Global spaces and the homotopy theory of stacks

TL;DR

This paper establishes an equivalence between global spaces and a localized category of sheaves on differentiable stacks, demonstrating the cohesive nature of this category and its relation to existing frameworks.

Contribution

It proves the equivalence of global spaces with a homotopy localization of sheaves on differentiable stacks and shows this category's cohesive structure and embedding properties.

Findings

Equivalence between global spaces and localized sheaves on stacks

Cohesive $mbda$-topos structure of the sheaf category

Full embedding of singular-cohesive $mbda$-topos

Abstract

We show that the $\infty$-category of global spaces is equivalent to the homotopy localization of the $\infty$-category of sheaves on the site of separated differentiable stacks, following a philosophy proposed by Gepner-Henriques. We further prove that this $\infty$-category of sheaves is a cohesive $\infty$-topos and that it fully faithfully contains the singular-cohesive $\infty$-topos of Sati-Schreiber.