nLab
derived smooth manifold

Idea

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.

Definition

… just a second …

hm, took me more than a second…

$(\mathrm{\infty }\mathrm{,1})$-smooth algebras – simplicial $C^{\mathrm{\infty }}$-rings

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 $A:\mathrm{CartSp}\to$ ∞Grpd.

In Spiv this is modeled in terms of a left Bousfield localization of the injective model structure on simplicial presheaves on ${\mathrm{CartSp}}^{\mathrm{op}}$.

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References

Derived smooth manifolds were defined in

• David Spivak, Quasi-smooth derived manifolds, PhD thesis, Berkeley (2007) (pdf)

This original version of the PhD thesis essentially looks at structured (∞,1)-toposes of ∞-stacks on a topological space for the pregeometry given by $𝒯=$ Diff.

The published version

• David Spivak, Derived smooth manifolds (arXiv)

essentially does the same, but with Diff replaced by CartSp. Only that CartSp does not quite satisfy all the axioms of a pregeometry.