**Disclaimer.** One might argue that my question is off topic as it is clearly a question about physics...

But I'd like a mathematically phrased answer,
and I expect that only a mathematician can offer an answer that I would deem satisfying.
I hope that this forum is the correct place to ask my question.

*By the way, FQHE = Fractional quantum Hall effect*

Consider a *FQH* fluid in topological phase, on a compact Riemann surface. That is a physical system and, presumably, there is a ** set** of possible states of that physical system.

I'm guessing that that set comes with an equivalence relation: the relation of being physically undistinguishable. I'm not (only) asking about the set of equivalence classes under the above equivalence relation: I'm asking about the actual set of physical states.

Ok, this might sound perverse, so let me justify... I'm secretly guessing that the set of possible states might actually form the objects of a category (groupoid? higher category?), and that the above mentioned relation is that of isomorphism in the category.
If that is indeed the case, then I change my question to: what is the category of possible physical states?

The next question (somewhat more relevant for actually physical applications), is the same one as above, but when the Riemann surface is allowed to have a boundary, e.g. when it is an open subset of ℂ.