The axioms of synthetic differential geometry are intended to pin down the minimum general abstract axioms 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 William Lawvere found 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.
Let $X$ be a scheme and $Sh(Sch/X)$ the big Zariski topos associated to $X$. Denote by $\mathbb{A}^1$ (the affine line) the ring object $T \mapsto \Gamma(T,\mathcal{O}_T)$, i.e. the functor represented by the $X$-scheme $\mathbb{A}^1_X \coloneqq X \times Spec(\mathbb{Z}[t])$. Then:
$\mathbb{A}^1$ is internally a local ring.
$\mathbb{A}^1$ is internally a field in the sense that any nonzero element is invertible.
Internally, any function $f : \mathbb{A}^1 \to \mathbb{A}^1$ is a polynomial function, i.e. of the form $f(x) = \sum_i a_i x^i$ for some coefficients $a_i : \mathbb{A}^1$. More precisely,
Furthermore, these coefficients are uniquely determined.
See affine line.
Since the big Zariski topos is cocomplete (being a Grothendieck topos), one can also get rid of the external disjunction and refer to the object $\mathbb{A}^1[X]$ of internal polynomials: The canonical ring homomorphism $\mathbb{A}^1[X] \to [\mathbb{A}^1,\mathbb{A}^1]$ (given by evaluation) is an isomorphism.
this is old material that needs attention
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.
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).