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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?

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    $\begingroup$ There's an interesting recent book by Paugam that you might like (although I've only looked briefly at a few chapters), "Towards the Mathematics of Quantum Field Theory". The goal of his book is to formulate QFT mathematically in the most "correct" way possible, which for him means in terms of SDG, diffeological spaces, and homotopical algebra. $\endgroup$ – Dan Petersen Nov 11 '14 at 20:34
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    $\begingroup$ I think that most mainstream work in Riemannian geometry ignores SDG. Though I'm only on the boundary of Riemannian geometry, my impression is that there really isn't a "foundational crisis". The biggest issues are related to things where category theory is mostly pretty useless, like analysis. $\endgroup$ – Andy Putman Nov 11 '14 at 21:07
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    $\begingroup$ As someone who doesn't actually know any SDG, it seems like one of SDG's most attractive practical aspects could/would be for proving theorems in an interactive proof assistant (at least this notion was why I bought an as-yet unread book on the topic). $\endgroup$ – Steve Huntsman Nov 11 '14 at 22:36
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    $\begingroup$ It's just a generalization of $C^{\infty}$-rings and the Weil functors. The relation to stacks is not very clear to me. However a fiber of a differentiable map is not a differentiable stack (in a natural way at least), while in the case of $C^{\infty}$-rings, it's a $C^{\infty}$-ring. $\endgroup$ – user40276 Nov 12 '14 at 0:37
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    $\begingroup$ I have a variant of this question which is too similar to warrant making a seperate thread for: what are some things in regular differential geometry that are crucial to understand if one wishes to understand differential geoemtry and which haven't (yet) been reformulated in sdg? $\endgroup$ – Trent Jun 18 '16 at 7:54
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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.

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    $\begingroup$ @user59001 it's not anything to do with model theory (see plato.stanford.edu/entries/model-theory) $\endgroup$ – David Roberts Nov 12 '14 at 2:49
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    $\begingroup$ @user59001: Urs means "model" in the sense that a group is a model of the axioms of group theory, or $\mathbb{R}^n$ is a model of the axioms of Euclidean geometry: you don't need to know any model theory to appreciate that. I think Urs' basic point can be summarized as follows: as mathematicians, we've gotten a lot of mileage out of thinking about what axioms our mathematical objects ought to satisfy (e.g. the group axioms). But now we can go one step up: we can think about what axioms our categories of mathematical objects ought to satisfy (e.g. the topos axioms). $\endgroup$ – Qiaochu Yuan Nov 12 '14 at 5:56
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    $\begingroup$ And in the same way that thinking axiomatically about groups lets us prove general facts about groups without having to work in any particular group, thinking axiomatically about various kinds of categories (e.g. categories that look like categories of "smooth spaces" in some way) lets us prove general facts about those categories without having to work in any particular such category. (The drawback, of course, being the same in both cases: sometimes you need a particular fact about a particular group or a particular category! But that drawback has always been around.) $\endgroup$ – Qiaochu Yuan Nov 12 '14 at 5:57
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    $\begingroup$ Yes, sorry, I should have used less jargon. As David Roberts and Qiaochu Yuan point out, the "models" that I was referring to are simply the categories (the toposes) which satisfy the axioms of SDG (or else those of differential cohesion). SDG in itself is the theory, in the formal sense of formal logic, obtained by starting with intuitionistic set theory and adding to it the Kock-Lawvere axiom. The act of "speaking in this logic" is what people refer to when they say that "in SDG every function is smooth". This logic now has "models" in terms of categories built in ordinary set theory. $\endgroup$ – Urs Schreiber Nov 12 '14 at 7:48
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    $\begingroup$ +QiaochuYuan makes a nice point above about how synthetic reasoning is about passing from axioms for objects to axioms for their categories. Another perspective on that is foundational: Everyone is familiar with how the ZFC axioms lay foundations for sets. Similarly the axioms of "intuitionistic type theory" lay foundations for sets that may have geometric structure (may be interpreted as sheaves of sets). In "synthetic" reasoning we are adding axioms to this intuitionistic type theory that further specify the nature of this geometry. This way SDG connects to the foundations of mathematics. $\endgroup$ – Urs Schreiber Nov 12 '14 at 8:01
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In theory, most anything can be expressed with SDG, and there has been some work in expressing some of GR in this context, but I am not sure if much has been done beyond proof-of-concept. You can google yourself for "synthetic differetial geometry and general relaltivity" and see what is out there.

SDG as an area of research is a different question. It definitely a niche group. And whereas I make contact with certain areas of this group, and do not, as I am categorically minded, need to be convinced that it is a good idea, this is not the case of many of those outside the group. I do not want to dissuade you from learning SDG, but do not learn it as a replacement of the classical theory, among other reasons, because much more has been developed in the classical setting. Knowing only the language of SDG could be in danger of making you unmarketable, whereas knowing the classical setting really well, AND knowing SDG could only be considered an asset. Then, e.g., you could translate anything classical into the synthetic setting yourself, prove what you want, and translate back.

