Homotopy equivalence of topological categories

Way back in the dark ages when I was doing my PhD, but couldn’t settle on a topic, I was looking at trying to understand the homotopy theory of topological groupoids and categories. I had no idea what was done, or how to do this, so I started working some things out in a pedestrian way (around 2008–09). One thing that I now understand that I was lacking was the concept of cohesion, i.e. how Lie groupoids and topological groupoids are different from spaces and groupoids represent homotopy types in their own ways. But I did manage to prove a version of Quillen’s Theorem A about when a functor geometrically realises to a homotopy equivalence, but starting from topological categories (i.e. categories internal to \mathbf{Top}, or rather \mathbf{CGH}). I abandoned this paper very close to being finished, as I started working on what would eventually become my thesis, and also because I was going around in circles a bit, and not sure it was worth releasing. Maybe the result wasn’t that stellar, but I think it’s not been done in this way before (and it is much easier to understand than comparable results in the literature). The paper (with the above title) is now on the arXiv as arxiv:2204.02778. Here’s the theorems I had proved back in the 00’s

Theorem A and Theorem A’ in arxiv:2204.02778

I then used this to show that a weak equivalence of topological categories (ff+eso in the numerable pretopology) geometrically realises to a homotopy equivalence (assuming some mild condition on the codomain).

What is new now is that of course I know a lot more about bicategorical localisation, and this result I can now say implies that the classifying space functor B\colon \mathbf{Cat}(\mathbf{CGH})\to \mathbf{CGH} extends along the Yoneda embedding to define a “classifying space” 2-functor from a suitable 2-category of topological stacks of categories to the 2-category of spaces, maps and homotopy classes of homotopies. This improves on contemporaneous work of Ebert, who defines a homotopy type for certain topological stacks of groupoids, but has to battle size issues, and so only defines it on a small subcategory of stacks. This work was done in a better way indepdently by Noohi, who associated a weak homotopy type to a (large class of) topological stacks. My extension of B is also a somewhat orthogonal generalisation of Ebert’s work, since Noohi works with topological stacks under the open cover topology, and on all spaces, whereas my setup works with the numerable topology, and compactly-generated Hausdorff spaces. It does, however, allow for the full 2-category including non-invertible 2-arrows, which is not covered by the usual familiar (\infty,1)-setup.

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