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Hypercovers in Differential Geometry

Cheyne Glass, Emilio Minichiello

TL;DR

The paper addresses the challenge of extracting cocycle data from higher sheaves in differential geometry by establishing strict hypercompleteness for a broad class of sites and showing that the plus construction suffices for sheafification in this context. It proves that the local projective model structure and the Čech projective model structure coincide on several geometrically meaningful sites, enabling explicit fibrant replacement and descent calculations via Čech hypercovers. Central to the approach are Lurie’s refinement techniques for hypercovers and Low’s formula for fibrant replacement, which together yield practical tools for cocycle construction, Čech–sheaf cohomology comparisons, and Verdier hypercovering results. The work provides concrete applications to higher differential geometry, including cocycle derivations, isomorphisms between Čech and sheaf cohomology, and tractable handling of truncated hypercovers, thereby enriching the computational toolkit for higher stacks in differential geometry.

Abstract

In this paper we provide a simple proof that for several sites of interest in differential geometry, the local projective model structure and the Čech projective model structure are equal. In particular, this applies to the site of smooth manifolds with open covers and the site of cartesian spaces with good open covers. As an application, we show that for a presheaf of sets on these sites, applying the plus construction once is enough to sheafify.

Hypercovers in Differential Geometry

TL;DR

The paper addresses the challenge of extracting cocycle data from higher sheaves in differential geometry by establishing strict hypercompleteness for a broad class of sites and showing that the plus construction suffices for sheafification in this context. It proves that the local projective model structure and the Čech projective model structure coincide on several geometrically meaningful sites, enabling explicit fibrant replacement and descent calculations via Čech hypercovers. Central to the approach are Lurie’s refinement techniques for hypercovers and Low’s formula for fibrant replacement, which together yield practical tools for cocycle construction, Čech–sheaf cohomology comparisons, and Verdier hypercovering results. The work provides concrete applications to higher differential geometry, including cocycle derivations, isomorphisms between Čech and sheaf cohomology, and tractable handling of truncated hypercovers, thereby enriching the computational toolkit for higher stacks in differential geometry.

Abstract

In this paper we provide a simple proof that for several sites of interest in differential geometry, the local projective model structure and the Čech projective model structure are equal. In particular, this applies to the site of smooth manifolds with open covers and the site of cartesian spaces with good open covers. As an application, we show that for a presheaf of sets on these sites, applying the plus construction once is enough to sheafify.
Paper Structure (21 sections, 87 theorems, 162 equations)

This paper contains 21 sections, 87 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Sites and Sheaves
  3. Examples of Sites
  4. Simplicial Presheaves
  5. Split Simplicial Presheaves
  6. The Čech and Local Model Structures
  7. The Čech Model Structure
  8. The Local Model Structure
  9. Strict Hypercompleteness
  10. The Plus Construction
  11. Applications
  12. Deriving Cocycles in Higher Differential Geometry
  13. Isomorphism between Čech and Sheaf Cohomology
  14. Verdier's Hypercovering Theorem
  15. Truncated Hypercovers
  16. ...and 6 more sections

Key Result

Lemma 2.5

Suppose that $j$ is a coverage on a small category $\mathcal{C}$ such that pullbacks along covering maps exist. If $r = \{ r_i: U_i \to U \}_{i \in I}$ is a covering family of $U$, then a collection $\{s_i \in X(U_i) \}$ of sections of a presheaf $X$ on $\mathcal{C}$ is a matching family for $r$ if

Theorems & Definitions (219)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5: Minichiello2025
  • Lemma 2.6: Minichiello2025
  • Definition 2.7
  • Lemma 2.8: Minichiello2025
  • Definition 2.9
  • Definition 2.10
  • ...and 209 more