Proper classes and Gorensteinness in extriangulated categories

Abstract

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. A notion of proper class in an extriangulated category is defined in this paper. Let C be an extriangulated category and ξ a proper class in C. We prove that C admits a new extriangulated structure. This construction gives extriangulated categories which are neither exact categories nor triangulated categories. Moreover, we introduce and study ξ-Gorenstein projective objects in C and demonstrate that ξ-Gorenstein projective objects share some basic properties with Gorenstein projective objects in module categories or in triangulated categories. In particular, we refine a result of Asadollahi and Salarian (2004) [1]. As an application, the admissible model structure on extriangulated categories is obtained.