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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 14th 2015
• (edited May 14th 2015)

Given a differential cohesive topos $\mathbf{H}$, then for each object $X \in \mathbf{H}$ there is the jet comonad $Jet_X \colon \mathbf{H}_{/X} \to \mathbf{H}_{/X}$, which is a right adjoint (to the infinitesimal disk bundle operator). Therefore by this proposition its Eilenberg-Moore category of coalgebras $EM(Jet_X)$ is itself a topos, over $\mathbf{H}_{/X}$.

Question 1: is the topos $EM(Jet_X)$ cohesive, if $\mathbf{H}$ is?

Actually, that seems unlikely due to the dependency on $X$. Reminds one of the tangent topos construction. Therefore:

Question 2: Do the toposes $EM(Jet_X)$ as $X$ ranges over $\mathbf{H}$ maybe glue into one big topos? If so, is that cohesive if $\mathbf{H}$ is?

This question is motivated from discussion of variational calculus here.

(Incidentally, by Marvan 86 we have that $EM(Jet_X) = PDE(X)$ is the category of partial differential equations with variables in $X$.)

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeMay 20th 2015

What exactly is $PDE(X)$?

In the Marvan article, he speaks of EM coalgebras as ’infinitely prolonged systems’ of PDEs with a given manifold of independent variables, $X$, say. So an object of the category is such an infinitely prolonged system, I take it. Then morphisms?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMay 20th 2015

What exactly is $PDE(X)$?

Vinogradov decribed it as the category whose objects are PDEs regarded as subspaces of jet spaces, and whose morphisms are those smooth functions between the jet spaces that preserves these subspaces and the “Cartan distribution”, which essentially means that they preserve the differential relations and map solutions to solutions. The last result of Marvan’s article is that this is equivalent to the category $EM(J^\infty)$. I’d think another way to say this is that objects are collections of sections that solve a given differential equation, and morphisms are differential operators between spaces of sections that map solutions to solutions.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeMay 20th 2015

So why does he insist on the ’infinitely prolonged’ part? Again

The infinitely prolonged equations are the objects of Vinogradov category

And presumably, they’re just imagining a 1-category.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 20th 2015
• (edited May 20th 2015)

This is just a technicality. Given a system of differential equations of $k$-th order, i.e. involving derivatives of order $k$, then taken at face value this defines and is defined by a subbundle of the $k$-th order jet bundle, not of the infinite-order jet bundle.

But the abstract theory works best with the infinite-order jet bundle.

One then observeres that every system of order-$k$ differential equations is equivalent to a system of infinite-order differential equations. The latter is the result of starting with the former and adding more equations, all the “differential consequences” of the previous equations, namely all the results of acting on both sides of a previously given differential equation with the total derivative operation, and acting on the result of that again, and so on.

The final result of this process is called the prolongation of the original equations.

• CommentRowNumber6.
• CommentAuthorMichael_Bachtold
• CommentTimeMay 20th 2015
• (edited May 20th 2015)

Urs is right, although I wouldn’t call it a technicality. Let me just add a few remarks. The situation is analogous in algebraic geometry: traditionally people would write down a system of polynomial equations which they wanted to solve. One can then manipulate those equations without changing their solutions by say multiplying both sides with something or subtracting equations etc. This led people to consider the set of all equations which can be derived from the original system. This is now called the ideal generated by the polynomials but one may think of it as “all algebraic consequences of the original system of equations”. With a system of partial differential equations we can do the same but moreover, we are allowed to take derivatives (total derivatives) of the equations. The zero set of this differential ideal is the infinite prolongation of the equation and may be thought of as a submanifold in the infinite order jet bundle. If you consider this submanifold intrinsically, i.e. forget about the inclusion in the specific jet space you get what Vinogradov calls a diffiety. Algebraically this amounts to considering the quotient algebra, which has the additional structure of the Cartan distribution (horizontal vector fields).

If one just stayed at the level of finite order jets one would not see those “higher order symmetries” which we are discussing in the other thread. Also, by adjoining those differential consequences one might find out that the system of PDEs is inconsistent, so its prolongation is empty and it has no solutions (not even formal power series solutions).

