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Meanwhile I have a good idea of how to give an elementary formalization, in terms of differential cohesion, of some of the key concepts in the formulation of local Lagrangian field theory on jets of field bundles. In particular I know that given a local Lagrangian $\mathbf{L} \colon E \to \mathbf{B}^n U(1)_{conn}$ then its homotopy stabilizer under $\mathbf{Aut}(E)$, equivalently: its quantomorphism n-group, is the group of symmetries and their higher conserved currents…
…only that this is true only with some extra restrictions. If $E$ is a field fiber so that $\mathbf{L}$ is a WZW term, then it is true, but then the symmetries are just “point symmetries”. This is interesting enough in itself and I used to be content with this, but it is not the full story.
More generally, when $E = Jet(F)$ is the jet bundle of a field bundle, then the quantomorphisms of $\mathbf{L}$, which are diagrams of the form
$\array{ Jet(F) && \underoverset{\simeq}{\phi}{\longrightarrow} && Jet(F) \\ & {}_{\mathllap{\mathbf{L}}}\searrow &\swArrow^{\eta}_{\mathrlap{\simeq}}& \swarrow_{\mathrlap{\mathbf{L}}} \\ && \mathbf{B}^n U(1)_{conn} }$should be constrained:
$\phi$ should be evolutionary;
$\eta$ should be horizontal.
Question: How may one impose these conditions by general abstract means, using differential cohesion and the jet comonad structure?
I suppose one should demand some kind of factorization through some universal maps of the (co-)monad, but I still don’t really see it.
Do you really mean horizontal and not vertical in 2.?
Yes. Infinitesimally $\eta$ becomes the exact term by which the Lie derivative of the Lagrangian along the infinitesial symmetry may fail to vanish. This exact term is supposed to be horizontally exact. That’s the condition that I am after.
Thanks, I didn’t read carefully. I don’t know about $\eta$, this sounds related to what Vinogradov et. al. call the Legendre form.
Concerning a more abstract formulation of evolutionary fields, not yet in terms of the jet comonad, but maybe useful (although you probably know this already):
One of the main structures on $Jet(F)$ is the Cartan distribution (also called higher order contact structure), which may be seen as a flat Ehresmann connection on the bundle $Jet(F)\to X$. It consists of horizontal vector fields on $Jet(F)$, i.e. those vector fields tangent to the image of the universal differential operator from sections of $F\to X$ to sections of $Jet(F)\to X$. Maybe there is a different formulation of this distribution in terms of the comonad. Anyway, leaves of the Cartan distribution are in one to one correspondence with sections of $F\to X$. Denote the module of horizontal fields with $CDer$ and denote with $CDiff$ the module of linear differential operators generated by $CDer$. Then the module of vertical vector fields $V$ on $Jet(F)$ (which is canonically isomorphic to vector fields on $Jet(F)$ modulo horizontal ones: $Der(Jet(F))/CDer$) is naturally a left $CDiff$ module. This can be thought of as analogous to a D-module structure or flat linear connection on $V$, with the difference that it’s only defined in direction of the leaves of the Cartan distribution. From this flat connection you can obtain the standard horizontal de Rham complex with coefficients in $V$:
$V\to V \otimes C\Omega^1 \to \ldots$Where $C\Omega$ denotes horizontal forms. Now, the 0th-cohomology of this complex are precisely the evolutionary fields.
(This observation is due to Vinogradov and is one piece in his conjectured secondary “cohomological” calculus. There is his book Cohomological Analysis of Partial Differential Equations and Secondary Calculus, but he has several shorter expositions of the main ideas.)
Thanks! That sounds good.
I’ll dig out the statement by Vinogradov that you are highlighting, thanks.
One first remark: that universal differential operator from sections of $\array{F \\ \downarrow \\ X}$ to sections of $\array{Jet(F) \\ \downarrow \\ X}$ is the one whose $(T_{inf} \dashv Jet)$-adjunct is given by pre-composing a section with (the image under $\sum_{\Im X}$ of) the $i_!$ image
$i_! i^\ast i_! X \stackrel{i_!(\epsilon_X)}{\longrightarrow} i_! X$of the counit $\epsilon$ of the $(i_! \dashv i^\ast)$-adjunction for base change along $i \colon X\to \Im X$. So possibly it is this structure morphism from which we are to construct the desired constraint.
