Balanced pairs on triangulated categories

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.