The systems I have in mind are for example Kitaev's toric code model (arXiv:quant-ph/9707021) and Kitaev's honeycomb model (arXiv:cond-mat/0506438). What I'm looking for is a mathematically rigorous definition of superselection sector, also known as quasiparticle type, in such anyon systems, which I've been unable to find in the literature.
(Actually I was able to make a rigorous definition of superselection sector that works basically for any model whose fusion rules are always of the form $x\times y = z$, i.e. there's only one term in the right-hand side. This encompasses the toric code but unfortunately not the honeycomb model, so it doesn't seem illuminating to spell out my definition here.)
Kitaev said in the latter paper that they "do not care about rigorous definition" of superselection sector, but it wasn't clear to me that an appropriate, rigorous definition can be made either. Granted, we can get away with many physical concepts (e.g. thermodynamic limit) without rigorously defining them, but the situation with superselection sector in anyon systems seems slightly different. This is because the way we approach such systems is that we'll first cook up a unitary braided fusion category (UBFC) out of the physical systems, and then apply the mathematical theorems concerning UBFC back to the physical systems. UBFC is itself rigorously defined and intensively studied in the mathematical literature. I'd have reservation about the applicability of the theorems unless I'm confident that everything is on solid ground.
What bothered me was that superselection sector is pretty much the only thing that has not been rigorously defined, yet it's supposed to be the starting point of the construction of a UBFC. Everything that follows, e.g. splitting/fusion spaces, can be made well-defined. One can then construct, as far as I can envision, everything we need in a UBFC: the objects, the hom-spaces, the composition map, the biproduct, the monoidal structure, the conjugation, the dual objects, the pivotal structure, etc.