Homological rigidity of quiver representations over ${\mathbb{F}}_1$

We establish a homological rigidity phenomenon for the category of representations of quivers over the virtual field ${\mathbb{F}}_1$, which is inherently non-additive and does not admit classical homological algebra tools. We prove that all higher Yoneda extension groups vanish beyond degree two for arbitrary quivers, including infinite ones. Consequently, the global dimension of the category is universally bounded by 2. Moreover, we obtain a complete classification of quivers according to their homological dimension, which is determined solely by the underlying orientation structure.