Sheaf Theory for \'Etale Geometric Stacks

Abstract

We generalize the notion of a small sheaf of sets over a topological space or manifold to define the notion of a small stack of groupoids over an \'etale topological or differentiable stack. We then provide a construction analogous to the \'etal\'e space construction in this context, establishing an equivalence of 2-categories between small stacks over an \'etale stack and local homeomorphisms over it. We go on to characterize small sheaves and gerbes. We show that ineffective data of \'etale stacks is completely described by the theory of small gerbes. Furthermore, it is shown that \'etale stacks (and in particular orbifolds) induce a small gerbe over their effective part, and all gerbes arise in this way. It follows that ineffective orbifolds, sometimes called non-reduced orbifolds, encode a canonical gerbe over their effective (or reduced) part. For nice enough classes of maps, for instance submersions, we show that \'etale stacks are equivalent to a 2-category of gerbed effective \'etale stacks. Along the way, we also prove that the 2-category of topoi is a full reflective sub-2-category of localic stacks.