An introduction to C-infinity schemes and C-infinity algebraic geometry - Joyce, D (International Press, 2012)
If X is a smooth manifold then the R-algebra C∞(X) of smooth functions c : X → R is a "C∞-ring". That is, for each smooth function ƒ : Rn → R there is an n-fold operation Φƒ : C∞(X)n → C∞(X) acting by Φƒ: (c1,...,cn) |→ f(c1,...,cn), and these operations Φƒ satisfy many natural identities. Thus, C∞(X) actually has a far richer structure than the obvious R-algebra structure. We explain a version of algebraic geometry in which rings or algebras are replaced by C∞-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are "C∞-schemes", a category of geometric objects generalizing manifolds, and whose morphisms generalize smooth maps. We also discuss "C∞-stacks", including Deligne-Mumford C∞-stacks, a 2-category of geometric objects generalizing orbifolds. We study quasicoherent and coherent sheaves on C∞-schemes and C-infinity stacks, and orbifold strata of Deligne-Mumford C∞-stacks. This enables us to use the tools of algebraic geometry in differential geometry, and to describe singular spaces such as moduli spaces occurring in differential geometric problems.