Intervals of $s$-torsion pairs in extriangulated categories with negative first extensions

TL;DR

This paper introduces a new framework for $s$-torsion pairs within extriangulated categories with negative first extensions, unifying concepts from $t$-structures and torsion pairs in abelian categories.

Contribution

It defines the notion of hearts of intervals of $s$-torsion pairs, establishing a bijection with $s$-torsion pairs in the heart, generalizing existing bijections in the field.

Findings

Unifies $t$-structure and torsion pair bijections.

Defines hearts of intervals in $s$-torsion pairs.

Establishes a bijective correspondence between intervals and $s$-torsion pairs.

Abstract

As a general framework for the studies of $t$-structures on triangulated categories and torsion pairs in abelian categories, we introduce the notions of extriangulated categories with negative first extensions and $s$-torsion pairs. We define a heart of an interval in the poset of $s$-torsion pairs, which naturally becomes an extriangulated category with a negative first extension. This notion generalizes hearts of $t$-structures on triangulated categories and hearts of twin torsion pairs in abelian categories. In this paper, we show that an interval in the poset of $s$-torsion pairs is bijectively associated with $s$-torsion pairs in the corresponding heart. This bijection unifies two well-known bijections: One is the bijection induced by HRS-tilt of $t$-structures on triangulated categories. The other is Asai--Pfeifer's and Tattar's bijections for torsion pairs in an abelian category, which is related to $\tau$-tilting reduction and brick labeling.