# Double Category of Topological Stacks

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.

• You already gave the answer to the question in your last paragraph. I don't think that anything else can be said. Jun 1, 2010 at 17:12
• I think I agree with Andre. Since inequivalent groupoids can give rise to equivalent stacks, the objects of your double category will have to be stacks together with some sort of extra structure, and it seems like that extra structure will have to be essentially equivalent to giving a presentation of them as a groupoid. Jun 1, 2010 at 17:21