The $C^\infty$-ring $C^\infty (X)$ of a manifold $X$
$C^\infty $-ringed spaces and $C^\infty $-schemes
Modules over $C^\infty$-rings and $C^\infty $-schemes
Deligne-Mumford $C^\infty $-stacks
Sheaves on Deligne-Mumford $C^\infty $-stacks
Orbifold strata of $C^\infty $-stacks
Appendix A. Background material on stacks
Glossary of Notation
If X is a manifold then the R-algebra C8(X) of smooth functions c:X?R is a C8-ring. That is, for each smooth function f:Rn?R there is an n-fold operation Ff:C8(X)n?C8(X) acting by Ff:(c1,…,cn)?f(c1,…,cn), and these operations Ff satisfy many natural identities. Thus, C8(X) actually has a far richer structure than the obvious R-algebra structure.
The author explains the foundations of a version of algebraic geometry in which rings or algebras are replaced by C8-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are C8-schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps. The author also studies quasicoherent sheaves on C8-schemes, and C8-stacks, in particular Deligne-Mumford C8-stacks, a 2-category of geometric objects generalizing orbifolds.
Many of these ideas are not new: C8-rings and C8-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”.
Dominic Joyce: University of Oxford, United Kingdom