An étalé space construction for stacks

Abstract

We generalize the notion of a sheaf of sets over a space to define the notion of a small stack of groupoids over an étale stack. We then provide a construction analogous to the étalé space construction in this context, establishing an equivalence of $2$–categories between small stacks over an étale stack and local homeomorphisms over it. These results hold for a wide variety of types of spaces, for example, topological spaces, locales, various types of manifolds, and schemes over a fixed base (where stacks are taken with respect to the Zariski topology). Along the way, we also prove that the $2$–category of topoi is fully reflective in the $2$–category of localic stacks.