I don't think HoTT is any position to be used in algebraic geometry as it stands currently. There may be a way in via a functor of points approach which is well suited for constructing Hilbert schemes. However I think it would be impossible to say anything concrete at this point. Homology and cohomology are usually seen as "easy" invariants about spaces to calculate however in HoTT it is all very new and nobody quite know the best way to go about reasoning with these things. Let alone thinking about something with extra structure like Hodge structure.
However representation theory, as you have cited, may be more tractable. There are good formal properties of HoTT which may allow it to reason in representation theoretic terms quite concretely. So if you are interested in studying quiver varieties you may just be able to get away with studying quiver representations. However this is all speculative at this point. I don't think there are any researchers looking into any of these things yet as I believe HoTT just is not sophisticated to carry out such reasoning.
But this is all my opinion. I would be very suprised if anybody says otherwise.
On Saturday, 22 September 2018 17:59:03 UTC+1, José Manuel Rodriguez Caballero wrote:
Recently, there was a post about the Euler characteristic of a type. In my case, I am interested in the Hodge structure of the Hilbert scheme of n points on a 2-dimensional torus. Does such a topological construction make sense in HoTT for an arbitrary type, under some general hypothesis?
Kind Regards,
Jose M