A hyperlinked and more detailed version of this question is at nLab:synthetic differential geometry applied to algebraic geometry. Repliers are kindly encouraged to copy-and-paste relevant bits of their reply here into that wiki page.
The axioms of synthetic differential geometry are intended to pin down the minimum abstract nonsense necessary for talking about the differential aspect of differential geometry using concrete objects that model infinitesimal spaces.
But the typical models for the axioms – the typical smooth toposes – are constructed in close analogy to the general mechanism of algebraic geometry: well-adapted models for smooth toposes use sheaves on C ∞Ring op (the opposite category of smooth algebras) where spaces in algebraic geometry (such as schemes) uses sheaves on CRing op.
In fact, for instance also the topos of presheaves on k−Alg op, which one may think of as being a context in which much of algebraic geometry over a field k takes place, happens to satisfy the axioms of a smooth topos (see the examples there).
This raises some questions.
Quesions:
To which degree do results in algebraic geometry depend on the choice of site CRing op or similar?
To which degree are these results valid in a much wider context of any smooth topos, or smooth topos with certain extra assumptions?
In the general context of structured (∞,1)-toposes and generalized schemes: how much of the usual lore depends on the choice of the (simplicial)ring-theoretic Zariski or etale (pre)geometry (for structured (∞,1)-toposes), how much works more generally?
More concretely:
To which degree can the notion of quasicoherent sheaf generalize from a context modeled on the site CRing to a more general context. What is, for instance, a quasicoherent sheaf on a derived smooth manifold? If at all? What on a general generalized scheme, if at all?
Closely related to that: David Ben-Zvi et al have developed a beautiful theory of integral transforms on derived ∞-stacks.
But in their construction it is always assumed that the underlying site is the (derived) algebraic one, something like simplicial rings.
How much of their construction actually depends on that assumption? How much of this work carries over to other choices of geometries?
For instance, when replacing the category of rings /affine schemes in this setup with that of smooth algebra / smooth loci, how much of the theory can be carried over?
It seems that the crucial and maybe only point where they use the concrete form of their underlying site is the definition of quasicoherent sheaf on a derived stack there, which uses essentially verbatim the usual definition QC(−):Spec(A)↦AMod.
What is that more generally? What is AMod for A a smooth algebra? (In fact I have an idea for that which I will describe on the wiki page in a moment. But would still be interested in hearing opinions.)
Maybe there is a more intrinsic way to say what quasicoherent sheaves on an ∞-stack are, such that it makes sense on more general generalized schemes.
