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

33 pages