However I don't want to discourage you. One possible solution is 
(differential?) cohesive homotopy type theory (which is at the moment even 
more undeveloped). This may allow you to talk about manifolds and their 
structure "synthetically" which would allow for definitions of de Rham 
cohomology and possibly with care allow you to talk about hilbert schemes 
of some torus. Pessemistically I would add that it would be at least 10 
years before any of this is considered.

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