nForum - Search Results Feed (Tag: synthetic-differential-geometry) 2020-08-13T15:46:16-04:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher higher order frame bundle https://nforum.ncatlab.org/discussion/6417/ 2015-01-05T01:11:28-05:00 2020-06-20T07:53:11-04:00 Urs https://nforum.ncatlab.org/account/4/ created a bare minimum at higher order frame bundle and cross-linked a bit

created a bare minimum at higher order frame bundle and cross-linked a bit

]]>
arithmetic jet space https://nforum.ncatlab.org/discussion/6122/ 2014-07-23T16:39:34-04:00 2017-06-22T04:39:06-04:00 Urs https://nforum.ncatlab.org/account/4/ created arithmetic jet space, so far only highlighting the statement that at prime pp these are X&times;Spec(&Zopf;)Spec(&Zopf; p)X \underset{Spec(\mathbb{Z})}{\times}Spec(\mathbb{Z}_p) ...

created arithmetic jet space, so far only highlighting the statement that at prime $p$ these are $X \underset{Spec(\mathbb{Z})}{\times}Spec(\mathbb{Z}_p)$ (regarded so in Borger’s absolute geometry by applying the Witt ring construction $(W_n)_\ast$ to it).

This is what I had hoped that the definition/characterization would be, so I am relieved. Because this is of course just the definition of synthetic differential geometry with $Spec(\mathbb{Z}_p)$ regarded as the $p$th abstract formal disk.

Well, or at least this is what Buium defines. Borger instead calls $(W_n)_\ast$ itself already the arithmetic jet space functor. I am not sure yet if I follow that.

I am hoping to realize the following: in ordinary differential geometry then synthetic differential infinity-groupoids is cohesive over “formal moduli problems” and here the flat modality $\flat$ is exactly the analog of the above “jet space” construction, in that it evaluates everything on formal disks. Moreover, $\flat$ canonically sits in a fracture suare together with the “cohesive rationalization” operation $[\Pi_{dR}(-),-]$ and hence plays exactly the role of the arithmetic fracture square, but in smooth geometry. I am hoping that Borger’s absolute geometry may be massaged into a cohesive structure over the base $Et(Spec(\mathbb{F}_1))$ that makes the cohesive fracture square reproduce the arithmetic one.

If Borger’s absolute direct image were base change to $Spec(\mathbb{Z}_p)$ followed by the Witt vector construction, then this would come really close to being true. Not sure what to make of it being just that Witt vector construction. Presently I have no real idea of what good that actually is (apart from giving any base topos for $Et(Spec(Z))$, fine, but why this one? Need to further think about it.)

]]>
jet group https://nforum.ncatlab.org/discussion/6416/ 2015-01-05T01:10:56-05:00 2015-01-16T20:18:56-05:00 Urs https://nforum.ncatlab.org/account/4/ created a bare minimum at jet group

created a bare minimum at jet group

]]>
infinitesimal and local - table https://nforum.ncatlab.org/discussion/4732/ 2013-02-06T15:04:48-05:00 2014-07-23T15:41:08-04:00 Urs https://nforum.ncatlab.org/account/4/ I felt like starting a table infinitesimal and local - table and included it into the relevant entries. So far it reads as follows: first order infinitesimal object infinitesimal ...

I felt like starting a table infinitesimal and local - table and included it into the relevant entries. So far it reads as follows:

first order infinitesimal object infinitesimal $\subset$ formal = arbitrary order infinitesimal $\subset$ local = stalkwise $\subset$ finite
derivative Taylor series germ function
tangent vector jet germ of curve curve
Lie algebra formal group local Lie group Lie group
Poisson manifold formal deformation quantization local strict deformation quantization strict deformation quantization

Can be further expanded, clearly.

]]>
formal disk https://nforum.ncatlab.org/discussion/6104/ 2014-07-19T12:47:41-04:00 2014-07-19T12:47:41-04:00 Urs https://nforum.ncatlab.org/account/4/ created formal disk with some default text, just so that the links from function field analogy – table point to something

created formal disk with some default text, just so that the links from function field analogy – table point to something

]]>