TITLE: Synthetic vs. classical differential geometry
QUESTION [55 upvotes]: 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?

REPLY [6 votes]: Tim de Laat's bachelor thesis might be of interest:
Synthetic Differential Geometry: An application to Einstein’s Equivalence Principle

This thesis is the result of my bachelor project in both Mathematics and Physics & Astronomy. The aim of this project was to give a satisfactory and rigorous formulation of the equivalence principle of the general theory of relativity (GR) in terms of synthetic differential geometry (SDG). SDG is a “natural” formulation of differential geometry in which the notion of “infinitesimals” is very important. Smooth infinitesimal analysis (SIA) is the mathematical analysis corresponding to these infinitesimals and it forms an entrance to SDG. Both SIA and SDG are formulated in terms of categories and topoi. As I was quite new to these subjects, I first needed to study them thoroughly before I could start studying SDG.