Double Category of Topological Stacks

Question

There are two equivalent ways of describing topological stacks.

1. One is the "stacky" definition, that is, a topological stack is a stack $\mathbb{X}$ on $Top$ (a Grothendieck universe thereof, if you'd like) equipped with the topology generated by open covers, such that $\mathbb{X}$ admits an atlas (representable epimorphism) $X \to \mathbb{X}$ from a topological space. This is equivalent to saying that $\mathbb{X}$ is 2-iso to the stackification of a pseudofunctor $Hom(blank,G)$ for some topological groupoid $G$. Topological stacks are then the full sub-2-category all stacks on $Top$ consisting of those stacks with an atlas.

2. One is a "groupoidy" definition. The bicategory $BunGpd$ has topological groupoids as objects, and a morphism $H \to G$ is a principal $G$-bundle over $H$ (and biequivariant maps of as 2-cells). This bicategory is equivalent to that of topological stacks.

However, 2.) can be naturally strengthened to a weak double category by declaring vertical morphisms to be continuous functors and horizontal arrows to be principal bundles.

Now, I have a preference to working in 1.) as the stacky-language is quite useful. However, 2.) manifestly has "more structure". My question is, is there a way to "beef up" topological stacks into a weak double category in a natural way? I'm not really satisfied with taking the objects to be topological stacks with a preferred atlas and declaring the vertical arrows to be maps which factor through these atlases.

Answer 1

A quick answer regarding 2. This could be approached by using a proarrow equipment, of which I know nothing bar their existence and the easy example of Prof - a weak double category with vertical arrows honest functors, and horizontal arrows profunctors. Given the connection (tenuous, perhaps) between the bicategory of (saturated) anafunctors/generalised morphisms between ordinary small groupoids and profunctors, I would think that a lot of this would go across without too much difficulty.