Diffeological principal bundles and principal infinity bundles
type: Item
instance of: scholarly article
title: Diffeological principal bundles and principal infinity bundles
zbMATH Open document ID: 1547.55007
author: Emilio Minichiello
published in: Journal of Homotopy and Related Structures
publication date: 2024-05-17T00:00:00Z
review text: 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}
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 55N35
Mathematics Subject Classification ID: 55R10
Mathematics Subject Classification ID: 53C99
Mathematics Subject Classification ID: 58A05
Mathematics Subject Classification ID: 58A40
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 55R65
zbMATH DE Number: 7850991
zbMATH Keywords: diffeological spaces
zbMATH Keywords: simplicial presheaves
zbMATH Keywords: Cartesian spaces
zbMATH Keywords: good open covers
MaRDI profile type: Publication
cites work: On the van Kampen theorem
cites work: Convenient categories of smooth spaces
cites work: Homotopy limits, completions and localizations
cites work: Local projective model structures on simplicial presheaves
cites work: Differential cohomology theories as sheaves of spectra
cites work: Q3961493
cites work: Principal \(\infty\)-bundles and smooth string group models
cites work: The \(D\)-topology for diffeological spaces
cites work: Tangent spaces and tangent bundles for diffeological spaces
cites work: Smooth classifying spaces
cites work: Hypercovers and simplicial presheaves
cites work: Function complexes in homotopical algebra
cites work: Universal homotopy theories
cites work: Čech cocycles for differential characteristic classes: an \(\infty \)-Lie theoretic construction
cites work: A homotopy theory for stacks
cites work: Diffeology
cites work: COSIMPLICIAL SPACES AND COCYCLES
cites work: Q4783274
cites work: Higher Topos Theory (AM-170)
cites work: Q4693740
cites work: The diffeology of Milnor's classifying space
cites work: Principal \(\infty \)-bundles: general theory
cites work: Principal \(\infty\)-bundles: presentations
cites work: Numerable open covers and representability of topological stacks
cites work: Categorical homotopy theory
cites work: Internal categories, anafunctors and localisations
cites work: Décalage and Kan's simplicial loop group functor
cites work: Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps
cites work: Q4863380
DOI: 10.1007/S40062-024-00347-4
description: scientific article; zbMATH DE number 7850991