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.
Categories: math.RT math.RA