Tilting pairs in extriangulated categories

TL;DR

This paper introduces tilting pairs in extriangulated categories, providing a unified framework that generalizes known results in module categories and offers new insights into the structure of these categories.

Contribution

It defines tilting pairs in extriangulated categories, characterizes them via a Bazzoni-type criterion, and establishes an Auslander-Reiten correspondence for classifying certain tilting subcategories.

Findings

Characterization of tilting pairs in extriangulated categories

Establishment of Auslander-Reiten correspondence for tilting pairs

Generalization of known module category results to broader categorical settings

Abstract

Extriangulated categories were introduced by Nakaoka and Palu to give a unification of properties in exact categories and extension-closed subcategories of triangulated categories. A notion of tilting pairs in an extriangulated category is introduced in this paper. We give a Bazzoni characterization of tilting pairs in this setting. We also obtain Auslander-Reiten correspondence of tilting pairs which classifies finite $\mathcal{C}$-tilting subcategories for a certain self-orthogonal subcategory $\mathcal{C}$ with some assumptions. This generalizes the known results given by Wei and Xi for the categories of finitely generated modules over Artin algebras, thereby providing new insights in exact and triangulated categories.