synthetic differential geometry applied to algebraic geometry

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 $C^\infty Ring^{op}$ (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 the question:

**Questions**

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?

It is curious that the field of algebraic geometry has induced, first with Alexander Grothendieck now with Jacob Lurie, so much category theory and higher category theory, while at the same time it is common practice in this field to effectively disregard one of the major guidelines that practitioners in pure category theory are fond of adhering to: that of separation of context and implementation. Bill Lawvere’s famous dichotomy between *theory* and *model* .

In fact, it seems that Lawvere dreamed up the axioms of synthetic differential geometry not without the idea of capturing central structures in algebraic geometry this way, too. But, possibly due to the very term chosen, synthetic differential geometry it has apparently always (if at all) attracted more the attention of those interested in ordinary differential geometry than those interested in algebraic geometry.

But at least in the light of Lurie’s notion of structured (∞,1)-toposes and generalized schemes, from the point of which ordinary algebraic geometry as well as derived algebraic geometry is just one realization of a more general concept of geometry, it seems to be worthwhile to reexamine the wealth of knowledge accumulated in algebraic geometry and see how much of it depends on just general context, how much on concrete implementation.

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 ∞-stack, as described at geometric ∞-function theory.

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 scheme 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 essetnially verbatim the usual definition $QC(-) : Spec(A) \mapsto A Mod$.

What is that more generally? What is $A Mod$ for $A$ a smooth algebra?

If that doesn’t have a good answer, 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.

Urs:

Here is a proposal for how one might answer this very generally:

One observation is that the monoidal Dold-Kan correspondence identifies (co)simplicial algebras with dg-algebras. To say dg-algebra in a general context we need to say module, which may be hard in generalized situations such as working over generalized smooth algebras. On the other hand, it is straightforward to speak of cosimplicial smooth algebras of course.

This is the idea underlying the discussion at ∞-quantity. To that we add the observation that cosimplicial algebras that under the monoidal Dold-Kan correspondence maps to a *graded commutative* dg-algebra may be thought of as a ∞-Lie algebroid, as explained there. In its ordinary incarnation (see L-infinity algebroid) this is a complex of modules over the degree 0 algebra with some extra structure. So in light of this a cosimplicial algebra that maps under monoidal Dold-Kan to something graded-commutative might be a good very general abstract nonsense substitute for complexes of modules.

So for $\ell A$ a smooth locus or similar, consider (∞,1)-category $\infty Lie Algd/\ell A$ of cosimplicial algebras *under* $A$ that have graded commutative Dold-Kan image.

Then maybe a good substitute for QC(-) is

$QC(-) : \ell A \mapsto \infty LieAld/\ell A
\,.$

Being an over category, this would match well with the reasoning at geometric function object.

Somebody points out the discussion here

on formulating finitely-presented conditions internally in a topos. In particular William Lawvere’s message (the third one from the top).

will go here…

Revised on October 17, 2009 11:25:15
by Urs Schreiber
(212.23.128.235)