I have a suspicion:
a flow
$Jet(F) \stackrel{\phi}{\longrightarrow} Jet(F)$being evolutionary is to mean that it preserves the Cartan distribution, and by what I just said in #5, but dualizing back through the $(T_{inf} \dashv Jet)$-adjunction, I am guessing now that this means that its image under $i_\ast$ stabilizes the universal differential operator, i.e. that we have
$\array{ && Jet( F) \\ & {}^{\mathllap{}}\swarrow &\swArrow_{\mathrlap{\simeq}}& \searrow^{\mathrlap{}} \\ Jet(Jet(F)) && \underset{Jet(\phi)}{\longrightarrow}&& Jet(Jet(F)) } \,.$If that is true, then my second order conjecture is that the horizontal $\eta$ are those killed by this precomposition, i.e. that for a quantomorphism
$\array{ Jet(F) && \stackrel{\phi}{\longrightarrow} && Jet(F) \\ & {}_{\mathllap{\mathbf{L}}}\searrow &\swArrow_{\mathrlap{\eta}}& \swarrow_{\mathrlap{\mathbf{L}}} \\ && \mathbf{B}^{p+1}U(1)_{conn} }$to be admissible, we need
$\left( \array{ && Jet( F) \\ & {}^{\mathllap{}}\swarrow &\swArrow& \searrow^{\mathrlap{}} \\ Jet(Jet(F)) && \stackrel{Jet(\phi)}{\longrightarrow} && Jet(Jet(F)) \\ & {}_{\mathllap{Jet \mathbf{L}}}\searrow &\swArrow_{\mathrlap{Jet(\eta)}}& \swarrow_{ \mathrlap{Jet\mathbf{L}}} \\ && Jet \mathbf{B}^{p+1}U(1)_{conn} } \right) \;\;\;\; \simeq \;\;\;\; \left( \array{ && Jet(F) \\ & {}^{\mathllap{}}\swarrow && \searrow^{\mathrlap{}} \\ Jet(Jet(F)) && = && Jet(Jet(F)) \\ & {}_{\mathllap{Jet \mathbf{L}}}\searrow && \swarrow_{\mathrlap{Jet \mathbf{L}}} \\ && Jet \mathbf{B}^{p+1}U(1)_{conn} } \right)$ah, my first conjecture in #6 still looks good to me, but the second one goes in the wrong direction.
The first should work out by the fact that the coproduct of the Jet-comonad is really nothing but the image under $Jet$ of the universal differential operator.
For the second suggestion in #6 though, it’s the wrong way around: I am supposed to say that $\mathbf{L}$ and its stabilizing homotopies are horizontal, hence are entirely supported, in the sense of differential forms, on the image of this universal differential operator, not that they are trivialized by restriction to that image.
Hm…
I just want to make an elementary remark, which may or may not be useful towards a successful formalization. Let $\Omega^p$ be the classifying stack of $p$-forms. Then, in my understanding of the notation, $[M, \Omega^p]$ is the space of $p$-forms on $M$, usually denoted by $\Omega^p(M)$. Let $E = F \times M$ be a product bundle (over $M$). I take that simple case just so I can write the space of sections of $E$ as $[M, F]$. Consider those maps $\alpha\colon [M, F] \to [M,\Omega^p]$ that are differential operators. The collection of all these differential operators $\alpha$ is in bijection with horizontal forms on $J^\infty(E)$. Now, any such $\alpha$ becomes a bundle map $\alpha\colon J^\infty E \to \Lambda^p T^* M$. Now, pulling back the bundle $\Lambda^p T^*M$ along $J^\infty E \to M$, it is easy to identify the pulled back bundle with a sub-bundle of $\Lambda^p T^* (J^\infty E)$. Thus, horizontal forms can be identified with a subset of forms on $J^\infty E$, which with the help of the classifying stack is $[J^\infty E, \Omega^p]$.
