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Main Author:	Minichiello, Emilio
Format:	Preprint
Published:	2022
Subjects:	Differential Geometry Algebraic Topology Category Theory
Online Access:	https://arxiv.org/abs/2202.11023
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author	Minichiello, Emilio
author_facet	Minichiello, Emilio
contents	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. 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.
format	Preprint
id	arxiv_https___arxiv_org_abs_2202_11023
institution	arXiv
publishDate	2022
record_format	arxiv
spellingShingle	Diffeological Principal Bundles and Principal Infinity Bundles Minichiello, Emilio Differential Geometry Algebraic Topology Category Theory 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. 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.
title	Diffeological Principal Bundles and Principal Infinity Bundles
topic	Differential Geometry Algebraic Topology Category Theory
url	https://arxiv.org/abs/2202.11023

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