An answer to a recent question motivated the following question:
(how) is category theory actually
useful in actual physics?
By "actual physics" I mean to refer to areas where the underlying theoretical principle has solid if not conclusive experimental justification, thus ruling out not only string theory (at least for the moment) but also everything I could notice on this nLab page (though it is possible that I missed something).
Note that I do not ask (e.g.) whether or not category theory has been used in connection with hypothetical models in physics. I've read Baez' blog from time to time over the decades and have already demonstrated knowledge of the existence of the nLab. I am dimly aware of stuff like (e.g.) the connection between between Hopf algebras and renormalization, but I have yet to encounter something that seems like it has a nontrivial category theoretic-component and cannot be expressed in some other more "traditional" language.
Note finally that I am ignorant of category theory beyond the words "morphism" and "functor" and (in my youth) "direct limit". So answers that take this into account are particularly welcome.
Best Answer
Fusion categories and module categories come up in topological states of matter in solid state physics. See the research, publications, and talks at Microsoft's Station Q.