  We prove that the category of diffeological spaces does not admit a model
structure transferred via the smooth singular complex functor from simplicial
sets, resolving in the negative a conjecture of Christensen and Wu. Embedding
diffeological spaces into sheaves of sets (not necessarily concrete) on the
site of smooth manifolds, we then prove the existence of a proper combinatorial
model structure on such sheaves transferred via the smooth singular complex
functor from simplicial sets. We show the resulting model category to be
Quillen equivalent to the model category of simplicial sets. We then show that
this model structure is cartesian, all smooth manifolds are cofibrant, and
establish the existence of model structures on categories of algebras over
operads. We use these results to establish analogous model structures on
simplicial presheaves on smooth manifolds, as well as presheaves valued in left
proper combinatorial model categories, and prove a generalization of the smooth
Oka principle established in arXiv:1912.10544. We finish by establishing
classification theorems for differential-geometric objects like closed
differential forms, principal bundles with connection, and higher bundle gerbes
with connection on arbitrary cofibrant diffeological spaces.
