摘要
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 classesξwith enoughξ-projectives andξ-injectives.Assume thatξ:=ξX=ξ^(Y) is the proper class induced by a balanced pair(X,Y).We prove that(C,Eξ,sξ)is an extriangulated category.Moreover,it is proved that(C,Eξ,sξ)is a triangulated category if and only if X=Y=0,and that(C,Eξ,sξ)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.
基金
Xianhui Fu was supported by YDZJ202101ZYTS168 and the NSF of China(12071064)
Jiangsheng Hu was supported by the NSF of China(12171206)
the Natural Science Foundation of Jiangsu Province(BK20211358)
Haiyan Zhu was supported by Zhejiang Provincial Natural Science Foundation of China(LY18A010032)
the NSF of China(12271481).
