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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 13th 2015
   • (edited May 13th 2015)
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   Author: Urs
   Format: MarkdownItexMeanwhile 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: 1. $\phi$ should be evolutionary; 1. $\eta$ 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.

   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 then its homotopy stabilizer under , 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 is a field fiber so that 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 is the jet bundle of a field bundle, then the quantomorphisms of , which are diagrams of the form

   should be constrained:

   1. should be evolutionary;

   2. 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.

2. • CommentRowNumber2.
   • CommentAuthorMichael_Bachtold
   • CommentTimeMay 13th 2015
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   Author: Michael_Bachtold
   Format: MarkdownItexDo you really mean horizontal and not vertical in 2.?

   Do you really mean horizontal and not vertical in 2.?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 13th 2015
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   Author: Urs
   Format: MarkdownItexYes. 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.

   Yes. Infinitesimally 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.

4. • CommentRowNumber4.
   • CommentAuthorMichael_Bachtold
   • CommentTimeMay 13th 2015
   • (edited May 13th 2015)
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   Author: Michael_Bachtold
   Format: MarkdownItexThanks, 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, I didn’t read carefully. I don’t know about , 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 is the Cartan distribution (also called higher order contact structure), which may be seen as a flat Ehresmann connection on the bundle . It consists of horizontal vector fields on , i.e. those vector fields tangent to the image of the universal differential operator from sections of to sections of . 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 . Denote the module of horizontal fields with and denote with the module of linear differential operators generated by . Then the module of vertical vector fields on (which is canonically isomorphic to vector fields on modulo horizontal ones: ) is naturally a left module. This can be thought of as analogous to a D-module structure or flat linear connection on , 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 :

   Where 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.)

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 13th 2015
   • (edited May 13th 2015)
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   Author: Urs
   Format: MarkdownItexThanks! 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.

   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 to sections of is the one whose -adjunct is given by pre-composing a section with (the image under of) the image

   of the counit of the -adjunction for base change along . So possibly it is this structure morphism from which we are to construct the desired constraint.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMay 13th 2015
   • (edited May 13th 2015)
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   Author: Urs
   Format: MarkdownItexI 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 $$ \left( \array{ && Jet( F) \\ & {}^{\mathllap{}}\swarrow &\swArrow& \searrow^{\mathrlap{}} \\ Jet(Jet(F)) && \stackrel{Jet(\phi)}{\longrightarrow} && Jet(Jet(F)) \\ & {}_{\mathllap{Jet \mathbf{L}}}\searrow &\swArrow_{\mathrlap{Jet(\eta)}}& \swarrow_{ \mathrlap{Jet\mathbf{L}}} \\ && Jet \mathbf{B}^{p+1}U(1)_{conn} } \right) \;\;\;\; \simeq \;\;\;\; \left( \array{ && Jet(F) \\ & {}^{\mathllap{}}\swarrow && \searrow^{\mathrlap{}} \\ Jet(Jet(F)) && = && Jet(Jet(F)) \\ & {}_{\mathllap{Jet \mathbf{L}}}\searrow && \swarrow_{\mathrlap{Jet \mathbf{L}}} \\ && Jet \mathbf{B}^{p+1}U(1)_{conn} } \right) $$

   I have a suspicion:

   a flow

   being evolutionary is to mean that it preserves the Cartan distribution, and by what I just said in #5, but dualizing back through the -adjunction, I am guessing now that this means that its image under stabilizes the universal differential operator, i.e. that we have

   If that is true, then my second order conjecture is that the horizontal are those killed by this precomposition, i.e. that for a quantomorphism

   to be admissible, we need

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMay 13th 2015
   • (edited May 13th 2015)
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   Author: Urs
   Format: MarkdownItexah, 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...

   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 of the universal differential operator.

   For the second suggestion in #6 though, it’s the wrong way around: I am supposed to say that 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…

8. • CommentRowNumber8.
   • CommentAuthorigor
   • CommentTimeMay 13th 2015
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   Author: igor
   Format: MarkdownItexI 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"?

   I just want to make an elementary remark, which may or may not be useful towards a successful formalization. Let be the classifying stack of -forms. Then, in my understanding of the notation, is the space of -forms on , usually denoted by . Let be a product bundle (over ). I take that simple case just so I can write the space of sections of as . Consider those maps that are differential operators. The collection of all these differential operators is in bijection with horizontal forms on . Now, any such becomes a bundle map . Now, pulling back the bundle along , it is easy to identify the pulled back bundle with a sub-bundle of . Thus, horizontal forms can be identified with a subset of forms on , which with the help of the classifying stack is .

   So, starting with , we end up with an element of . Is this not enough to come up with a decent “formalization”?

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeMay 13th 2015
   • (edited May 13th 2015)
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   Author: Urs
   Format: MarkdownItexHi 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](http://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos#StructureSheaves)_. 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").

   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 over the base . But this is not quite equivalent to differential operators from to the trivial bundle over with fiber the sheaf .

