Singular chains on Lie groups and the Cartan relations I

Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$. We show that the following categories are naturally equivalent. The category $\mathsf{M}\mathsf{o}\mathsf{d}(C(G))$, of sufficiently smooth modules over the DG-algebra of singular chains on $G$. The category $\mathsf{R}\mathsf{e}\mathsf{p}(Tg)$ of representations of the DG-Lie algebra $T\mathfrak{g}$, which is universal for the Cartan relations. This equivalence extends the correspondence between representations of $G$ and representations of $\mathfrak{g}$. In a companion paper we show that in the compact case, the equivalence can be extended to an ${\mathsf{A}}_{\mathrm{\infty }}$ equivalence of DG-categories.