So, starting with $\alpha \colon [M, F] \to [M, \Omega^p]$, we end up with an element of $[J^\infty E, \Omega^p]$. Is this not enough to come up with a decent “formalization”?
Hi Igor, thanks!
There are some subtleties hidden there, let’s see:
For sure the traditional concept of horizontal forms is equivalent to bundle maps $J^\infty E \to \Lambda^p T^\ast \Sigma$ over the base $\Sigma$. But this is not quite equivalent to differential operators from $F \to \Sigma$ to the trivial bundle over $\Sigma$ with fiber the sheaf $\mathbf{\Omega}^p$.
The issue here, together with its solution, is discussed a bit at differential cohesion – structure sheaves.
The solution for the case of bare $p$-forms is to apply the etalification coreflection $Et$ from prop. 8 there, i.e.
$Et \left( \array{ \Sigma \times \mathbf{\Omega}^p \\ \downarrow \\ \Sigma } \right)$is a corrected stacky way to get the exterior form bundle from the moduli sheaf of all forms.
That serves to get the horizontal forms themselves. But now next we want to get also the horizontal differential into the picture and want to generalize from globally defined $p$-forms $\mathbf{\Omega}^p$ to $p$-form connections $\mathbf{B}^p U(1)_{conn}$.
Hm, so we could consider
$Et \left( \array{ \Sigma \times \mathbf{B}^p U(1)_{conn} \\ \downarrow \\ \Sigma } \right)$and then maps into that from
$Et \left( \array{ J^\infty (F) \\ \downarrow \\ \Sigma } \right) \,.$Will that come out right? I don’t see it presently. Somehow the universal Cartan distribution has to enter the picture, otherwise it seems to me we will only get the naive horizontal differential, not the correct one (the “total derivative”).
I’d have to get a better idea of what “etalification” is before deciding whether that might make sense or not. But I would 100% agree that the Cartan distribution on $J^\infty F$ must enter into the definition of horizontal (or for that matter vertical) forms on $J^\infty F$.
Behind the scenes, Igor kindly points out to me that with horizontal $p$-forms understood as bundle maps $\tilde \alpha \colon Jet(F) \to \Lambda^p T^\ast \Sigma$, hence as differential operators $\alpha \colon F \to \Lambda^p T^\ast \Sigma$, then the correct horizontal differential is just the composition with the de Rham differential as differential operators, i.e. the composite
$\widetilde{d_H \alpha} \;\colon\; Jet(F) \to Jet(Jet(F)) \stackrel{Jet(\tilde \alpha)}{\longrightarrow} Jet(\Lambda^p T^\ast \Sigma) \stackrel{\tilde d_{\sigma}}{\longrightarrow} \Lambda^p T^\ast \Sigma \,.$That’s neat.
Now this means that we will get the correct stacky version of the horizontal differential if we formulate the Deligne complex and everything internal to the Eilenberg-Moore category of $Jet_{\Sigma}$-coalgebras of the slice $\mathbf{H}_{/\Sigma}$ of our differential cohesive topos $\mathbf{H}_{}$. (By the result of Marvan 86, this Eilenberg-Moore category is the category $PDE(X)$ of partial differential equations with variables in $X$.(!))
Moreover, by this proposition and using that the jet comonad is a right adjoint, $PDE(X)$ is itself a topos over $\mathbf{H}_{/X}$.(!)
Now the big question: is $PDE(\mathbf{H})$ suitably cohesive if $\mathbf{H}$ is??
If it is, then we are in business: then we simply formulate the differential coefficients $\mathbf{B}\mathbb{G}_{conn}$ not in $\mathbf{H}$ but in $PDE(\mathbf{H})$.
This question maybe deserves a separate thread here.
