On Friday at 3pm Mike Freedman is speaking in the Topological Matter summer school. This is a “pre-talk”, trying to fill in some background. I’ll try to explain what:
1) deformation classes of local gapped Hamiltonians are (and why they ought to correspond to both “topological matter” and “higher categories”), and
2) what the torus trick is, in smooth manifold topology.
I have no idea yet how Mike is planning on putting these two ingredients together.