There is a category of "smooth categories", where the objects and the morphisms don't form sets, but manifolds (and there are some other conditions that I won't repeat here). Important examples are Lie groups and Lie groupoids such as the action groupoid of a homogeneous space.
The action groupoid of a finite $G$-set is equivalent to the stabiliser group of a point, seen as a 1-object category. I want to know whether this is the case in the smooth setting as well.
Given two equivalent smooth categories, what can we say about their object spaces? Doesn't the equivalence imply a homotopy equivalence? Or even a diffeomorphism?