Compactly Generated Stacks: A Cartesian Closed Theory of Topological Stacks

TL;DR

This paper introduces a new bicategory of topological stacks, called compactly generated stacks, which is complete and Cartesian closed, providing a more flexible framework for topological stacks with better categorical properties.

Contribution

The paper constructs the bicategory of compactly generated stacks, establishing its equivalence with classical topological stacks admitting locally compact Hausdorff atlases and with topological groupoids.

Findings

The bicategory of compactly generated stacks is complete and Cartesian closed.

Equivalence between compactly generated stacks and classical stacks with certain atlases.

Homotopy equivalence for stacks presented by the same groupoid over locally compact Hausdorff spaces.

Abstract

A convenient bicategory of topological stacks is constructed which is both complete and Cartesian closed. This bicategory, called the bicategory of compactly generated stacks, is the analogue of classical topological stacks, but for a different Grothendieck topology. In fact, there is an equivalence of bicategories between compactly generated stacks and those classical topological stacks which admit locally compact Hausdorff 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.