There is some conventional wisdom that an equivalence of categories is akin to a homotopy equivalence between topological spaces. If I were forced to explain this wisdom, I'd fail miserably, but there are some concrete instances where a category is equivalent to a category which seems significantly smaller. The example which comes from my field is the equivalence of the category of flat $G$-bundles over a reasonable topological space $M$ (here $G$ is a Lie group), and homomorphisms from the fundamental group of $M$ to $G.$
In the homotopy theoretic world, extreme examples of this "smallness" behavior would be contractible topological spaces. But, there are much more refined invariants than homotopy equivalence for topological spaces.
My question is this: is there a gradation of fineness of "equivalence" for categories which mirrors the situation for topological spaces, and further, for manifolds.
For example, if a homotopy equivalence is like an equivalence of categories, what is the equivalent notion of a homeomorphism for a pair of categories. If our topological spaces are manifolds, what about a diffeomorphism. If they are symplectic manifolds, what about symplectomorphisms, etc. Of course, this should involve decorating our categories with more structure, but I'm interested in any answer, especially since some category theory experts frequent this site.
Perhaps this involves things like A-infinity categories and various other gadgets that I don't yet understand, but I would love if that were a good reason to want to understand them.
As always, I appreciate any answers and thanks for reading.