I have been reading the anyon theory by Kitaev and Wang. I have two possibly related questions:

- Why is the Pentagon equation/axiom sufficient for characterizing associative relations?
- Are there anyon theories with more than two (or three) elementary (non-composite) excitations?

My impression has been that there is no anyon theory with more than two non-composite elementary excitations. This seems to be the reason that the Pentagon equation (together with braiding structures, of course) is sufficient in describing anyon theories. Is this naive intuition correct? Could someone enlighten me on this subject?

cross-posted on SE.