Is there a "universal connection" on the universal $G$-bundle? Let $G$ a Lie group, $X$ a smooth manifold. Let $EG \rightarrow BG$ be the (topological) universal $G$-bundle.
We know for every (topological) principal $G$-bundle $P \rightarrow X$
there is a (continuous) classifying map $f_P : X \rightarrow BG$
such that $P \approx f_P^*EG$.
I was wondering if there exists a (perhaps infinite-dimensional) smooth structure on $EG \rightarrow BG$,
and a principal $G$-connection $\omega$ on $EG$,
such that every smooth principal $G$-bundle on $X$ equipped with connection
is a pullback of $EG \rightarrow BG$ equipped with $\omega$, along some smooth map $X\rightarrow BG$?
 A: This may not be exactly what you had in mind, but another more modern reference for the universal connection on the universal principal $G$-bundle with connection in the language of abstract homotopy theory can be found here. In this paper the diffeological spaces or differentiable spaces of Mostow are replaced with simplicial (pre)sheaves (if I understand correctly).
In this context, the universal principal $G$-bundle with connection is a map of simplicial sheaves $E_{\nabla}G\to B_{\nabla}G$ where $E_{\nabla}G(M)$ is the nerve of the groupoid of trivial $G$-bundles with connection over $M$ and $B_{\nabla}G$(M) is the nerve of the groupoid of all principal $G$-bundles with connection over $M$. The map itself is the canonical map associated with the homotopy quotient $E_{\nabla}G//G$ (here $G$ is the discrete simplicial sheaf represented by G) composed with an equivalence $E_{\nabla}G//G\xrightarrow{\sim}B_{\nabla}G$. In this sense it truly does exhibit the structure of a principal $G$-bundle as the quotient of a "space" (simplicial sheaf) by the free action of a "Lie group" (simplicial sheaf represented by a Lie group). The universal connection is a map of simplicial sheaves $\Theta^{\mathrm{univ}}:E_{\nabla}G\to \Omega^{1}\otimes\mathfrak{g}$ where $\Omega^{1}\otimes\mathfrak{g}$ is the discrete simplicial sheaf that assigns to a manifold $M$ the set $\Omega^{1}(M:\mathfrak{g})$. Concretely, $\Theta^{\mathrm{univ}}$ maps a trivial bundle $P\to M$ with connection $\omega\in\Omega^{1}(P;\mathfrak{g})$ to the pullback $s^{*}\omega$ by a global section $M\xrightarrow{s}P$.
It is proven in the paper that given any honest principal $G$-bundle $P\to M$ with honest connection $\omega\in\Omega^{1}(P;\mathfrak{g})$, there exists a unique map of simplicial sheaves $P\xrightarrow{f} E_{\nabla}G$ such that $\omega = f^{*}\Theta^{\mathrm{univ}}$. It is implicit that one must first upgrade $P$ and $X$ to (discrete representable) simplicial sheaves, and further upgrade $\omega$ to a map of simplicial sheaves $P\xrightarrow{\omega}\Omega^{1}\otimes\mathfrak{g}$ in order for any of this to make sense, but a similar "upgrading" is done in the framework using diffeological spaces.
The advantage of this language is that, at the expense of introducing a complicated framework, the proof of existence, i.e. the construction of the map $f$ and the proof of its desired properties, becomes an almost tautological affair of tracing the relevant definitions.
A: It appears there's an nLab article on the universal connection; it mentions references describing both the case for bounded dimension (including the paper linked in the comments above), and a paper describing the unbounded dimension case using differentiable spaces.
