AI 中文总结
本研究构建有限非奇异半单纯集及其胞层的微分分次模型，建立范畴等价并推广图与一阶微分演算的对应，刻画胞层的微分分次模描述，还研究了联络与曲率的性质及粘合准则。
AI 中文摘要
我们构建了有限非奇异半单纯集及其胞层的微分分次模型。对有限非奇异半单纯集\textit{S}，我们构造了一个外微分分次代数（DGA）\textit{Ω}^\bullet_\textit{S}，将有限简单有向图与一阶微分演算（FODC）之间的经典反等价关系推广到更高次数。我们刻画了这一构造的本质像，得到了一个范畴等价，该等价在一维情形下恢复了图与一阶微分演算的对应关系。对于固定的\textit{S}，我们随后刻画了一类微分分次\textit{Ω}^\bullet_\textit{S}-模的范畴，该范畴等价于\textit{S}上的胞层范畴，从而对通常的关联代数描述给出了微分精细化。最后，我们研究了该框架下的联络与曲率。联络由逐边线性映射描述，其在二维单形上的曲率为直接边输运与复合边输运的差。当边输运可逆时，该曲率在每个2-单形上为零可等价地视为一种粘合准则，用于将给定的边输运延拓为\textit{P}_\textit{S}上的联络层。
英文摘要
We develop a differential graded model for finite non-singular semisimplicial sets and their cellular sheaves. To a finite non-singular semisimplicial set \(S\), we associate an exterior DGA \(Ω^\bullet_S\), extending the classical anti-equivalence between finite simple directed graphs and first-order differential calculi to higher degrees. We characterize the essential image of this construction, obtaining an equivalence of categories that recovers the graph-FODC correspondence in dimension one. For a fixed \(S\), we then characterize a category of differential graded \(Ω^\bullet_S\)-modules equivalent to the category of cellular sheaves on $S$, thereby giving a differential refinement of the usual incidence-algebra description. Finally, we study connections and curvature in this framework. Connections are described by edgewise linear maps and their curvature on a 2-dimensional simplex is the difference between direct and composite edge transport. When the edge transports are invertible, vanishing of this curvature on every \(2\)-simplex can equivalently be seen as a gluing criterion to extend the given edge transports to a connection sheaf on \(P_S\).
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