Balanced pairs on triangulated categories

TL;DR

This paper introduces balanced pairs in triangulated categories, establishes a correspondence with proper classes, and characterizes when these categories are extriangulated, triangulated, or exact, providing numerous examples of extriangulated categories.

Contribution

It defines balanced pairs in triangulated categories and links them to proper classes, expanding the understanding of extriangulated categories beyond classical cases.

Findings

Balanced pairs correspond bijectively to proper classes with enough projectives and injectives.

The constructed categories are extriangulated, triangulated, or exact depending on the properties of the pairs.

Many examples of extriangulated categories that are neither exact nor triangulated are provided.

Abstract

Let $\mathcal{C}$ be a triangulated category. We first introduce the notion of balanced pairs in $\mathcal{C}$, and then establish the bijective correspondence between balanced pairs and proper classes $\xi$ with enough $\xi$-projectives and enough $\xi$-injectives. Assume that $\xi:=\xi_{\mathcal{X}}=\xi^{\mathcal{Y}}$ is the proper class induced by a balanced pair $(\mathcal{X},\mathcal{Y})$. We prove that $(\mathcal{C}, \mathbb{E}_\xi, \mathfrak{s}_\xi)$ is an extriangulated category. Moreover, it is proved that $(\mathcal{C}, \mathbb{E}_\xi, \mathfrak{s}_\xi)$ is a triangulated category if and only if $\mathcal{X}=\mathcal{Y}=0$; and that $(\mathcal{C}, \mathbb{E}_\xi, \mathfrak{s}_\xi)$ is an exact category if and only if $\mathcal{X}=\mathcal{Y}=\mathcal{C}$. As an application, we produce a large variety of examples of extriangulated categories which are neither exact nor triangulated.