Dominic Joyce 
Algebraic Geometry over $C^/infty $-Rings [PDF ebook] 

If $X$ is a manifold then the $/mathbb R$-algebra $C^/infty (X)$ of smooth functions $c:X/rightarrow /mathbb R$ is a $C^/infty $-ring. That is, for each smooth function $f:/mathbb R^n/rightarrow /mathbb R$ there is an $n$-fold operation $/Phi _f:C^/infty (X)^n/rightarrow C^/infty (X)$ 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".