Abelian quotients of triangulated categories

TL;DR

This paper investigates abelian quotients of triangulated categories, providing criteria for when quotient functors are representable, and explores their properties, especially in 2-Calabi-Yau categories, with applications to cluster-tilting subcategories.

Contribution

It offers explicit descriptions and criteria for abelian quotients of triangulated categories and characterizes when such quotients are cluster-tilting subcategories.

Findings

Quotient functors are representable under certain conditions.

Criteria established for when a functor is a quotient functor.

In 2-Calabi-Yau categories, J is often a cluster-tilting subcategory.

Abstract

We study abelian quotient categories A=T/J, where T is a triangulated category and J is an ideal of T. Under the assumption that the quotient functor is cohomological we show that it is representable and give an explicit description of the functor. We give technical criteria for when a representable functor is a quotient functor, and a criterion for when J gives rise to a cluster-tilting subcategory of T. We show that the quotient functor preserves the AR-structure. As an application we show that if T is a finite 2-Calabi-Yau category, then with very few exceptions J is a cluster-tilting subcategory of T.