I see James Borger gave a lecture series Witt vectors, lambda-rings, and arithmetic jet spaces, so I’ve linked to that from his page. Might be something new there if we ever go back to #11 and wonder whether differential cohesion has something to say there.

]]>I wouldn’t think that being a universe is preserved by the cofree PDE functor, because by definition of being cofree, a map $\mathcal{E} \longrightarrow F (U)$ is equivalently a map $U(\mathcal{E}) \longrightarrow U$ and this classifies a bundle over the underlying bundle $U(\mathcal{E})$, but does not classify a non-trivial (non-co-free) PDE on the sections of that bundle.

One way to get ones hands on the universe in $PDE_\Sigma(\mathbf{H})$ is to use that by the jet comonadicity and monadic descent, and using that $\mathbf{H}$ has a site of definition $FormalSmoothManifold$, it follows that $\mathrm{PDE}_\Sigma(\mathbf{H})$ has site of definition the comma category $FormalSmoothManifold/\Im \Sigma$.

From this one gets the description of a universe as the presheaf that sends any representable to the collection of small types over it. Not that this is necessarily particularly useful, but at least it may reduce any feeling of mystery regarding the universe in $PDE_\Sigma(\mathbf{H})$.

]]>That is what I meant. So being a universe is not preserved by the $F$?

Anyway, there must be a universe there somewhere in $\mathrm{PDE}_\Sigma(\mathbf{H})$, and it initially struck me as odd that such a thing would be a PDE, but I guess it’ll be some complicated homotopy colimit.

]]>Do you mean a universe $U \in \mathbf{H}$ will give a universe $F \Sigma^\ast U \in \mathrm{PDE}_\Sigma(\mathbf{H})$? I wouldn’t think that’s right. But maybe I misunderstand what you mean.

]]>Ah, OK thanks.

And I guess I would just chase along those arrows to find the homotopy PDE which is the object classifier of $\mathrm{PDE}_\Sigma(\mathbf{H})$.

]]>Ah, I think that was the wrong question.

The good perspective seems to be this: given the cohesion in $\mathbf{H}$ we find differential hexagons there, which we may send along $\mathbf{H} \stackrel{\Sigma^\ast}{\longrightarrow} \mathbf{H}_{/\Sigma} \stackrel{F}{\longrightarrow} \mathrm{PDE}_\Sigma(\mathbf{H})$. Up there one finds that the differential form coefficients in the hexagon may decompose along another, a “variational” Hodge filtration. The induced filtration on the hexagons is what governs the homotopy PDE theory.

You’ll recognize what I just said in the notes that I had sent you. I’ll be further expanding on this in the next days.

]]>Did you ever resolve the question starting this thread about the cohesiveness of $EM(Jet_X)$ or some global version of it for all $X$?

Restricting to the local case, $EM(Jet_X) = PDE(X)$, so there’s some kind of universal PDE with variables ranging over $X$ which acts as an object classifier.

]]>No doubt there’s a connection somewhere: Artin L-function - Frobenius morphism - Lambda-ring.

A cursory glance finds Davis and Wan looking to use Witt vectors to understand continuation of L-functions:

]]>The aim of this paper is to re-examine this L-function from a new point of view via Witt vectors in the hope that it will provide new insight into this mysterious meromorphic continuation problem.

Sorry for the slow reply, somehow I almost missed the last messages.

Yes, this is, if maybe not a mystery then at least a good question. This might have a simple answer once one really thinks about it.

As you may have noticed, I am not jumping all that eagerly on the train which you are pointing out here.

I do think you are absolutely right that it would be most worthwhile to think of all the Lambda-ring business in terms of D-geometry / differential cohesion.

I wish somebody finds time to think about this! Myself, for the moment, in as far as number theory is concerned I am concentrating on another approach, as you know, and that keeps me busy enough.

That other approach, to recall, is based on two observations:

a) (with Domenico Fiorenza): the extended TQFT with coefficients in higher spans “phased” over a moduli stack $\mathbf{B}^n U(1)$ which to the point assingns a given higher Chern-Simons bundle $\mathbf{B}G \to \mathbf{B}^n U(1)$ will in codimension-1 assign the corresponding theta-line, and the functoriality in codimension-1 expresses this as a higher modular functor;

b) (with suggestions from Minhyong Kim): The system of sections of that theta-line, something like a corrected exponentiated eta-invariant, is the by far best differential geometric analog of the Artin L-functions.

I think this clearly suggests a connection to Langlands’ genuine program, a connection which is geometric but is rather different (it seems) from the popular “geometric Langlands” program. Right now the most pressing questions in this “other” geometric connection don’t seem to feature structures related to Lambda-rings quite as prominently. That is the reason why I am not pouring all my energy into Lambda-rings right now…

]]>So the mystery is why should it be that the arithmetic case is so similar to the geometric in that there is:

(1) A relative cohesion that involves shape and flat corresponding to torsion approximation and formal completion,

and

(2) Something jet space like, suggestive of a differential cohesion which would give rise to the relative cohesion of (1),

and yet there isn’t precisely

$\mathbf{H}_{reduced} \hookrightarrow \mathbf{H} \longrightarrow \mathbf{H}_{infinitesimal}.$Given that in the geometric case, these arrows are related via a fiber sequence, one might expect some connection in the arithmetic case between the jet-space/Witt vectors account and the torsion approximation/formal completion account.

