. The above discussion is also considered in the particular case of an $(n+2)$-cluster tilting subcategory of $\mathcal{C}$ since in this case we know that $\mathcal{X}_{n+1}^{\vee}=\mathcal{C}=\mathcal{X}_{n+1}^{\wedge}.$ Finally, by considering the category of conflations of a exact category, we show that it is possible to get an abelian category from these quotients.