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 (the opposite category of smooth algebras) where spaces in algebraic geometry (such as schemes) uses sheaves on CRing.
In fact, for instance also the topos of presheaves on , which one may think of as being a context in which much of algebraic geometry over a field takes place, happens to satisfy the axioms of a smooth topos (see the examples there).
This raises the question:
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?
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?
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 .
What is that more generally? What is for a smooth algebra?
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.
Then maybe a good substitute for QC(-) is
QC(-) : \ell A \mapsto \infty LieAld/\ell A \,.
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…