Almost Split Triangles and Morphisms Determined by Objects in Extriangulated Categories

Abstract

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