Algebraic Geometry over 𝐶^{∞}-rings

Source abstract

If X X is a manifold then the R \mathbb R -algebra C ∞ ( X ) C^\infty (X) of smooth functions c : X → R c:X\rightarrow \mathbb R is a C ∞ C^\infty - ring . That is, for each smooth function f : R n → R f:\mathbb R^n\rightarrow \mathbb R there is an n n -fold operation Φ f : C ∞ ( X ) n → C ∞ ( X ) \Phi _f:C^\infty (X)^n\rightarrow C^\infty (X) acting by Φ f : ( c 1 , … , c n ) ↦ f ( c 1 , … , c n ) \Phi _f:(c_1,\ldots ,c_n)\mapsto f(c_1,\ldots ,c_n) , and these operations Φ f \Phi _f satisfy many natural identities. Thus, C ∞ ( X ) C^\infty (X) actually has a far richer structure than the obvious R \mathbb R -algebra structure. We explain the foundations of a version of algebraic geometry in which rings or algebras are replaced by C ∞ C^\infty -rings. As schemes are the basic objects in algebraic geometry, the new basic objects are C ∞ C^\infty - schemes , a category of geometric objects which generalize manifolds, and whose morphisms generalize smooth maps. We also study quasicoherent sheaves on C ∞ C^\infty -schemes, and C ∞ C^\infty - stacks , in particular Deligne–Mumford C ∞ C^\infty -stacks , a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: C ∞ C^\infty -rings and C ∞ C^\infty -schemes have long been part of synthetic differential geometry. But we develop them in new directions. In Joyce (2014, 2012, 2012 preprint), the author uses these tools to define d-manifolds and d-orbifolds , ‘derived’ versions of manifolds and orbifolds related to Spivak’s ‘derived manifolds’ (2010).