   The issue here, together with its solution, is discussed a bit at differential cohesion – structure sheaves.

   The solution for the case of bare -forms is to apply the etalification coreflection from prop. 8 there, i.e.

   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 -forms to -form connections .

   Hm, so we could consider

   and then maps into that from

   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”).

10. • CommentRowNumber10.
    • CommentAuthorigor
    • CommentTimeMay 13th 2015
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    Author: igor
    Format: MarkdownItexI'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 $J^\infty F$.

    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 must enter into the definition of horizontal (or for that matter vertical) forms on .

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2015
    • (edited May 14th 2015)
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    Author: Urs
    Format: MarkdownItexBehind 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](http://ncatlab.org/nlab/show/jet%20comonad#Marvan86), this Eilenberg-Moore category is the category $PDE(X)$ of partial differential equations with variables in $X$.(!)) Moreover, by [this proposition](http://ncatlab.org/nlab/show/topos+of+algebras+over+a+monad#ToposProperty) 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](http://nforum.mathforge.org/discussion/6592/is-the-emtopos-of-the-jetcomonad-cohesive-if-the-base-is/?Focus=53367#Comment_53367).

    Behind the scenes, Igor kindly points out to me that with horizontal -forms understood as bundle maps , hence as differential operators , then the correct horizontal differential is just the composition with the de Rham differential as differential operators, i.e. the composite

    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 -coalgebras of the slice of our differential cohesive topos . (By the result of Marvan 86, this Eilenberg-Moore category is the category of partial differential equations with variables in .(!))

    Moreover, by this proposition and using that the jet comonad is a right adjoint, is itself a topos over .(!)

    Now the big question: is suitably cohesive if is??

    If it is, then we are in business: then we simply formulate the differential coefficients not in but in .

    This question maybe deserves a separate thread here.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2015
    • (edited May 14th 2015)
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    Author: Urs
    Format: MarkdownItexAh, 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](http://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos#StructureSheaves) 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 $\iota$.

    Ah, we don’t need that is cohesive. We simply send all our ingredients along

    where the second map is etalification and the third is the free Jet-coalgebra functor (the inverse image of the geometric morphism by which sits over ).

    Then notably is our coefficient.

    Now for our field bundle, then a horizontal Lagrangian is nothing but a map

    in , i.e. is just the naive thing, but done not internal to but internal to .

    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 , constrained such that the automorphisms of come from under .

13. • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 16th 2015
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    Author: David_Corfield
    Format: MarkdownItexSo 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](http://www.staff.science.uu.nl/~igoni101/preprints/) 'Analogues of coverings and the fundamental group for the category of partial differential equations'. That [Diffiety school](http://school.diffiety.org/) sounds like it's aiming to be quite a movement.

    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.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2015
    • (edited May 16th 2015)
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    Author: Urs
    Format: MarkdownItexYes. 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.

    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.

15. • CommentRowNumber15.
    • CommentAuthorRodMcGuire
    • CommentTimeMay 16th 2015
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    Author: RodMcGuire
    Format: MarkdownItex> That [Diffiety school](http://school.diffiety.org/) sounds like it's aiming to be quite a movement. ?? As an organization it seems to have died in 2010. The [Diffity Institute](http://diffiety.ac.ru/) says: > Since July 15, 2010 the site is frozen. Updates are possible for [Sophus Lie e-library](http://diffiety.ac.ru/djvu/index.htm) only. > For Diffiety schools and Current Geometry conferences see [The Levi-Civita Institute](http://www.levi-civita.org/). > For Moscow seminar see [site on the Geometry of Differential Equations](http://gdeq.org).

    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.

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2015
    • (edited May 16th 2015)
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    Author: Urs
    Format: MarkdownItexLuckily 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.

    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.

17. • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 17th 2015
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    Author: David_Corfield
    Format: MarkdownItexSo do you reckon you can take their constructions and reinterpret them in differential cohesive terms, e.g., their [secondary calculus](http://www.impan.pl/~jagrab/vitagliano.pdf)? By the way, I see that Gabriele Vezzosi who works with Toën once [worked](http://www.dma.unifi.it/~vezzosi/papers/) with Vinogradov.

    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.

18. • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2015
    • (edited May 18th 2015)
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    Author: Urs
    Format: MarkdownItexDavid, 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](http://ncatlab.org/nlab/show/variational+bicomplex#MoreOntheHorizontalComplex)_ (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 * [Variational bicomplex -- Elementary formalization in differential cohesion](http://ncatlab.org/nlab/show/variational+bicomplex#FormalizationInDifferentialCohesion).

    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

    • Variational bicomplex – Elementary formalization in differential cohesion.
19. • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 18th 2015
    • (edited May 18th 2015)
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    Author: David_Corfield
    Format: MarkdownItexWhat do '$\kappa$ generically' and '$\iota$ generically mean?

    What do ’ generically’ and ’ generically mean?

20. • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2015
    • (edited May 18th 2015)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThe full sentence (beginning before the bullet list [here](http://ncatlab.org/nlab/show/variational%20bicomplex#FormalizationInDifferentialCohesion)) 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.)

    The full sentence (beginning before the bullet list here) is “we write 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 ” and depending on how far away from we start, they apply the necessary re-identifications, as indicated. The key is that applies étalification, which amounts to the horizontality constraint, while doesn’t. So under the differentials are purely horizontal, while under 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.)

21. • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2015
    • PermaLink
    Author: Urs
    Format: MarkdownItexLet's come back to the evolutionary vertical vector fields. According to, for instance, ["Homological Methods in Equations of Mathematical Physics"](http://arxiv.org/abs/math/9808130), 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.

    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 inside some jet bundle 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 in the category of -coalgebras.

22. • CommentRowNumber22.
    • CommentAuthorMichael_Bachtold
    • CommentTimeMay 19th 2015
    • (edited May 19th 2015)
    • PermaLink
    Author: Michael_Bachtold
    Format: MarkdownItexRe #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)

    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)

23. • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2015
    • (edited May 19th 2015)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks 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](https://dl.dropboxusercontent.com/u/12630719/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 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 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 is a topos over , hence over , it is canonically enriched in . For a differential equation, then its -valued automorphism group is the direct image of the PDE-internal automorphism group in , let’s write 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 -stacks are equivalent to -algebras.

    The infinitesimal group -stack of a group -stack may be axiomatized in differential cohesion as the homotopy fiber of the infinitesimal shape unit . In the standard model of formal smooth -groupoids this produces excactly an infinitesimal -stack as considered by Pridham and Lurie (namely an -stack on formal duals of local Artin algebras, aka “Weil algebra” in SDG jargon).

24. • CommentRowNumber24.
    • CommentAuthorMichael_Bachtold
    • CommentTimeMay 19th 2015
    • PermaLink
    Author: Michael_Bachtold
    Format: MarkdownItexThanks 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.

    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 ?

    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.

25. • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2015
    • (edited May 19th 2015)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> 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.

    evolutionary vector fields should correspond to the first order infinitesimal neighbourhood of the identity in ?

    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.

26. • CommentRowNumber26.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 19th 2015
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexI see Igonin in [Notes on symmetries of PDEs and Poisson structures](http://www.staff.science.uu.nl/~igoni101/preprints/sym-poisson.pdf) 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](http://www.staff.science.uu.nl/~igoni101/preprints/cfg.pdf), this stays at the ordinary Lie group level.

    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.

27. • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2015
    • PermaLink
    Author: Urs
    Format: MarkdownItexYes, 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](http://arxiv.org/abs/math/9808130). To find what "we" consider as higher symmetries one has to go to the literature on BV-BRST complexes.

    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.

28. • CommentRowNumber28.
    • CommentAuthorMichael_Bachtold
    • CommentTimeMay 19th 2015
    • PermaLink
    Author: Michael_Bachtold
    Format: MarkdownItex>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.

    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.

29. • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2015
    • (edited Sep 10th 2015)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI 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.

    I am stuck with the following technical thing:

    There is a coverage (Grothendieck pretopology) on the category

    of PDEs with free variables ranging in some manifold , induced by its full embedding into the topos

    (the Cahiers topos sliced over the infinitesimal shape of ).

    What I’d like to have is that a covering of bundles over 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.

30. • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2015
    • PermaLink
    Author: Urs
    Format: MarkdownItexAh, 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.

    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.

31. • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2015
    • (edited Sep 16th 2015)
    • PermaLink
    Author: Urs
    Format: MarkdownItexNow 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.

    Now I am trying to get the following, or to see if it works in the first place.

    Let again be the Cahiers topos, its infinitesimal shape modality, , and the induced comonad of base change along the unit of at . I write for its category of coalgebras, and since is epi, so by comonadic descent this is equivalently .

    This restricts to the traditional jet comonad on , where its coalgebras are PDEs with free variables ranging in : . For this statement to make sense, I must mean some type of infinite-dimensional smooth manifolds when writing . 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 just a tad by taking it to be the category of coalgebras of acting on , i.e. to allow all (infinite dimensional) manifolds and bundles to have infinitesimal directions.

    Now, since is also a site of definition for the slice of the Cahiers topos over , it is natural to wonder if there is a site structure on this , such that we have the following equivalence on the right, making the square commute:

    Does this work? I am thinking to go about it as follows.

    Using that , it follows that this topos has a site of definition given by the comma category with coverage induced from the canonical one on by forgetting the maps to .

    Now take to be equipped with the coverage whose covers are those reflected by the functor . This is indeed a coverage, because on preserves pullbacks, being the restriction of the right adjoint .

    With this I’d think that is a dense subsite of . First of all it is indeed a subcategory: given a morphism out of a formal smooth manifold, then also its image under is a formal smooth manifold mapping into , hence under the object is in the inclusion of , 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 inside which 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.