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

Abstract

AbstractAs 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 generalises 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 the 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 labelling.