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We prove that the category of diffeological spaces does not admit a modelstructure transferred via the smooth singular complex functor from simplicialsets, resolving in the negative a conjecture of Christensen and Wu. Embeddingdiffeological spaces into sheaves of sets (not necessarily concrete) on thesite of smooth manifolds, we then prove the existence of a proper combinatorialmodel structure on such sheaves transferred via the smooth singular complexfunctor from simplicial sets. We show the resulting model category to beQuillen equivalent to the model category of simplicial sets. We then show thatthis model structure is cartesian, all smooth manifolds are cofibrant, andestablish the existence of model structures on categories of algebras overoperads. We use these results to establish analogous model structures onsimplicial presheaves on smooth manifolds, as well as presheaves valued in leftproper combinatorial model categories, and prove a generalization of the smoothOka principle established in arXiv:1912.10544. We finish by establishingclassification theorems for differential-geometric objects like closeddifferential forms, principal bundles with connection, and higher bundle gerbeswith connection on arbitrary cofibrant diffeological spaces.

Projective model structures on diffeological spaces and smooth sets and the smooth Oka principle