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.)

]]>created a bare minimum at *jet group*

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

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.

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