Balanced pairs on triangulated categories

Abstract

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