  We introduce a basic geometric-categorical notion of a smooth simplicial set.
Loosely, this is to Chen/diffeological spaces of Chen-Souriau as simplicial
sets are to spaces. Given a Frechet Lie group $G$, and a chosen Grothendieck
universe $\mathcal{U} $ of a certain type, we give a new construction of a
classifying space of $G$: $|BG ^{\mathcal{U} }|$, so that $BG ^{\mathcal{U} }$
is a smooth Kan complex. When $G$ in addition has the homotopy type of CW
complex, there is a homotopy equivalence $BG \simeq |BG ^{\mathcal{U} }|$,
where $BG$ is the usual Milnor classifying space. This leads to our main
application that for $G$ an infinite dimensional Lie group, having the homotopy
type of CW complex, there is a universal Chern-Weil homomorphism:
\begin{equation*}
  \mathbb{R} [\mathfrak g] ^{G} \to H ^{*} (BG,
  \mathbb{R}), \end{equation*} satisfying naturality, and generalizing the
classical Chern-Weil homomorphism for finite dimensional Lie groups. As one
basic example we give a full statement and proof of Reznikov's conjecture,
which in particular gives an elementary proof of a theorem of Kedra-McDuff, on
the topology of $BHam (\mathbb{CP} ^{n} ) $. We also give a construction of the
universal coupling class for all possibly non-compact symplectic manifolds.
