Étale Stacks as Prolongations

Abstract

In this article, we derive many properties of étale stacks in various contexts, and prove that étale 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 étale 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 étale topological stacks, and Zariski étale stacks, and we also sketch a proof of an analogous characterization of Deligne-Mumford algebraic stacks. We go on to characterize effective étale stacks as precisely those stacks arising as the prolongations of sheaves. It follows that étale stacks (and in particular orbifolds) induce a small gerbe over their effective part, and all gerbes over effective étale 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.

65 pages. Contents of Section 2 appears also in arXiv:1011.6070. Notable changes from first version: Changed title. Added a section on computations of presentations for moduli stacks in terms of groupoids. Moved background material to the appendix. Removed Section 5