Balanced Pairs on Triangulated Categories☆

作者：Fu, Xianhui; Hu, Jiangsheng; Zhang, Dongdong; Zhu, Haiyan^*

来源：Algebra Colloquium, 2023, 30(03): 385-394.

DOI：10.1142/S1005386723000329

• triangulated category
• proper class
• balanced pair
• extriangulated category

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.

• 单位
  杭州师范大学; 东北师范大学; 浙江工业大学; 浙江师范大学