The Diffeological \v{C}ech-de Rham Obstruction

Abstract

Using higher topos theory, we explore the obstruction to the \v{C}ech-de Rham map being an isomorphism in each degree for diffeological spaces. In degree 1, we obtain an exact sequence which interprets Iglesias-Zemmour's construction from "\v{C}ech-de Rham Bicomplex in Diffeology" in $\infty$-stack cohomology. We obtain new exact sequences in all higher degrees. These exact sequences are constructed using homotopy pullback diagrams that include the $\infty$-stack classifying higher $\mathbb{R}$-bundle gerbes with connection. We also obtain a conceptual and succinct proof that the $\infty$-stack cohomology of the irrational torus $T_K$ for $K \subset \mathbb{R}$ a diffeologically discrete subgroup, agrees with the group cohomology of $K$ with values in $\mathbb{R}$. Finally, for a Lie group $G$, we prove that the groupoid of diffeological principal $G$-bundles with connection one obtains via higher topos theory is equivalent to the groupoid of diffeological principal $G$-bundles with connection defined in Waldorf's "Transgression to Loop Spaces and its Inverse, I".