Ah, we don’t need that $PDE(\mathbf{H})_X$ is cohesive. We simply send all our ingredients along
$\iota \;\colon\; \mathbf{H} \stackrel{\Sigma^\ast}{\longrightarrow} \mathbf{H}_{/\Sigma} \stackrel{Et}{\longrightarrow} \mathbf{H}_{/\Sigma} \stackrel{free}{\longrightarrow} PDE(\mathbf{H})_{\Sigma}$where the second map is etalification and the third is the free Jet-coalgebra functor (the inverse image of the geometric morphism by which $PDE(\mathbf{H})_{\Sigma}$ sits over $\mathbf{H}$).
Then notably $\iota \mathbf{B}^{p+1}U(1)_{conn} \in PDE(\mathbf{H})_{\Sigma}$ is our coefficient.
Now for $F \in \mathbf{H}_{/\Sigma} \stackrel{free}{\to} PDE(\mathbf{H})_{\Sigma}$ our field bundle, then a horizontal Lagrangian is nothing but a map
$\mathbf{L} \colon F \longrightarrow \iota \mathbf{B}^{p+1}U(1)_{conn}$in $PDE(\mathbf{H})_\Sigma$, i.e. is just the naive thing, but done not internal to $\mathbf{H}$ but internal to $PDE(\mathbf{H})_\Sigma$.
Homotopies between such maps encode now the correct horizontal differentials.
The correct quantomorphism/current group now is the naively formulated quantomorphism group, but internal to $PDE(\mathbf{H})_\Sigma$, constrained such that the automorphisms of $F$ come from $\mathbf{H}_{\Sigma}$ under $\iota$.
So this idea of a category of pdes goes back to Vinogradov? He seems to feature most places you see the term, such as Igonin’s ’Analogues of coverings and the fundamental group for the category of partial differential equations’. That Diffiety school sounds like it’s aiming to be quite a movement.
Yes. As far as I am aware (thanks also to discussion with Zoran yesterday), the history is roughly like this:
Vinogradov et al set of diffiety theory in the context of differential geometry
Marvan understands the full comonadic abstract theory behind it all, but is being ignored
following Grothendieck’s crystals, Beilinson-Drinfeld set up special cases of diffieties (namely linear and affine) in the context of algebraic geometry and call it D-geometry
Lurie in 09 and Gaitsgory in 14 recover aspects of the general comonadic theory behind this. (Lurie points out that forming Jets is comonadic, Gaisgory checks that D-modules, hence “linear algebraic diffeties” are (co-)monadic over their base).
From a suitably abstract perpective of an infinitesimal shape modality, all this follows immediately.
That Diffiety school sounds like it’s aiming to be quite a movement.
?? As an organization it seems to have died in 2010. The Diffity Institute says:
Since July 15, 2010 the site is frozen. Updates are possible for Sophus Lie e-library only.
For Diffiety schools and Current Geometry conferences see The Levi-Civita Institute.
For Moscow seminar see site on the Geometry of Differential Equations.
Luckily mathematical ideas are connected to institutes and servers etc., no more than, say, spirituality is connected to the church. It helps to have a warm shelter, but that shelter is not the substance of the idea.
So do you reckon you can take their constructions and reinterpret them in differential cohesive terms, e.g., their secondary calculus?
By the way, I see that Gabriele Vezzosi who works with Toën once worked with Vinogradov.
David, I suppose it should all go through, but so far I have only a subset of the existing constructions and properties elementarily axiomatized. Will have to see how far this may be pushed.
I have expanded the entry a bit more. Added some further remarks and pointers to traditional literature to More on the horizontal differential complex (thanks to Igor Khavkine for providing more pointers!!).
Then I have typed out the elementary axiomatization of the horizontal complex, as far as I understand it momentarily, at
What do ’$\kappa$ generically’ and ’$\iota$ generically mean?
The full sentence (beginning before the bullet list here) is “we write $\iota$ generically for…”
I am using either letter for any of the morphisms displayed, the morphism in question is determined by the type of its argument.
These morphisms just “regard everything canonically as a PDE over $\Sigma$” and depending on how far away from $PDE(\mathbf{H})_{\Sigma}$ we start, they apply the necessary re-identifications, as indicated. The key is that $\iota$ applies étalification, which amounts to the horizontality constraint, while $\kappa$ doesn’t. So under $\iota$ the differentials are purely horizontal, while under $\kappa$ they involve both horizontal and vertical contributions.
