Categorical properties of topological and differentiable stacks

Abstract

The focus of this PhD research is on the theory of topological and differentiable stacks. There are two main themes of this research. The first, is the creation of the theory of compactly generated stacks, which solve many categorical shortcomings of the theory of classical topological stacks. In particular, they are Cartesian closed. Secondly, this thesis develops the theory of small sheaves and stacks over étale topological and differentiable stacks, and explores in particular, the connection between gerbes and ineffective isotropy data. In the first part of the thesis, a convenient 2-category of topological stacks is constructed which is both complete and Cartesian closed. This 2-category, called the 2-category of compactly generated stacks, is the analogue of classical topological stacks, but for a different Grothendieck topology. In fact, there is an equivalence of 2-categories between compactly generated stacks and those classical topological stacks which admit locally compact atlases. Compactly generated stacks are also equivalent to a bicategory of topological groupoids and principal bundles, just as in the classical case. If a classical topological stack and a compactly generated stack have a presentation by the same topological groupoid, then they restrict to the same stack over locally compact Hausdorff spaces and are homotopy equivalent. In the second part of the thesis, 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 étale topological or differentiable 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. We go on to characterize small sheaves and gerbes. We show that ineffective data of étale stacks is completely described by the theory of small gerbes. In particular, it is shown that étale stacks (and in particular orbifolds) induce a small gerbe over their effective part, and all gerbes arise in this way. For nice enough classes of maps, for instance submersions, we show that étale stacks are equivalent to a 2-category of gerbed effective étale stacks. Along the way, we also prove that the 2-category of topoi is a full reflective sub-2-category of localic stacks.