Almost split triangles and morphisms determined by objects in extriangulated categories

Abstract

Let (C,E,s) be an Ext-finite, Krull-Schmidt and k-linear extriangulated category with k a commutative artinian ring. We define an additive subcategory Cr (respectively, Cl) of C in terms of the representable functors from the stable category of C modulo s-injectives (respectively, s-projectives) to k-modules, which consists of all s-projective (respectively, s-injective) objects and objects isomorphic to direct summands of finite direct sums of all third (respectively, first) terms of almost split s-triangles. We investigate the subcategories Cr and Cl in terms of morphisms determined by objects, and then give equivalent characterizations on the existence of almost split s-triangles.