N ov 2 01 2 An introduction to C ∞-schemes and C ∞-algebraic geometry

If X is a manifold then the R-algebra C(X) of smooth functions c : X → R is a C-ring. That is, for each smooth function f : R → R there is an n-fold operation Φf : C (X) → C(X) acting by Φf : c1, . . . , cn 7→ f(c1, . . . , cn), and these operations Φf satisfy many natural identities. Thus, C (X) actually has a far richer structure than the obvious R-algebra structure. In [7] the author set out the foundations of a version of algebraic geometry in which rings or algebras are replaced by C-rings, focussing on C-schemes, a category of geometric objects which generalize manifolds, and whose morphisms generalize smooth maps, quasicoherent and coherent sheaves on C-schemes, and C-stacks, in particular Deligne–Mumford C-stacks, a 2-category of geometric objects which generalize orbifolds. This paper is a survey of [7]. C-rings and C-schemes were first introduced in synthetic differential geometry, see for instance Dubuc [3], Moerdijk and Reyes [15] and Kock [11]. Following Dubuc’s discussion of ‘models of synthetic differential geometry’ [2] and oversimplifying a bit, symplectic differential geometers are interested in Cschemes as they provide a category C∞Sch of geometric objects which includes smooth manifolds and certain ‘infinitesimal’ objects, and all fibre products exist in C∞Sch, and C∞Sch has some other nice properties to do with open covers, and exponentials of infinitesimals. Synthetic differential geometry concerns proving theorems about manifolds using synthetic reasoning involving ‘infinitesimals’. But one needs to check these methods of synthetic reasoning are valid. To do this you need a ‘model’, some category of geometric spaces including manifolds and infinitesimals, in which you can think of your synthetic arguments as happening. Once you know there exists at least one model with the properties you want, then as far as synthetic differential geometry is concerned the job is done. For this reason C-schemes were not developed very far in synthetic differential geometry. Recently, C-rings and C-ringed spaces appeared in a very different context, as part of Spivak’s definition of derived manifolds [18], which are an extension to differential geometry of Jacob Lurie’s ‘derived algebraic geometry’ programme. The author [8–10] is developing an alternative theory of derived differential geometry which simplifies, and goes beyond, Spivak’s derived man-