A derived smooth manifold is a structured generalized space given by a manifold whose structure sheaf of “smooth functions” is a higher categorical generalized smooth algebra.
… just a second …
hm, took me more than a second…
An smooth (∞,1)-algebra is an (∞,1)-algebra over the algebraic theory CartSp. This is the (∞,1)-category analog of the notion of smooth algebra: a product-preserving (∞,1)-functor ∞Grpd.
In Spiv this is modeled in terms of a left Bousfield localization of the injective model structure on simplicial presheaves on .
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Derived smooth manifolds were defined in
This original version of the PhD thesis essentially looks at structured (∞,1)-toposes of ∞-stacks on a topological space for the pregeometry given by Diff.
The published version
essentially does the same, but with Diff replaced by CartSp. Only that CartSp does not quite satisfy all the axioms of a pregeometry.