Hypercovers in Differential Geometry

TL;DR

This paper proves the equivalence of local and Čech projective model structures in differential geometry sites, simplifying sheafification processes for presheaves of sets.

Contribution

It provides a straightforward proof of model structure equivalence in key differential geometry sites, enabling simpler sheafification with a single plus construction.

Findings

Local and Čech projective model structures are equal in relevant sites.

Applying the plus construction once sheafifies presheaves of sets.

Simplifies sheafification process in differential geometry contexts.

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 \v{C}ech 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.