Diffeological principal bundles and principal infinity bundles (Q6541388)

This paper studies diffeological spaces as a certain kind of discrete simplicial presheaves on the site of cartesian spaces with the coverage of good open covers.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] is concerned with preliminaries about diffeological spaces.\N\N\item[\S 3] turns to diffeological principal \(G\)-bundles, defining \(G\)-cocycles and establishing a bundle construction-type theorem (Theorem 3.15).\N\N\item[\S 4] gives a brisk introduction to sheaf theory, explaining the Baez-Hoffnung Theorem [\textit{J. C. Baez} and \textit{A. E. Hoffnung}, Trans. Am. Math. Soc. 363, No. 11, 5789--5825 (2011; Zbl 1237.58006), Proposition 24]\N\N\item[\S 5] reviews the Čech model structure on simplicial presheaves over cartesian spaces, providing a cofibrant replacement of a diffeological space as the nerve of a diffeological category (Proposition 5.20), which is compared to two other diffeological categories \(\check{C}\left( X\right) \)\ [\textit{D. Krepski} et al., ``Sheaves, principal bundles, and Čech cohomology for diffeological spaces'', Preprint, \url{arXiv:2111.01032}] and \(B//M\)\ [\textit{P. Iglesias-Zemmour}, Isr. J. Math. 259, No. 1, 239--276 (2024; Zbl 1537.55011)]. These three diffeological categories yield three separate notions of Čech cohomology, which are compared in \S 5.3.\N\N\item[\S 6] deals with the main result of this paper that if \(G\)\ is a diffeological group and \(X\)\ is a diffeological space, then the nerve of the category of principal \(G\)-bundles\ on \(X\)\ is weakly homotopy equivalent to the nerve of the category of \(G\)-principal\ \(\infty\)-bundles over \(X\).\N\N\item[Appendix A] compares several categories of concrete sheaves on various sites, which are shown to be all equivalent.\N\end{itemize}