Projective model structures on diffeological spaces and smooth sets and the smooth Oka principle (Q6621502)

The category \textsf{Diffeo} of diffeological spaces is a Grothendieck quasitopos, being complete and cocomplete, cartesian closed, and locally cartesian closed category. A closely related notion is that of smooth sets, which are sheaves of sets on the site of smooth manifolds and open covers. A diffeological space is a smooth set \(F\) that is a concrete sheaf. Smooth sets contain diffeological spaces as a full subcategory. The category of smooth sets is a Grothendieck topos, so it inherits all the nice properties of diffeological spaces and, in addition, it is a balanced category.\N\NIn complete analogy to topological spaces, one can define a (smooth) singular fnctor \textsf{Smsing}, which endows the categories of smooth sets and diffeological spaces with a relative category structure. Continuing the analogy to topological spaces, one can then inquire whether the resulting relative categories of smooth sets and diffeological spaces can be promoted to model categories by creating the class of fibrations using the functor \textsf{Smsing}, and whether this turns \ into a right Quillen equivalence of model categories.\N\NThe main result of the first part of this paper (\S 2--\S 6) is that the answer in the case of diffeological spaces is negative (Theorem 1.1), which is in sharp contrast to Kihara's [\textit{H. Kihara}, J. Homotopy Relat. Struct. 14, No. 1, 51--90 (2019; Zbl 1459.58001)] model structure on diffeological spaces constructed using a different singular complex structure. The main result of the second part of the paper (\S 7--\S 11) is that for smooth sets we do indeed get a Quillen equivalence of model categories, with rather nice properties of involved model structures (Theorem 1.2). The third part of the paper (\S 12--\S 14) extends Theorem 1.2 to the case of sheaves and presheaves valuued in a left proper combinatorial model category \textsf{V}, such as simplicial sets or chain complexes (Theorem 1.3), establishing a generalization of the smooth Oka principle [\textit{D. Berwick-Evans} et al., ``Classifying spaces of infinity-sheaves'', Preprint, \url{arXiv:1912.10544}]. The author applies these results to establish classification theorems for differential-geometric objects like closed differential forms, principla bundles with connection, and higher bundle gerbes with connection on arbitrary cofibrant diffeological spaces.