Hypercovers in Differential Geometry

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.

💡 Research Summary

The paper “Hypercovers in Differential Geometry” establishes that for several sites of central interest in differential geometry—most notably the site of smooth manifolds with open covers and the site of Cartesian spaces with good open covers—the local projective model structure and the Čech projective model structure on simplicial presheaves coincide. The authors begin by reviewing the basic notions of sites, coverages, and sheaves, and then introduce simplicial presheaves together with the three standard model structures: the objectwise (projective) model structure, its left Bousfield localization at local weak equivalences (the local projective model structure), and the Čech projective model structure obtained by localizing at the Čech nerve maps of ordinary open covers.

Historically, Dugger–Hollander–Isaksen showed that the fibrant objects in the local projective model structure satisfy hyperdescent, a condition stronger than Čech descent, and that in general the two model structures are not Quillen equivalent. A site for which they are equivalent is called hypercomplete; if the two model structures are literally equal, the site is called strictly hypercomplete. The paper’s main result (Theorem 5.9) proves that a wide class of differential‑geometric sites—paracompact Hausdorff spaces with finite covering dimension, topological manifolds, smooth manifolds, Cartesian spaces with good covers, and Stein manifolds—are strictly hypercomplete.

The proof hinges on a lemma due to Jacob Lurie (referred to as “Lurie’s Lemma”) which asserts that any hypercover can be refined level‑wise by an ordinary open cover. By iteratively applying this refinement, the authors construct a chain of refinements that shows every hypercover is weakly equivalent, in the sense of the model structure, to a Čech nerve of an ordinary cover. Consequently, the left Bousfield localization at Čech nerves already forces all local weak equivalences, and the two model structures coincide. This argument uses only elementary model‑categorical tools (cofibrantly generated projective structures, Bousfield localization) and avoids the heavy ∞‑categorical machinery (post‑nikov towers, homotopy dimension) employed in earlier proofs by Hoyois and Carchedi.

A second major contribution (Theorem 6.15) concerns the plus construction for presheaves of sets. Traditionally, one must apply the plus construction twice to obtain a sheaf. By exploiting strict hypercompleteness, the authors show that on any of the aforementioned sites a single application of the plus construction already yields a sheaf. The key observation is that the plus construction can be interpreted as a fibrant replacement of a discrete simplicial presheaf, and the explicit fibrant replacement formula of Zhen Lin Low (involving homotopy colimits over hypercovers) collapses to a single step when hypercovers are equivalent to Čech nerves.

The paper also discusses several applications. It shows how strict hypercompleteness simplifies cocycle constructions for higher bundles (e.g., bundle gerbes), provides a streamlined proof of Verdier’s hypercovering theorem, and yields direct isomorphisms between Čech cohomology and sheaf cohomology in the differential‑geometric setting. The extensive appendices supply background on augmented simplicial presheaves, Verdier sites, the detailed proof of Lurie’s Lemma, homotopical categories of fibrant objects, and a historical overview of hypercovers.

Overall, the work delivers a concise, model‑categorical proof that many natural sites in differential geometry are strictly hypercomplete, thereby unifying Čech descent and local weak equivalences and greatly simplifying the construction of higher sheaves and their cohomology in these contexts.