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The Diffeological Čech-de Rham Obstruction

Emilio Minichiello

Abstract

Using higher topos theory, we explore the obstruction to the Čech-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 "Čech-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".

The Diffeological Čech-de Rham Obstruction

Abstract

Using higher topos theory, we explore the obstruction to the Čech-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 "Čech-de Rham Bicomplex in Diffeology" in -stack cohomology. We obtain new exact sequences in all higher degrees. These exact sequences are constructed using homotopy pullback diagrams that include the -stack classifying higher -bundle gerbes with connection. We also obtain a conceptual and succinct proof that the -stack cohomology of the irrational torus for a diffeologically discrete subgroup, agrees with the group cohomology of with values in . Finally, for a Lie group , we prove that the groupoid of diffeological principal -bundles with connection one obtains via higher topos theory is equivalent to the groupoid of diffeological principal -bundles with connection defined in Waldorf's "Transgression to Loop Spaces and its Inverse, I".
Paper Structure (10 sections, 47 theorems, 140 equations)

This paper contains 10 sections, 47 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Smooth Sheaves and Diffeological Spaces
  3. Simplicial Presheaves
  4. The Irrational Torus
  5. The Dold-Kan Correspondence
  6. Examples of $\infty$-stacks
  7. The Čech de Rham Obstruction
  8. Diffeological Principal Bundles with Connection
  9. Totalization
  10. Proof of Theorem \ref{['th hopullback diagram']}

Key Result

Theorem 2.7

Given a diffeological space $X$ and a diffeological group $G$, the functor is an equivalence of groupoids.

Theorems & Definitions (112)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7: minichiello2022diffeological
  • Definition 2.8
  • Theorem 2.9: baez2009convenient
  • Corollary 2.10
  • ...and 102 more