To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of Lee's series) but I don't have any background in categorical logic or model theory.
I've recently come across some interesting surveys and articles on synthetic differential geometry (SDG) that made the approach seem very appealing. Many of the definitions become very elegant, such as the definition of the tangent bundle as an exponential object. The ability to argue rigorously using infinitesimals also appeals to the physicist in me, and seems to yield more intuitive proofs.
I just have a few questions about SDG which I hope some experts could answer. How much of modern differential geometry (Cartan geometry, poisson geometry, symplectic geometry, etc.) has been reformulated in SDG? Have any physical theories such as general relativity been reformulated in SDG? If so, is the synthetic formulation more or less practical than the classical formulation for computations and numerical simulations?
Also, how promising is SDG as an area of research? How does it compare to other alternative theories such as the ones discussed in comparative smootheology?
Best Answer
One point of synthetic differential geometry is that, indeed, it is "synthetic" in the spirit of traditional synthetic geometry but refined now from incidence geometry to differential geometry. Hence the name is rather appropriate and in particular highlights that SDG is more than any one of its models, such as those based on formal duals of C-infinity rings ("smooth loci"). Indeed, traditional algebraic geometry with formal schemes is another model for SDG and this is where the origin of the theory lies: William Lawvere was watching Alexander Grothendieck's work and after abstracting the concept of elementary topos from what Grothendieck did with sheaves, he next abstracted the Kock-Lawvere axioms of SDG from what Grothendieck did with infinitesimal extensions, formal schemes and crystals/de Rham spaces. The idea of SDG is to abstract the essence of all these niceties, formulate them in terms of elementary topos theory, and hence lay mathematical foundations for differential geoemtry that are vastly more encompassing than either algebraic geometry or traditional differential geometry alone. For instance there are also models in supergeometry, in complex analytic geometry and in much more exotic versions of "differential calculus" (such as Goodwillie calculus, see below).
Regarding applications, a curious fact that remains little known is that Lawvere, while widely renowned for his work in the foundations of mathematics, has from the very beginning and throughout the decades been directly motivated by, actually, laying foundations for continuum physics. See here for commented list of pointers and citations on that aspects. In particular SDG was from the very beginning intended to formalize mechanics, that's why one of the earliest texts on the topic is titled "Toposes of laws of motion" (referring to SDG toposes).
A little later Lawvere tried another approach to such foundations, not via the KL-axioms this time, but via "axiomatic cohesion". One may recover SDG in axiomatic cohesion in a way that realizes it in close parallel to modern D-geometry with axiomatic de Rham stacks, jet-bundles, D-modules and all. I like to call this differential cohesion but of course it doesn't matter what one calls it.
Viewed from this perspective the scope of models for the SDG axiomatics becomes more powerful still. For instance Goodwille tangent calculus is now also part of the picture, in terms of synthetic tangent cohesion. Another model is in spectral derived geometry that knows about structures of relevance in arithmetic geometry, chromatic homotopy theory and class field theory, this is discussed at differential cohesion and idelic structure.
All this synthetic reasoning is maybe best viewed from the general perspective that it is useful in mathematics to stratify all theory as much as possible by the hierarchy of assumptions and axioms needed, try to prove as much as possible from as little assumptions as necessary and pass to fully concrete models only at the very end. If you are interested only in one specific model, such as derived geometry over $C^\infty$-rings, then such synthetic reasoning may offer some guidance but might otherwise seem superfluous. The power of the synthetic method is in how it allows to pass between models, see their similarities and differences, and prove model-independent theorems. As in "I don’t want you to think all this is theory for the sake of it, or rather for the sake of itself. It’s theory for the sake of other theory." (Lurie, ICM 2010) Synthetic geometry is "inter-geometric", to borrow a term-formation from Mochizuki. If you run into something like the function field analogy then it may be time to step back and ask if such analogy between different flavors of geometry maybe comes from the fact that they all are models for the same set of "synthetic" axioms.