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeMay 20th 2015

Thanks both. So now, according to the analogy I observed between

$Jet \coloneqq i^\ast i_\ast \;\colon\; \mathbf{H}_{/X} \to \mathbf{H}_{/X} \,.$

and

$\Box_W: \mathbf{H}_{/W} \to \mathbf{H}_{/W} \,.$

both arising as dependent product then context extension for the units of modalities, $\ast$ and $\Im$, in the Aufhebung table, what are the ’PDEs of necessity’?

In other words, what are the coalgebras for $\Box_W: \mathbf{H}_{/W} \to \mathbf{H}_{/W}$? This would require a way of mapping from each fibre of world-dependent type to the sections of the fibration. Easy examples would come from $W^{\ast} B$, for $B \in \mathbf{H}$. There we could send $(w, b)$ to $(w, f)$, where $f: w' \mapsto (w', b)$.

But is there a way to capture these coalgebras in full generality?

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMay 20th 2015
• (edited May 20th 2015)

Generally, for $i \colon X \longrightarrow Y$ an epimorphism, the coalgebras for $i^\ast i_\ast$ are equivalent to the algebras of $i^\ast i_!$, and by the Bénabou–Roubaud theorem for monadic descent (and using that the codomain fibration satisfies all the required properties), this are equivalently just the objects $E$ over $Y$, with $i^\ast E$ regarded with its canonical coalgebra structure.

That’s why for instance in Lurie’s notes on crystals the jet bundle is defined as an object over $\Im X$, not over $X$.

In the necessity example this says that the “necessary” types are indeed those which are constant on the type of possible worlds.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeMay 20th 2015

the “necessary” types are indeed those which are constant on the type of possible worlds.

Great. If there’s anything in the philosophical literature that comes close to this idea it’s rigid designators which are said to be terms that refer to the ’same’ entity across possible worlds. Kripke suggested proper names and natural kinds do this.

But that’s about terms in an untyped setting. Here we want to mention these constant types. Perhaps they could be called ’rigid types’ whose terms are ’rigid designators’.

On the other hand, there’s usually an addition

A rigid designator designates the same object in all possible worlds in which that object exists and never designates anything else,

to specify that the invariant object may not exist in all worlds.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeMay 20th 2015
• (edited May 20th 2015)

Something that does exist in worlds $W$ but not in a more general worlds $\tilde W$, under some embedding $i \colon W \hookrightarrow \tilde W$, is naturally formalized as a type in the image of $i_!$.

So a type necessary for worlds in $W$ but non-existent for worlds not in the image of $i \colon W \to \tilde W$ would be one in the image of $i_! \circ W^\ast$.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeMay 20th 2015

Interpreting your jet comonad-PDE constructions in the arithmetic setting gives arithmetic jet spaces, arithmetic differential operators and arithmetic partial differential equations, etc.?

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeMay 20th 2015

That’s an excellent point, thanks. This certainly looks light it ought to be true when things are set up correctly. Whether it is strictly true with the definitions available in the literature would need to be checked…

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeMay 20th 2015

Oh look, Buium on page 9 of Differential calculus with integers speaks of

an object of a natural category underlying a geometry more general than algebraic geometry which we refer to as δ-geometry . This geometry is an arithmetic analogue of the categorical setting in  and also an arithmetic analogue of the Ritt-Kolchin δ-algebraic geometry [52, 9].

and  is by Vinogradov et al.

• CommentRowNumber14.
• CommentAuthorDavid_Corfield
• CommentTimeMay 23rd 2015

I know you a thousands others things to do, but I might set up a page ’under construction’ to jot down ideas as to whether differential cohesion applied to arithmetic gives Buium-ish results. What would be a good name for that to fit in with other pages?

It should presumably link in with differential cohesion and idelic structure and arithmetic jet space and Borger’s absolute geometry. Or maybe it could just be an extension of one of these. Were you going to expand on the jet comonad story?

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeMay 25th 2015

I suppose the page arithmetic jet space would be a good point to further develop. The page jet comonad is there mainly to offer a means to point directly to the comonadic aspect, without losing the reader in more general jet bundle discussion.

• CommentRowNumber16.
• CommentAuthorDavid_Corfield
• CommentTimeMay 26th 2015
• (edited May 26th 2015)

Never as easy as I hope. For one thing, there’s the local-global issue (Borger teaming up with Buium). Manin describes some aspects of this in Numbers as Functions. Maybe for a latter stage there’s also the choice of $\mathbb{S}$ rather than $\mathbb{Z}$. Anyway, it doesn’t seem to me terribly clear what the relevant toposes should be.

