Balanced Pairs on Triangulated Categories☆

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 xi with enough xi-projectives and xi-injectives. Assume that xi := xi X = xi(Y) is the proper class induced by a balanced pair (X;Y). We prove that (C; E-xi; s(xi)) is an extriangulated category. Moreover, it is proved that (C; E-xi; s(xi)) is a triangulated category if and only if X = Y = 0, and that (C; E-xi; s(xi)) 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.