Étale stacks as prolongations

Abstract

We give a complete and categorical characterization of étale stacks (generalized orbifolds) in various geometric contexts, including differentiable stacks and topological stacks. This characterization states that étale stacks are precisely those stacks that arise as prolongations of stacks on a site of spaces and local homeomorphisms, and makes no mention of an atlas or groupoid presentation. Moreover, we show that the bicategory of étale differentiable stacks and local diffeomorphisms is equivalent to the bicategory of stacks on the site of smooth manifolds and local diffeomorphisms. An analogous statement holds for other flavors of manifolds (topological, Ck, 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, which are analogues of Deligne-Mumford stacks for the Zariski topology, and we also sketch a proof of an analogous characterization of classical Deligne-Mumford algebraic stacks (which we prove in full in [7] by different methods). We go on to characterize effective étale stacks (e.g. reduced orbifolds) as precisely those stacks arising as the prolongations of sheaves of sets (rather than stacks of groupoids). 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. We show that well known Lie groupoids from foliation theory literature can be reproduced from this framework as natural presentations for moduli stacks of certain geometric structures. For example, there exists a classifying stack for Riemannian metrics, presented by Haefliger's groupoid RΓ [13] and submersions into this stack classify Riemannian foliations, and similarly for symplectic structures, with the role of RΓ replaced with ΓSp. We also prove some unexpected results, for example: the category of smooth n-manifolds and local diffeomorphisms has binary products.