(This isn’t really explained in the entry at the moment, but it follows from unwinding the definitions.)
Let’s come back to the evolutionary vertical vector fields.
According to, for instance, “Homological Methods in Equations of Mathematical Physics”, theorem 3.26, def. 3.30, then evolutionary vector fields are equivalently the infinitesimal symmetries, i.e. the infinitesimal diffeomorphisms of some differential equation $\mathcal{E}$ inside some jet bundle $J^\infty E$ which preserve the Cartan distribution.
Right?
If that is true, then the elementary axiomatization of this is immediate with the main result of Marvan86: there these symmetries are identified with the automorphisms of $\mathcal{E}$ in the category of $J^\infty$-coalgebras.
Re #21:
Right?
yes that’s the idea.
If that is true, then the elementary axiomatization of this is immediate…
It’s not so immediate to me, how does one formulate “infinitesimal symmetry” (i.e. vector field) for objects of the Eilenberg Moore category? (I assume that this can be done in any topos with differential cohesion, and probably it is written somewhere in the nlab)
Thanks for your reply.
yes
Good, thanks
that’s the idea.
Which remaining fine print would you highlight?
how does one formulate “infinitesimal symmetry” (i.e. vector field) for objects of the Eilenberg Moore category?
Yes, this is one of the constructions that are implied by differential cohesion. On the $n$Lab there are remarks at Lie differentiation, a more comprehensive discussion is in section 5.3.5, Infinitesimal neighbourhoods and Lie differentiation of dcct (pdf).
By the fact that $\mathrm{PDE}(\mathbf{H})_\Sigma$ is a topos over $\mathbf{H}_{/\Sigma}$, hence over $\mathbf{H}$, it is canonically enriched in $\mathbf{H}$. For $\mathcal{E} \in \mathrm{PDE}(\mathbf{H})_\Sigma$ a differential equation, then its $\mathbf{H}$-valued automorphism group is the direct image of the PDE-internal automorphism group in $\mathbf{H}$, let’s write $\mathbf{Aut}_{\mathbf{H}}(\mathcal{E})$ for that. This is already the formalization of Vinogradov’s group of symmetries.
Now Lie algebras are simply the first order infinitesimal neighbourhoods of the neutral element in Lie groups, regarded as infinitesimal groups. This is described in much detail way back in Anders Kock’s books on synthetic differential geometry. The homotopy-theoretic version of this statement is the theorem by Pridham and Lurie that suitably cohesive pointed connected infinitesimal $\infty$-stacks are equivalent to $L_\infty$-algebras.
The infinitesimal group $\infty$-stack of a group $\infty$-stack $G$ may be axiomatized in differential cohesion as the homotopy fiber of the infinitesimal shape unit $G \longrightarrow \Im_{(1)}G$. In the standard model of formal smooth $\infty$-groupoids this produces excactly an infinitesimal $\infty$-stack as considered by Pridham and Lurie (namely an $\infty$-stack on formal duals of local Artin algebras, aka “Weil algebra” in SDG jargon).
Thanks for the continuing effort to try and explain these things. I didn’t answer earlier since I haven’t caught up with all the things you wrote.
So summarising #21 and #24 (although I haven’t understood all the details): evolutionary vector fields should correspond to the first order infinitesimal neighbourhood of the identity in $\mathbf{Aut}_{\mathbf{H}}(\mathcal{E})$?
Which remaining fine print would you highlight?
Hm, maybe not fine print, but things that I’m curious about. One thing is, if I recall correctly, that there are examples of such infinitesimal symmetries of PDEs that don’t possess a flow (on the space of solutions). I’d have to dig out the explicit counterexamples to recall in which sense exactly they don’t integrate. So I’m slightly curious as to what happens in this general setting with that.
