Diffeological Principal Bundles and Principal Infinity Bundles

TL;DR

This paper explores the cohomology of diffeological spaces using ch models and establishes a connection between diffeological principal bundles and higher er bundles through homotopy equivalence.

Contribution

It introduces a framework for ch cohomology in diffeology and links classical principal bundles to inite principal bundles via homotopy theory.

Findings

ch cohomology aligns with existing notions in diffeology.

Nerve of diffeological principal G-bundles is weak homotopy equivalent to inite G-bundles.

Bridges diffeological bundle theory with higher topos theory.

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 \v{C}ech 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 \v{C}ech 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.