Abelian categories arising from cluster-tilting subcategories II: quotient functors

TL;DR

This paper investigates conditions under which ideal quotients of extriangulated categories become abelian, providing new characterizations of cluster-tilting subcategories and their relation to abelian quotients in 2-Calabi-Yau categories.

Contribution

It introduces a criterion for when an ideal quotient of an extriangulated category is abelian and offers a new homological characterization of cluster-tilting subcategories.

Findings

The ideal quotient becomes abelian when the functor is dense and full.

A new homological characterization of cluster-tilting subcategories is provided.

In 2-Calabi-Yau categories, cluster tilting subcategories correspond to abelian quotients.

Abstract

In this paper, we consider a kind of ideal quotient of an extriangulated category such that the ideal is the kernel of a functor from this extriangulated category to an abelian category. We study a condition when the functor is dense and full, in another word, the ideal quotient becomes abelian. Moreover, a new equivalent characterization of cluster-tilting subcategories is given by applying homological methods according to this functor. As an application, we show that in a connected 2-Calabi-Yau triangulated category B, a functorially finite, extension closed subcategory T of B is cluster tilting if and only if B/T is an abelian category.