  Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$. We
show that the following categories are naturally equivalent. The category
$\mathsf{Mod}(C(G))$, of sufficiently smooth modules over the DG-algebra of
singular chains on $G$. The category $\mathsf{Rep}(Tg)$ 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 $\mathsf{A}_\infty$
equivalence of DG-categories.