As far as comparative smootheologies are concerned, all of these concepts can be embedded into the topos of sheaves over the category of smooth manifolds. Here you are still missing infinitesimals- you'd have to go to the Cahiers topos to do SDG. Rather, these approaches allow you to deal with infinite-dimensional objects via their functor of points, as well as poor quotients of manifolds. The latter however, is much more better dealt with using the language of stacks, and you are definitely losing information by not going the stacky-route. You would be much better to focus your energy on learning about sheaves, topoi, and eventually higher sheaves and higher topos theory, as this is much more general than these approaches, e.g. diffeological spaces can be recasted in this language anyhow. There are higher categorical concepts that play a role physics, e.g. gerbes and higher gerbes (with connections). I would suggest taking a look at the work of Schreiber:

http://ncatlab.org/nlab/show/Urs+Schreiber

Finally, (and unfortunately) where you are based geographically can have a big influence as to "what is a promising research area". I would still learn as much as possible, in all the available approaches, and learn how to go back and forth between them. The more ways you have to attack something, the better.

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    $\begingroup$ Thanks for taking the time to reply. I really appreciate the advice! $\endgroup$ – ಠ_ಠ Nov 11 '14 at 20:02
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    $\begingroup$ No problem. Good luck! I wouldn't just take my advice. I am in the beginning of my career. Ask your adviser, ask other professors you know. Reach out and ask people in the SDG community (e.g. ask on nforum.mathforge.org or the category-theory mailing list). Ask people with a more classical background too, who know the mathematical climate better. $\endgroup$ – David Carchedi Nov 11 '14 at 20:06
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You might also find the work of Joyce useful, which builds upon the work of Dubuc (E. J. Dubuc. C∞-schemes. Amer. J. Math., 103(4):683–690, 1981) and Moerdijk and Reyes (I. Moerdijk and G. E. Reyes. Models for smooth infinitesimal analysis. Springer-Verlag, New York, 1991 ). The idea is to do "Hartshorne Algebraic Geometry" over $C^{\infty}$-rings and $C^{\infty}$-schemes, which eventually leads to the notion of d-manifolds and d-orbifolds (derived differential geometry).

Joyce, Algebraic Geometry over C-infinity rings (or for the survey: Click me)

For the ~800 page book on d-manifolds and d-orbifolds have a look at d-manifolds and d-orbifolds, but I would highly recommend to have a look at either of these surveys first:

An introduction to d-manifolds and derived differential geometry (Short)

D-manifolds, d-orbifolds and derived differential geometry: a detailed summary (More detialed)

The beauty of d-manifolds and d-orbifolds is that there exist truncation functors from algebraic ($\mathbb{C}$-schemes and $\mathbb{C}$-stacks with obstruction theories) as well as symplectic geometry (Kuranishi spaces) and so d-manifolds (d-orbifolds) are in particular suited to study moduli problems in algebraic and symplectic geometry. (For instance Gromov-Witten theory, Lagrangian Floer cohomology, etc.)

So depending on whether you count d-manifolds and d-orbifolds as advancements of SDG or not, you might answer your second question (whether SDG is a promising research area) with yes.

For your first question: I know that this is not the reformulation of symplectic geometry you had in mind, but d-orbifolds actually had their beginnings in the idea to improve the notion of Kuranishi spaces. As d-manifolds and d-orbifolds admit virtual cycles, one can moreover study and formulate Gromov-Witten theory for instance within this framework. It is also possible to define a notion of blowups in the context of d-manifolds (d-orbifolds) and so if you consider d-manifolds (d-orbifolds) SDG-related, yes, some concepts have been reformulated within SDG.

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Another reference that might be useful for you is R. P. Kostecki, Differential Geometry in Toposes , ms. University of Warsaw (2009) (pdf). He takes the reader in 90 pages from Zenon's paradox to an understanding of the classical constructions in differential geometry including Riemannian structure from the synthetic perspective. He also has an extensive bibliography referencing e.g. applications in GRT or symplectic geometry. So the text gives a good idea what to expect from SDG.

Few researchers with Anders Kock as a notable exception have worked exclusively on SDG so it is probable that one has to move into category theory or higher categorical geometry as well. The techniques and concepts of SDG, as already pointed out in detail by the other answers, have manifold points of contact with the active research at the frontiers of modern geometry though.

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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.

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Two points that others haven't touched on:

General relativity is about a 4-dimensional manifold with tensors satisfying some differential equations and inequalities. You can reformulate it using SDG, but the numerical computations in your chosen coordinate system will be the same. The insight from reformulating classical mechanics along Lagrangian or Hamiltonian lines is much greater than the insight from reformulating general relativity with SDG.

It might be more fertile to use SDG in formulating new physical theories, e.g. general relativity with quantum fields. That might allow you to use the strengths of SDG in dealing with structures that are not ordinary manifolds.

The rigor of SDG is appealing, but it may not be intuitive. Consider the argument:

If $x>0$, then $P$. $\ $ If $x< 1$, then $P$. $\ $ So $P$.

This looks odd, and most people would find it easier to read this instead:

If $x>0$, then $P$. $\ $ If $x\le0$, then $P$. $\ $ So $P$.

In SDG, only the first is a valid argument. So using SDG may require a different mathematical style also. That may help you set up numerical computations, but the change with the new rigor may be bigger than you expected.

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    $\begingroup$ I actually find the first version easier to read because I find it more aesthetically pleasing. $\endgroup$ – k.stm Nov 18 '14 at 12:43

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