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

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 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.

Stratifying systems and Jordan-Hölder extriangulated categories

“…(iii) Independently, Adachi-Tsukamoto [AT22] have recently investigated similar concepts in the extriangulated setting, namely those of mixed standardisable sets and mixed (bi)stratifying systems, which generalise work of Dlab-Ringel [DR92], [ES03] and the works discussed above. In [AT22], when working in an extriangulated category (A, E, s) equipped with a negative first extension structure E −1 in the sense of [AET21], the authors define and study sets of objects Φ = {Φ 1 , . .…”

“…(ii) If A is equipped with a negative first extension structure E −1 in the sense of [AET21], then the condition of A(P i , −) being left exact on F(Φ) is equivalent to E −1 (P i , −) being right exact on F(Φ). Alternatively, E −1 (P i , −) F (Φ) = 0 gives a sufficient condition for the left exactness of A(P i , −)| F (Φ) .…”