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stub for Sullivan construction (I got annoyed that the entry didn’t exist, but also don’t feel like doing it justice right now)
Is Sullivan construction related to the fact that a category of spaces is opposite a category of algebras? So starting with a particular kind of algebra, you get a particular kind of space?
Yes, sure it’s related. Only that here instead of just “algebras of functions” you take “dg-algebras of differential forms.”
One day I hope I will know enough math kung fu to be able to work out the category of spaces opposite the category of noncommutative dg-algebras.
the category of spaces opposite the category of noncommutative dg-algebras.
Well, this will be something like the category of oo-groupoids whose spaces of k-morphisms may be non-commutative spaces.
At least that’s true for cosimplicial noncommutative algebras. And under forming normalized cochain complexes these turn into noncommutative dg-algebras.
Now I am not sure if we have a non-commutative Dold-Kan correspondence that asserts that this passage is an equivalence. So i can’t quite decide off the top of my head if every NC dg-algebra will come, up to equivalence, from a cosimplcial NC algebra (but I guess it does??).
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