Diffeological principal bundles and principal infinity bundles

Abstract

In this paper, we study diffeological spaces as certain kinds of discrete simplicial presheaves on the site of cartesian spaces with the coverage of good open covers. The Čech model structure on simplicial presheaves provides us with a notion of \(\infty \)-stack cohomology of a diffeological space with values in a diffeological abelian group A. We compare \(\infty \)-stack cohomology of diffeological spaces with two existing notions of Čech cohomology for diffeological spaces in the literature Krepski et al. (Sheaves, principal bundles, and Čech cohomology for diffeological spaces. (2021). arxiv:2111 01032 [math.DG]), Iglesias-Zemmour (Čech-de-Rham Bicomplex in Diffeology (2020). http://math.huji.ac.il/piz/documents/CDRBCID.pdf). Finally, we prove that for a diffeological group G, that the nerve of the category of diffeological principal G-bundles is weak homotopy equivalent to the nerve of the category of G-principal \(\infty \)-bundles on X, bridging the bundle theory of diffeology and higher topos theory.