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”.