Balanced pairs on triangulated categories
Xianhui Fu, Jiangsheng Hu, Dongdong Zhang, Haiyan Zhu

TL;DR
This paper introduces balanced pairs in triangulated categories, establishes a correspondence with proper classes, and characterizes when these categories are extriangulated, triangulated, or exact, providing numerous examples of extriangulated categories.
Contribution
It defines balanced pairs in triangulated categories and links them to proper classes, expanding the understanding of extriangulated categories beyond classical cases.
Findings
Balanced pairs correspond bijectively to proper classes with enough projectives and injectives.
The constructed categories are extriangulated, triangulated, or exact depending on the properties of the pairs.
Many examples of extriangulated categories that are neither exact nor triangulated are provided.
Abstract
Let be a triangulated category. We first introduce the notion of balanced pairs in , and then establish the bijective correspondence between balanced pairs and proper classes with enough -projectives and enough -injectives. Assume that is the proper class induced by a balanced pair . We prove that is an extriangulated category. Moreover, it is proved that is a triangulated category if and only if ; and that is an exact category if and only if . As an application, we produce a large variety of examples of extriangulated categories which are neither exact nor…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Rings, Modules, and Algebras
