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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.

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    $\begingroup$ honestly, my impression (which is one of a largely ignorant bystander) is that all those people saying that higher categories are like topological spaces lie slightly (in most implementations of the idea of higher category, higher categories behave like homotopy types of spaces, not like spaces themselves). I even saw a MO comment somewhere to that effect... $\endgroup$
    – user138661
    Apr 25, 2019 at 21:12
  • $\begingroup$ ...so to use their terminology, "on the nose" the answer to your question would no, there is no obvious categorical analogue of geometric topology. Maybe there are some other contexts where your question makes sense and has a positive answer. But I should be shutting up by this point. $\endgroup$
    – user138661
    Apr 25, 2019 at 21:12
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    $\begingroup$ You can associate a space to a category: the geometric realisation of the nerve. This is called the classifying space. Functors induce maps, and natural transformations homotopies. But you can apply anything of topology what you want to these spaces. Is that what you are after? $\endgroup$
    – Thomas Rot
    Apr 25, 2019 at 21:14
  • $\begingroup$ I'm interested in something more functorial Thomas, but maybe I just don't know what I'm talking about. Perhaps in all the instances I care about, an isomorphism of stacks is the right notion, but the point of this question is to gain some intuition about some line in the sand whose location I don't understand. $\endgroup$ Apr 25, 2019 at 21:19

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