There’s the $Et(Spec(\mathbb{Z}))$/$Et(Spec(\mathbb{F}_1))$ construction from Borger’s absolute geometry, and the $A Mod_{\mathfrak{a}comp}^{op}$/$A Mod_{\mathfrak{a}tors}^{op}$ construction from differential cohesion and idelic structure.

Interesting to see how often things come down to poor quotienting, Via what Google Books provides of the book Arithmetic Differential Equations, Buium sets off in the Introduction with an account of the problem of quotienting the wrong way, the “basic pathology”. He terms two solutions as ’invariant theory’ and ’groupoid theory’.

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeMay 27th 2015

For future reference, Borger speaks in here of $W_\ast = W_{\infty \ast}$, as the ’arithmetic jet space functor’, which has a comonad structure.

The category of $\Lambda$-spaces is the same as the category of spaces equipped with an action of the comonad $W_\ast$. It is therefore a topos…

Is there going to be a derived version of all this, an $E_\infty$ arithmetic jet space functor? At E-∞ geometry, it speaks of the (∞,1)-topos over the (∞,1)-site of formal duals of $E_\infty$ rings, equipped with the etale topology. So the equivalent of Borger’s $Et(Spec(\mathbb{Z}))$. There’s some relative differential cohesion somewhere about here providing a comonad, $\Im$?

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeMay 27th 2015

Thanks for re-highlighting these things. I should say that I’ll be absorbed with something non-arithmetic until June 16, then I’ll be back in arithmetic mode for at least one month. I’ll get back to all this then.

• CommentRowNumber19.
• CommentAuthorDavid_Corfield
• CommentTimeMay 28th 2015
• (edited May 28th 2015)

No problem. I may jot down some thoughts form time to time, which you can sweep through on June 16.

If there is an $\mathbb{S}$-equivalent of $\Lambda$-rings, we might look at Morava and Santhanam’s B-comonad here or in Rezk’s paper.

• CommentRowNumber20.
• CommentAuthorDavid_Corfield
• CommentTimeJun 2nd 2015

When we get back to this, I’m going to be wondering whether there’s an arithmetic jet comonad induced through a map $X \mapsto \Im(X)$.

Borger (Lambda-rings and the field with one element. p. 8) tells us that for $S$, the spectrum of the ring of integers of any number field or any smooth curve over a finite field, there is a map from the topos of spaces over $S$ to another topos of spaces which can be thought of as spaces over a $\Lambda$-version of $S$, $Sp(S/\Lambda_S)$. In the case of $S = Spec(\mathbb{Z})$, this would have to be the mythical $Spec(\mathbb{F}_1)$, so $S/\Lambda_S$ doesn’t exist as such in this set-up.

But is there than a setting in which some kind of $\Im(X) := X/\Lambda_X$ can be defined, to allow the jet comonad construction from induced maps between slices, $\mathbf{H}/X$ and $\mathbf{H}/\Im(X)$? Can $X/\Lambda_X$ be defined as a generalized space?

• CommentRowNumber21.
• CommentAuthorDavid_Corfield
• CommentTimeJun 9th 2015
• (edited Jun 11th 2015)

Morning wild speculative connections before admin tasks: Where Le Bruyn points out

the exciting prospect of extending or modifying Borger’s $\lambda$-rings approach to $\mathbb{F}_1$-geometry to other categories $rings_G$ of commutative rings with suitable morphisms to/from a collection of rings (for any conjugacy class of a cofinite subgroup of $G$) such that the Dress-Siebeneicher-Witt functor $W_G$ is a right adjoint functor to the forgetful functor $rings_G \to rings$,

should we look to Stickland on Tambara functors for which there are examples

…related to Witt rings in the sense of Dress and Siebeneicher

and for which

We suspect that the proper context for our results is really a theory of global Tambara functors, similar to Webb’s theory of global Mackey functors.

Which maybe all points to something globally equivariant (Global homotopy theory)

This kind of algebraic structure has been studied under different names (e.g., ‘global Mackey functor’, ‘inflation functor’,… ),

and so to Charles Rezk’s cohesion.