Also I recall Vinogradov saying, that the whole cohomology of the complex I mentioned in #4 should be considered as the symmetries of the PDE. I never figured out an intuitive way of how to think of the higher cohomology classes, but it seems tempting to try an interpretation along the lines of “higher infinitesimal identities”, or maybe they say something about the deformation of the boundary of a solution… I don’t know, but I wonder again if this general approach leads to new insights.
evolutionary vector fields should correspond to the first order infinitesimal neighbourhood of the identity in $\mathbf{Aut}_{\mathbf{H}}(\mathcal{E})$?
Yes!
Hm, maybe not fine print, but things that I’m curious about. One thing is, if I recall correctly, that there are examples of such infinitesimal symmetries of PDEs that don’t possess a flow (on the space of solutions). I’d have to dig out the explicit counterexamples
Please let me know when you find the reference that you are thinking of. But it makes me wonder since it seems that the statement that evolutionary vector fields are infinitesimal symmetries is due to Vinogradov himself (or at least to his school, as cited in #21)
Also I recall Vinogradov saying, that the whole cohomology of the complex I mentioned in #4 should be considered as the symmetries of the PDE.
Yes, I was wondering about that ever since your #4. While I see that you say the intuitive interpretation of the relevant cohomology groups is elusive, could you recap for me one or two facts that Vinogradov does amplify about these cohomology groups? Just so that I get a feeling for what might be going on.
I see Igonin in Notes on symmetries of PDEs and Poisson structures says:
In these notes we try to describe the theory of (generalized or higher) symmetries of PDEs in the most general, compact, and coordinate-free form,
but it seems there’s no appearance of higher Lie theory. And in his Analogues of coverings and the fundamental group for the category of partial differential equations, this stays at the ordinary Lie group level.
Yes, many variationalists say “higher symmetry” to refer to simply to symmetries that involve higher order elements of the jet filtration (i.e. that act on and produce higher order derivatives of the fields), for instance leading up to def. 3.29 in arXiv:9808130.
To find what “we” consider as higher symmetries one has to go to the literature on BV-BRST complexes.
many variationalists say “higher symmetry” to refer to simply to symmetries that involve higher order elements of the jet filtration
yes, this is a source of possible confusion in this context of higher categories. Also the terminology “secondary foo” is not really very helpful. But to get an intuition it is useful to read “secondary vector field” as “vector field on the space of solutions of a PDE” etc.
Urs, concerning your questions in #25 I’ll try to come back to it. Currently I’m kept busy by some “external” activities and since I haven’t thought about this in some years I need to refresh my memory and find the relevant references. So apologies if I don’t answer in the coming days.
I am stuck with the following technical thing:
There is a coverage (Grothendieck pretopology) on the category
$PDE_\Sigma = EM(J^\infty_\Sigma|_{FrechetMfd_{/\Sigma}})$of PDEs with free variables ranging in some manifold $\Sigma$, induced by its full embedding into the topos
$FormalSmoothSet_{/\Im \Sigma} \simeq EM(J^\infty_\Sigma)$(the Cahiers topos sliced over the infinitesimal shape of $\Sigma$).
What I’d like to have is that a covering $\{U_i \to E\}_i$ of bundles over $\Sigma$ remains a covering after regarding bundles as the cofree PDEs on them.
Intuitively it seems this ought to be true, but I have trouble finding a proof.
The cofree PDE functor is right adjoint not left adjoint. And while applying its left adjoint to such a would-be epi yields an epi, the left adjoint is not faithful. So it doesn’t just follow abstractly, it seems.
Ah, I was being stupid. The left adjoint is conservative, by monadicity, and since it preserves all limits and colimits, in particular it reflects epis. And that’s it. Sorry for the noise.
Now I am trying to get the following, or to see if it works in the first place.
