General heart construction for twin torsion pairs on triangulated categories

TL;DR

This paper introduces a new general construction of an abelian category called the 'general heart' using twin torsion pairs on triangulated categories, unifying previous approaches and broadening the framework for categorical quotients.

Contribution

It provides a novel, unified method to construct abelian categories from twin torsion pairs, extending prior constructions related to t-structures and cluster tilting subcategories.

Findings

Unified construction of abelian categories from twin torsion pairs

Generalizes previous heart constructions for t-structures and cluster tilting

Broadens the scope of categorical quotients in triangulated categories

Abstract

In our previous article, we constructed an abelian category from any torsion pair on a triangulated category. This generalizes the heart of a $t$-structure and the ideal quotient by a cluster tilting subcategory. Recently, generalizing the quotient by a cluster tilting subcategory, Buan and Marsh showed that an integral preabelian category can be constructed as a quotient, from a rigid object in a triangulated category with some conditions. In this article, by considering a pair of torsion pairs, we make a simultaneous genralization of these two constructions.