About extending the Chern-Weil classification to the simplicial principal bundles

After introducing the simplicial manifolds, such as the different ways of defining the differential forms on them, we summarized a canonical way of calculating the characteristic classes of a $G$-principal bundle by computing them on the classifying bundle $EG⟶BG$. Finally, we calculated the first Pontryagin class on the classifying bundle of the Lie matrix groups and showed that for certain of them, the computed form is equal to the symplectic form on $BG$ given by some authors up to a constant coefficient.