• CommentRowNumber22.
• CommentAuthorDavidRoberts
• CommentTimeJun 10th 2015

Another intriguing thread, this in an upcoming talk by Annette Huber on differential forms in algebraic geometry at the combined conference in Porto where Urs is:

We explain how the use of the h-topology (introduced by Suslin and Voevodsky in order to study motives) gives a very good object also in the singular case, at least in characteristic zero. The approach unifies other ad-hoc notions and simplies many proofs

• CommentRowNumber23.
• CommentAuthorDavid_Corfield
• CommentTimeJun 11th 2015

So I guess I’m at the stage of wondering whether the second map followed by the third map in this from Borger’s absolute geometry is like $\Im$

$Et(Spec(\mathbb{Z})) = Sh(Spec(\mathbb{Z})_{et}) \stackrel{\longleftarrow}\stackrel{\longrightarrow}{\stackrel{\longleftarrow}{\stackrel{\longrightarrow}{}}} Sh(Spec(\mathbb{F}_1)_{et}) = Et(Spec(\mathbb{F}_1)).$
• CommentRowNumber24.
• CommentAuthorDavid_Corfield
• CommentTimeJun 12th 2015
• (edited Jun 12th 2015)

If cohesion makes sense in the 1-topos case, is there a notion of differential cohesion already there, or do we have to move up to the $(\infty, 1)$ situation?

Does the $Sh_{\infty}$ situation give the $X/\Lambda_X$ from #20 the chance to exist as a space?

What happens if you have a site which isn’t cohesive, and then another site which has all the properties of being an infinitesimal neighborhood of it? Here such neighborhoods are only defined for cohesive sites. But can go you go ahead with some relative differential constructions all the same?

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeJun 13th 2015

Yes, differential cohesion may well be considered in the 1-topos case (in contrast to cohesion, which is qualitatively different in 1-toposes from its nature in $\infty$-toposes), and yes one may have “differentially cohesive infinitesimal neighbourhoods” without having cohesion, as is the case in most algebraic and arithmetic situations. (At least in dcct there is an explicit mentioning of this non-cohesive case, maybe on the $n$Lab the distinction hasn’t been brought out sufficiently.)

Regarding the question about $X/\Lambda_X$ being a “space” or not, you may have to remind me what kind of object you say it is meant to be by dafault and what it would take for it to count as a “space” in your sense here?

• CommentRowNumber26.
• CommentAuthorDavid_Corfield
• CommentTimeJun 14th 2015

differential cohesion may well be considered in the 1-topos case

Ah, I never saw that said anywhere. Which of the structures will then be induced?

But perhaps the first question to ask is whether we are in fact dealing here in the diagram reproduced in #23 with a case of “differentially cohesive infinitesimal neighbourhoods”.

• CommentRowNumber27.
• CommentAuthorDavid_Corfield
• CommentTimeJun 14th 2015

Looking around nLab, I see there is something that confuses me. At differential cohesive (infinity,1)-topos we have

Given a cohesive $(\infty,1)$-topos $\mathbf{H}$ we say that an infinitesimal cohesive neighbourhood of $\mathbf{H}$ is another cohesive $(\infty,1)$-topos $\mathbf{H}_{th}$ equipped with an adjoint quadruple of adjoint (∞,1)-functors of the form

$(i_! \dashv i^* \dashv i_* \dashv i^!) : \mathbf{H} \stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^*}{\leftarrow}}{\stackrel{\overset{i_*}{\hookrightarrow}}{\underset{i^!}{\leftarrow}}}} \mathbf{H}_{th}$

But these can be distinct, no? So why do the maps between toposes in infinitesimal cohesive (infinity,1)-topos have to form ambidexterous adjunctions?

By the way, I see Pierre Cartier will speak here on ’Analogies entre les espaces de jets et les vecteurs de Witt’.

