![]() For instance Goodwille tangent calculus is now also part of the picture, in terms of synthetic tangent cohesion. Viewed from this perspective the scope of models for the SDG axiomatics becomes more powerful still. I like to call this differential cohesion but of course it doesn't matter what one calls it. 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. 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).Ī little later Lawvere tried another approach to such foundations, not via the KL-axioms this time, but via " axiomatic cohesion". See here for commented list of pointers and citations on that aspects. 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. 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). ![]() 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. 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. 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"). 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. 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?Īlso, 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? I just have a few questions about SDG which I hope some experts could answer. ![]() ![]() The ability to argue rigorously using infinitesimals also appeals to the physicist in me, and seems to yield more intuitive proofs. Many of the definitions become very elegant, such as the definition of the tangent bundle as an exponential object. I've recently come across some interesting surveys and articles on synthetic differential geometry (SDG) that made the approach seem very appealing. 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. To provide context, I'm a differential geometry grad student from a physics background.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |