but we shall concentrate on the applications to lattice gauge theory extending the work of Phillips and Stone to arbitrary simplicial sets and all dimensions. For an arbitrary simplicial set $S$ and a given Lie group $G$ together with a set of parallel transport functions in their sense, we construct a prismatic set $\bar{P}(S)$ of the same homotopy type as $S$ and a classifying map from $\bar{P}(S)$ to a prismatic version of the standard model for $BG$. This is one of our main results (Theorem 8.1). Geometrically, for $S$ a simplicial complex, $\bar{P}(S)$ is closely related to the nerve of the covering by stars of vertices (Theorem 5.1). In turn this gives a principal $G$-bundle with a connection and thus in principle gives rise via the usual Chern-Weil and Chern-Simons theory to explicit formulas for characteristic classes (Corollary 8.2). We shall return to this elsewhere. One can see [@CrS], [@CnS], [@D], [@F3], [@W] for further information about Chern-Simons Theory.

The paper is organized as follows:

In chapter 2, prismatic sets are defined and their various geometric realizations are studied.

The third chapter introduces the prismatic triangulation of a simplicial map and in particular of a simplicial set. Furthermore, we comment 