For the past few decades the study of so-called topological orders has remained an active area of research. By definition (following the motivation section of this recent invitation to the subject, for the sake of anchoring ourselves to a common reference point), a topological order is a gapped quantum liquid without symmetry, where a gapped quantum liquid is a specific type of quantum phase, which itself is an equivalence class of a quantum state of matter at zero temperature. Fairly recently, these were classified in terms of mildly dualizable, monoidal, Karoubi-complete n-categories with trivial center.
In the physics literature, however, there is some (admittedly not much) discussion of topological order at finite temperature, and so I was wondering: is there anything known on the mathematical side about these (in terms of higher categories)?
(Or, I suppose, another way of asking the same thing: is there a clear way in which the zero temperature condition is encoded within the higher categorical language of Johnson-Freyd’s classification scheme?)