]]>Regarding finiteness of $\mathfrak{a}$, in Lambda-rings and the field with one element Borger describes a variant in 7.1

$\Lambda_{S, E}$-spaces. Let $S$ be a scheme of finite type over $Z$, and let $E$ be a set of regular closed points of codimension 1.

What he says in section 7.7 sounds interesting.

]]>So we have

$?? \hookrightarrow A Mod^{op} \longrightarrow A Mod^{op}_{\mathfrak{a} comp} \simeq A Mod^{op}_{\mathfrak{a} tor}.$$\mathfrak{a}$ is only finitely generated, so we wouldn’t have all primes at once?

If, as in Remark 4, this $A Mod^{op}$ concerns augmented E-∞ rings over the sphere spectrum, we are looking (like Glasman #42) for a spectral version of $\Lambda$-rings?

]]>One thing to maybe keep in mind is that

in the differential geometric context then there is this homotopy cofiber sequence of $\infty$-toposes (here)

$\mathbf{H}_{reduced} \hookrightarrow \mathbf{H} \longrightarrow \mathbf{H}_{infinitesimal}$and $\Im$ is the monad induced on the left, while the comonad $\flat^{rel}$ induced on the right produces infinitesimal neighbourhoods

we know what $\flat^{rel}$ is in “$E_\infty$-arithmetic geometry”, namely it’s what is denoted $\flat_{\mathfrak{a}}$ here.

So from this point of view we have the right morphism of the sequence above and are asking for the left one, suspecting that if it exists then it should induce the global Witt vector thing.

]]>Regarding raising all this to the level of spectra, I see in Saul Glasman’s research statement the goal of

]]>Building a categorically robust theory of Witt vectors of commutative ring spectra.

If differential cohesion and idelic structure is intended to make inter-geometry more systematic and there it claims

the statement of the geometric Langlands correspondence is that there is a natural correspondence between $\Pi_{inf}[\Sigma, \mathbf{B}G]$ and $[\Pi_{inf}\Sigma, \mathbf{B}{}^L G]$,

which in the new notation is $\Im[\Sigma, \mathbf{B}G]$ and $[\Im\Sigma, \mathbf{B}{}^L G]$, what happens if we use the prospective $\Im$ of #38?

]]>curious, at least, that in the other thread(s) related to complex volumes algebraic K-theory shows up closely related to the Artin L-function, pointers now at *Borel regulator – Relation to complex volume and Bloch group*

(Hisham Sati highlighted this connection to me)

]]>I wonder if there’s anything to be gained by something like $\mathbf{B} = Sh_{\infty}(Spec(\mathbb{F}_1)_{et})$ in

$\mathbf{H} \coloneqq Sh_\infty\left(SmthMfd, \mathbf{B} \right) \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} \mathbf{B}$]]>

I suppose so, but not sure if this extends to a functor $\Im$ on some gros site. But quite possibly it does! Need to think about it. Good point.

]]>Can’t what I have as $S$ in #35 be any space?

]]>Yes, so I think it is clear that Borger’s base change is (at least analogous to) that along the unit component of an infinitesimal shape modality at a fixed arithmetic base scheme.

An evident question is then whether there is also a global version where the full infinitesimal shape modality exists.

]]>Busy day today. I remember convincing myself that 10.1 of that Borger paper was useful. Where we have $Sp_S = Sh(Aff)/S = Sh(AffRel_S)$ which for affine $S$ is $Sh(Aff_S)$.

Now, why did that seem important? Something about a comonad on $Sh(Aff)/S$ induced by $S \to \Im(S)$. So that needed to be on $Sp_S$.

In the other paper, in section 1.2, $W_\ast$ seems to be a map from $Sp_S$ to $Sp_{S/\Lambda}$. Presumably he uses the same notation when composed with forgetting the $\lambda$ structure, so as a comonad on $Sp_S$, his $v^\ast \circ v_\ast$.

]]>Okay, now I may concentrate on this a bit more.

Regarding #20, #23: it seems to me that by Borger 10, (12.8.2) the adjoint $(W_n)_\ast$ in Borger’s adjoint triple/quadruple is not like $\Im$ itself but is like pushworward along the unit map $Spec R \to \Im(Spec R)$ for the fixed base ring $R$.

That also explains why this adjoint system is not idempotent.

]]>It was what you said elsewhere that addresses what I was wondering about:

if $\mathbf{H}$ is infinitesimally cohesive over $\infty\mathrm{Grpd}$, then the inclusion $\infty Grpd \hookrightarrow \mathbf{H}$ also exhibits $\mathbf{H}$ as being differentially cohesive over $\infty Grpd$. But the converse does not in general hold.

But anyway, what am I of course really after at the moment is what the story is with Borger’s arithmetic.

]]>regarding #27: not sure if I understand the question properly, but if this is, as in #31, about the relation between “infinitesimally cohesive” and “differentially cohesive” then, yes, the point to notice is that there is a priori no relation. The former is a property that may be satisfied by cohesion, the second is extra structure with which a topos or $\infty$-topos (which is or is not cohesive itself) may be equipped.

]]>