Description
If $X$ is a manifold then the $mathbb R$-algebra $C^infty (X)$ of smooth functions $c:Xrightarrow mathbb R$ is a $C^infty $-ring. That is, for each smooth function $f:mathbb R^nrightarrow mathbb R$ there is an $n$-fold operation $Phi _f:C^infty (X)^nrightarrow C^infty (X)$ acting by $Phi _f:(c_1,ldots ,c_n)mapsto f(c_1,ldots ,c_n)$, and these operations $Phi _f$ satisfy many natural identities. Thus, $C^infty (X)$ actually has a far richer structure than the obvious $mathbb R$-algebra structure. The author explains the foundations of a version of algebraic geometry in which rings or algebras are replaced by $C^infty $-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are $C^infty $-schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps. The author also studies quasicoherent sheaves on $C^infty $-schemes, and $C^infty $-stacks, in particular Deligne-Mumford $C^infty$-stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: $C^infty$-rings and $C^infty $-schemes have long been part of synthetic differential geometry. But the author develops them in new directions. In earlier publications, the author used these tools to define d-manifolds and d-orbifolds, “derived” versions of manifolds and orbifolds related to Spivak’s “derived manifolds”.
