\'Etale Stacks as Prolongations

TL;DR

This paper characterizes etale stacks categorically as prolongations of stacks on spaces, explores their properties across various contexts, and connects them to gerbes, Lie groupoids, and moduli stacks, with applications in foliation theory.

Contribution

It provides a categorical characterization of etale stacks as prolongations, links them to gerbes and effective stacks, and applies these insights to moduli stacks and foliation theory.

Findings

Etale stacks are characterized as prolongations of stacks on spaces.

Effective etale stacks are exactly those arising from sheaves.

The category of smooth manifolds with local diffeomorphisms has binary products.

Abstract

In this article, we derive many properties of \'etale stacks in various contexts, and prove that \'etale stacks may be characterized categorically as those stacks that arise as prolongations of stacks on a site of spaces and local homeomorphisms. Moreover, we show that the bicategory of \'etale differentiable stacks and local diffeomorphisms is equivalent to the 2-topos of stacks on the site of smooth manifolds and local diffeomorphisms. An analogous statement holds for other flavors of manifolds (topological, $C^k,$ complex, super...), and topological spaces locally homeomorphic to a given space $X.$ A slight modification of this result also holds in an even more general context, including all \'etale topological stacks, and Zariski \'etale stacks, and we also sketch a proof of an analogous characterization of Deligne-Mumford algebraic stacks. We go on to characterize effective \'etale stacks as precisely those stacks arising as the prolongations of sheaves. It follows that \'etale stacks (and in particular orbifolds) induce a small gerbe over their effective part, and all gerbes over effective \'etale stacks arise in this way. As an application, we show that well known Lie groupoids arising in foliation theory give presentations for certain moduli stacks. For example, there exists a classifying stack for Riemannian metrics, presented by Haefliger's groupoid $R\Gamma$ and submersions into this stack classify Riemannian foliations, and similarly for symplectic structures, with the role of $R\Gamma$ replaced with $\Gamma^{Sp}.$ We also prove some unexpected results, for example: the category of smooth $n$-manifolds and local diffeomorphisms has binary products.