AI 中文总结
研究有限域\(\mathbb{F}_1\)上箭图表示范畴的同调刚性，证明任意箭图高于二阶的 Yoneda 扩张群消失，范畴整体维数有界于 2，并依据同调维数对箭图完全分类，分类由底层定向结构决定。
AI 中文摘要
我们为有限域\(\mathbb{F}_1\)上箭图表示的范畴建立了一种同调刚性现象，该范畴本质上是非加法的且不适用经典同调代数工具。我们证明对于任意箭图（包括无限箭图），所有高于二阶的 Yoneda 扩张群都消失。因此，该范畴的整体维数普遍有界于 2。此外，我们根据同调维数对箭图进行了完全分类，这仅由底层定向结构决定。
英文摘要
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.
