The whole "foundations of math" part is extremely exciting!
But I'm not an mathematician so I don't understand much (if anything). More in the programming languages "department" of interests.
But if something is "good enough" to describe whole math it could be useful for getting strong guaranties about programs, I guess. So I would be quite interested how this HOT stuff could be useful in practice. What could type systems based on this enable?
My limited understanding is that you get additionally "paths" between types and type constructor which denote equivalence between them.
But I have no clue what one could do with that (besides doing math, of course).