• CommentRowNumber28.
• CommentAuthorDavid_Corfield
• CommentTimeJun 15th 2015

And Andre Joyal at CT2015, Witt vectors and the James construction

The Witt vectors construction is a comonad on the category of commutative rings  . We show that the comonad is cofreely cogenerated by a pointed endo-functor. The proof uses an abstract version of the James construction in topology  and the theory of Tall and Wraith 

• CommentRowNumber29.
• CommentAuthorDavidRoberts
• CommentTimeJun 15th 2015

Don’t forget the paper on the arXiv today about Witt rings of real varieties…

• CommentRowNumber30.
• CommentAuthorDavid_Corfield
• CommentTimeJun 15th 2015

Witt rings of real varieties…

That’s the second sense of Witt ring, no?

• CommentRowNumber31.
• CommentAuthorDavid_Corfield
• CommentTimeJun 16th 2015

Re my #27, I guess that’s the difference between infinitesimal and differential. So do any infinitesimal extensions give rise to an interesting jet comonad?

How about the Jet topos constructions? In the case of $J_1$ (tangent topos) for $\infty Grpd$, so parameterized spectra, the (co)monad $\Im$ takes $A \to B$ to $0 \to B$?

So does anything interesting happen back and forth between $T(\infty Grpd)/(A \to B)$ and $T(\infty Grpd)/(0 \to B)$?

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeJun 16th 2015
• (edited Jun 16th 2015)

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.

• CommentRowNumber33.
• CommentAuthorDavid_Corfield
• CommentTimeJun 16th 2015

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.

• CommentRowNumber34.
• CommentAuthorUrs
• CommentTimeJun 17th 2015
• (edited Jun 17th 2015)

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.

• CommentRowNumber35.
• CommentAuthorDavid_Corfield
• CommentTimeJun 17th 2015

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 $v^\ast \circ v_\ast$.

• CommentRowNumber36.
• CommentAuthorUrs
• CommentTimeJun 17th 2015

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.

• CommentRowNumber37.
• CommentAuthorDavid_Corfield
• CommentTimeJun 17th 2015

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

• CommentRowNumber38.
• CommentAuthorUrs
• CommentTimeJun 17th 2015

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.

• CommentRowNumber39.
• CommentAuthorDavid_Corfield
• CommentTimeJun 17th 2015

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

$\mathbf{H} \coloneqq Sh_\infty\left(SmthMfd, \mathbf{B} \right) \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} \mathbf{B}$
• CommentRowNumber40.
• CommentAuthorUrs
• CommentTimeJun 17th 2015
• (edited Jun 17th 2015)

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)

• CommentRowNumber41.
• CommentAuthorDavid_Corfield
• CommentTimeJun 18th 2015
• (edited Jun 18th 2015)

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?

• CommentRowNumber42.
• CommentAuthorDavid_Corfield
• CommentTimeJun 18th 2015

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.

• CommentRowNumber43.
• CommentAuthorUrs
• CommentTimeJun 18th 2015
• (edited Jun 18th 2015)

One thing to maybe keep in mind is that

1. 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

2. 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.

• CommentRowNumber44.
• CommentAuthorDavid_Corfield
• CommentTimeJun 18th 2015

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?

• CommentRowNumber45.
• CommentAuthorDavid_Corfield
• CommentTimeJun 18th 2015

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.

• CommentRowNumber46.
• CommentAuthorDavid_Corfield
• CommentTimeJun 19th 2015

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.

• CommentRowNumber47.
• CommentAuthorUrs
• CommentTimeJun 25th 2015
• (edited Jun 25th 2015)

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…

• CommentRowNumber48.
• CommentAuthorDavid_Corfield
• CommentTimeJun 25th 2015

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.

• CommentRowNumber49.
• CommentAuthorDavid_Corfield
• CommentTimeSep 22nd 2015

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.

• CommentRowNumber50.
• CommentAuthorUrs
• CommentTimeSep 22nd 2015

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.

• CommentRowNumber51.
• CommentAuthorDavid_Corfield
• CommentTimeSep 22nd 2015

Ah, OK thanks.

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

• CommentRowNumber52.
• CommentAuthorUrs
• CommentTimeSep 22nd 2015

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.

• CommentRowNumber53.
• CommentAuthorDavid_Corfield
• CommentTimeSep 22nd 2015

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.

• CommentRowNumber54.
• CommentAuthorUrs
• CommentTimeSep 22nd 2015

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 $PDE_\Sigma(\mathbf{H})$.

• CommentRowNumber55.
• CommentAuthorDavid_Corfield
• CommentTimeJan 22nd 2017

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.