Abelian categories arising from cluster tilting subcategories

TL;DR

This paper generalizes the concept of cluster tilting subcategories from triangulated categories to extriangulated categories, establishing conditions under which the quotient category is abelian, thus broadening the scope of cluster-tilting theory.

Contribution

It introduces pre-cluster tilting subcategories in extriangulated categories and characterizes cluster tilting subcategories via the abelian property of the quotient.

Findings

C is cluster tilting iff B/C is abelian

Generalization from triangulated to extriangulated categories

Provides new framework for cluster-tilting theory

Abstract

For a triangulated category T, if C is a cluster-tilting subcategory of T, then the quotient category T\C is an abelian category. Under certain conditions, the converse also holds. This is an very important result of cluster-tilting theory, due to Koenig-Zhu and Beligiannis. Now let B be a suitable extriangulated category, which is a simultaneous generalization of triangulated categories and exact categories. We introduce the notion of pre-cluster tilting subcategory C of B, which is a generalization of cluster tilting subcategory. We show that C is cluster tilting if and only if B/C is abelian.