Let again $\mathbf{H}$ be the Cahiers topos, $\Im \colon \mathbf{H}\to \mathbf{H}$ its infinitesimal shape modality, $\Sigma \in SmoothMfd \hookrightarrow \mathbf{H}$, and $J^\infty_\Sigma \coloneqq (\eta_\Sigma)^\ast (\eta_\Sigma)_\ast \colon \mathbf{H}_{/\Sigma} \to \mathbf{H}_{/\Sigma}$ the induced comonad of base change along the unit of $\Im$ at $\Sigma$. I write $PDE_\Sigma(\mathbf{H}) \coloneqq EM(J^\infty_\Sigma)$ for its category of coalgebras, and since $\eta_\Sigma$ is epi, so by comonadic descent this is equivalently $PDE_\Sigma(\mathbf{H}) \simeq \mathbf{H}_{/\Im \Sigma}$.
This $J^\infty_\Sigma$ restricts to the traditional jet comonad on $SmoothMfd_{/\Sigma}\hookrightarrow \mathbf{H}_{/\Sigma}$, where its coalgebras are PDEs with free variables ranging in $\Sigma$: $EM(J^\infty_\Sigma|_{SmoothMfd}) \simeq \mathrm{PDE}_\Sigma$. For this statement to make sense, I must mean some type of infinite-dimensional smooth manifolds when writing $SmoothMfd$. I am inclined to use pro-objects in finite dimensional manifolds, since that’s just big enough to accomodate jet bundles, but Fréchet manifolds or whatever other concept might do just as well.
With that understood, it’s not a big deal to declare that we generalize $PDE_\Sigma$ just a tad by taking it to be the category of coalgebras of $J^\infty_\Sigma$ acting on $FormalSmoothMfd_{/\Sigma}$, i.e. to allow all (infinite dimensional) manifolds and bundles to have infinitesimal directions.
Now, since $FormalSmoothMfd_{/\Sigma}$ is also a site of definition for the slice $\mathbf{H}_{/\Sigma}$ of the Cahiers topos over $\Sigma$, it is natural to wonder if there is a site structure on this $PDE_\Sigma$, such that we have the following equivalence on the right, making the square commute:
$\array{ \mathbf{H}_{/\Sigma} &\stackrel{\overset{U}{\longleftarrow}}{\underset{F}{\longrightarrow}}& PDE_\Sigma(\mathbf{H}) = EM(J^\infty_\Sigma) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ Sh(FormalSmoothMfd_{/\Sigma}) &\stackrel{\overset{U_!}{\longleftarrow}}{\underset{F_!}{\longrightarrow}}& Sh(PDE_\Sigma) }$Does this work? I am thinking to go about it as follows.
Using that $PDE_\Sigma(\mathbf{H}) \simeq \mathbf{H}_{/\Im \Sigma}$, it follows that this topos has a site of definition given by the comma category $FormalSmoothMfd/\Im \Sigma$ with coverage induced from the canonical one on $FormalSmoothMfd$ by forgetting the maps to $\Im \Sigma$.
Now take $PDE_\Sigma$ to be equipped with the coverage whose covers are those reflected by the functor $U \colon PDE_\Sigma \longrightarrow FormalSmoothMfd_{/\Sigma}$. This is indeed a coverage, because $U$ on $PDE_\Sigma$ preserves pullbacks, being the restriction of the right adjoint $(\eta_\Sigma)^\ast$.
With this I’d think that $FormalSmoothMfd/\Im \Sigma$ is a dense subsite of $PDE_\Sigma$. First of all it is indeed a subcategory: given a morphism $K \longrightarrow \Im \Sigma$ out of a formal smooth manifold, then also its image under $(\eta_\Sigma)^\ast$ is a formal smooth manifold mapping into $\Sigma$, hence under $PDE_\Sigma(\mathbf{H})\simeq \mathbf{H}_{\Im \Sigma}$ the object $K \to \Im \Sigma$ is in the inclusion of $PDE_\Sigma$, by the above remark.
And that this subcategory is indeed a dense subsite should follow from the fact that it is indeed a site of definition even of $PDE_\Sigma(\mathbf{H})$ inside which $PDE_\Sigma$ sits. This is maybe the point that needs more thorough thinking…
This would give the equivalence on the right in the above diagram. With this then the commutativity of the diagram of left adjoints may be checked on representables, and then the commutativily of the diagram of right adjoints follows by uniqueness of adjoints.
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