Balanced pairs and recollements in extriangulated categories with negative first extensions

TL;DR

This paper introduces the concept of balanced pairs in extriangulated categories with negative first extensions, establishing a bijection with certain proper classes and exploring their behavior in recollements, generalizing previous results.

Contribution

It defines balanced pairs in extriangulated categories with negative first extensions and shows their correspondence with proper classes, extending known theories and applications.

Findings

Established a bijective correspondence between balanced pairs and proper classes with enough projectives and injectives.

Demonstrated that balanced pairs in a category can induce balanced pairs in related categories within a recollement.

Generalized existing results from abelian and triangulated categories to extriangulated categories.

Abstract

A notion of balanced pairs in an extriangulated category with a negative first extension is defined in this article. We prove that there exists a bijective correspondence between balanced pairs and proper classes $\xi$ with enough $\xi$-projectives and enough $\xi$-injectives. It can be regarded as a simultaneous generalization of Fu-Hu-Zhang-Zhu and Wang-Li-Huang. Besides, we show that if $(\mathcal A ,\mathcal B,\mathcal C)$ is a recollement of extriangulated categories, then balanced pairs in $\mathcal B$ can induce balanced pairs in $\mathcal A$ and $\mathcal C$ under natural assumptions. As a application, this result gengralizes a result by Fu-Hu-Yao in abelian categories. Moreover, it highlights a new phenomena when it applied to triangulated categories.