Balanced Pairs on Triangulated Categories

Abstract

Let [Formula: see text] be a triangulated category. We first introduce the notion of balanced pairs in [Formula: see text], and then establish the bijective correspondence between balanced pairs and proper classes [Formula: see text] with enough [Formula: see text]-projectives and [Formula: see text]-injectives. Assume that [Formula: see text] is the proper class induced by a balanced pair [Formula: see text]. We prove that [Formula: see text] is an extriangulated category. Moreover, it is proved that [Formula: see text] is a triangulated category if and only if [Formula: see text], and that [Formula: see text] is an exact category if and only if [Formula: see text]. As an application, we produce a large variety of examples of extriangulated categories which are neither exact nor triangulated.