Can we see a finite dimensional Hilbert space, $H$ as a groupoid if we include the unitary endomorphisms of $H$? It would be like a category with a single object and just isos.
If so, can we take a category of all such objects and put them together to form a 2category of groupoids?
Ben
This is not viewing the Hilbert space as a groupoid, it's looking at a category whose only object is that space as a groupoid. You can do this, as you can with literally any object in any category, by taking the group of automorphisms. You could put all such categories together into a sub-2-category of the 2-category of groupoids, if you wanted to. But again, this possibility exists in absolutely every category. It's generally of limited interest.