Finite semisimplicial sets, differential graded algebras, and cellular sheaves

Abstract

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 \(\Omega^\bullet_S\), 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 \(\Omega^\bullet_S\)-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 \(P_S\).