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Where did we get to with those discussions on what kinds of thickened points there are, or alternatively, what kinds of cohesive $(\infty, 1)$-categories there are?
So, there are plain, smooth and synthetic differential versions of $\infty-Grpd$ and $super \infty-Grpd$. Was the idea that these are just low truncations of versions of some ’generalized’ super $\infty-Grpd$?
There was then the question of when the tangent to a cohesive $(\infty, 1)$-topos is also a cohesive $(\infty, 1)$-topos?
We had $T(\infty Grpd) = parameterized spectra$, but what are $T(super \infty Grpd)$ and $T^2(\infty Grpd) = T(parameterized spectra)$?
And, does $T(smooth \infty-Grpd)$ equal smooth parameterized spectra?
Hi David,
I still don’t have much news on these questions, good as they are…
I was thinking that would make a fun story for philosophers - ’The history of the point’, from Euclid’s ’That which has no part’ to the thickened point underlying a cohesive $(\infty, 1)$-topos.
Hi David,
yes certainly. I would be more than happy to further expand on this. It’s just somehow right now it feels more urgent to me to work out more of the synthetic description of QFT in cohesive infinity-toposes, and that and its examples such as the “brane bouquet” is mostly absorbing me at the moment.
We should find a student to look more into the topic of mapping the space of cohesion! :-)
Sure there are golden nuggets everywhere, and who better than you to choose where to prospect.
I wonder though, in a spirit of speculation, if you had an army of postdocs, where they might be pointed. You’d think it would be profitable to have some head off to homotopy theory in search of stable cohesion. And there must be a way to link up with number theory.
I wonder if
Arithmetic Chow groups are refinements of ordinary Chow groups analogous to how ordinary differential cohomology refines ordinary cohomology,
might be a clue.
Hmm, if the analogy above is roughly right and from here
…Arakelov motivic cohomology…relates to motivic cohomology in the sense of Voevodsky roughly in the same way that the arithmetic Chow groups of Gillet and Soule relate to ordinary Chow groups,
and if analogies compose, then is Holmstrom onto some kind of cohesion?
Any experts on Arakelov motivic cohomology on hand to explain I and II?
Could the refinement of algebraic K-theory to differential algebraic K-theory of Bunke/Tamme fit with those refinements in #5 and #6 above? This was mentioned as a possible case of cohesive thickening. And Bunke and Tamme write
differential algebraic K-theory may also be seen as a variant of arithmetic algebraic K-theory,
so somehow there’s a relation between the differential and the arithmetic which brings us back to #5.
Maybe. I need to study this.
Hmm, I see
Arakelov motivic cohomology is a generalization of arithmetic K-theory and arithmetic Chow groups.
This is closely related to the Bunke-Tamme article which we discussed recently.
Let me tell you what we are after next in this context:
given that we have smooth (cohesive) versions of ordinary cohomology and of K-theory spectra, next we want a smooth version of the elliptic cohomoloy spectra. This should be not just probed by smooth manifolds, but also be parameterized over elliptic curves, clearly. Therefore we expect to build smooth elliptic cohomology in the $\infty$-topos
$Sh_\infty(SmoothMfd, Sh_\infty(Schemes)) \longrightarrow Sh_\infty(Schemes)$which is cohesive over the $\infty$-topos over a suitable big site of suitable schemes.
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