I don’t know why he calls them monads though.

I believe this is to refer to something like Leibniz’s monads, but not, in any case, to monads in the sense of category theory. There is a lot in the *Science of Logic* about the issue of Leibniz’s monads and infinitesimal units, but I’ll spare us the details.

Yes, I have just added this to *infinitesimal disk bundle*.

Re #15-18: Thanks David and Urs! That helped. Although I still have some basic “obstructions” in my understanding of cohesive types, which I might ask in a separate thread.

Concerning #25: it seems that infinitesimal disk bundle are basically the same as what Kock calls the “bundle of k-monads” in Synthetic Geometry of Manifolds p. 39. (I don’t know why he calls them monads though.)

]]>chasing references, I found Kock 80 which has not the comonad structure on $Jet$, but does have the statement that $Jet(-)$ is right adjoint to forming infinitesimal disk bundles (and I have created this entry now).

]]>Marvan’s name appears 18 times in that paper I mentioned in #20.

]]>for the moment the link is here: Marvan 86, a nice article.

(my understanding at the moment is that it is okay to link to the pdf)

]]>I have contacted Michal Marvan, and he has sent me a scan of his original article (I am in the process of checking if I may share it here). In there he in fact proves that the Eilenberg-Moore category of coalgebras over the jet comonad is equivalently the category of differential equations with variables in the given base manifold.

]]>Regarding your first paragraph: right, so by adjunction, the sections of $i^\ast i_\ast E$ over some base $X$ are bundles maps $i^\ast i_! X \to E$, hence are certain “generalized sections” of the original $E$.

When $i \colon X \to \int X$ is the shape unit, then $i^\ast i_! X = X \times_{\int X} X$ is a kind of bundle over $X$ whose fibers are the homotopy types of smooth collections of based path spaces of paths emanating at that point and smoothly parameterized by their endpoint. For whatever that’s worth. That tells one in principle what the sections of any $i^\ast i_\ast E$ are in this case, though right now I don’t recognize it as anything. But possibly this is some important concept and we should think about it.

Regarding the article that you point to: thanks for the reference. Yes, those jet bundles of superbundles which these authors dicuss should be precisely what the Jet comonad in super formal smooth homotopy types produces when applied to these bundles.

]]>Re #18, so it’s a process of taking something bundle like and returning another bundle where the fibre at a point is now sections through fibres of ’adjoining’ points. So for necessity, a necessary man at this world picks out a man under variation across worlds, such as ’the tallest man’. So they’re like germs. With shape, you’d have at a point all sections for the component of the point?

For $Rh$ this would send a fibre of entities to a fibre of entities varying over points which only differ in the super directions. Could they be called super-jets?

I see talking about an ’infinite jet superbundle’, p. 17 of On the (non)removability of spectral parameters in Z2-graded zero-curvature representations and its applications. But that’s just over a real manifold. I guess we’d want it over a superspace.

]]>Michael, re #13,

I believe I see how to prove that the abstractly defined Jet comonad has the coproduct operation that it is supposed to have.

There is now a remark 5.3.88 in the dcct pdf which indicates how, dually, the product of the left adjoint monad works. I think from this it is pretty clear how it goes, but of course this remark is not yet a proof of anything.

]]>David, re #12,

that’s a neat analogy that you are observing there. I’ll think about this.

]]>Michael, David,

thanks for catching typos! Have fixed them now.

Regarding the path inclusion: think of $\Pi_{inf}X \coloneqq \Im(X)$ as the result of forming a new groupoid from the groupoid $X$ by adding in further isomorphisms between all objects (points of $X$) that are infinitesimal neighbours. These isomorphisms are hence infinitesimal paths. The canonical map $X \to \Pi_{inf}(X)$ sends all the existing objects and morphsims of $X$ “to themselves” in $\Pi_{inf}X$ and doesn’t hit any of these new non-trivial isomorphisms between infinitesimal neighbours.

This is a standard story from algebraic geometry. I think this is where the word “crystal” in “crystalline cohomology” comes from, in that one visualizes $\Pi_{inf}X$ as looking like a crystalline version of $X$, where all these new infinitesimal paths are little edges of the crystal.

I don’t know for sure if this is the historical visualization behind Grothendieck’s “crystalline”, but certainly texts on crystalline cohomology draw these kinds of pictures.

]]>Re #14,

Shouldn’t it read: “collapsing infinitesimal paths to constant paths”

I think Urs would point out that identifying infinitesimally close points does not mean collapsing them as in a strict quotient. As with the shape modality, identifying path connected points in a space, one retains the information of different identifications, and identifications between identifications, etc.

]]>Michael I was going to point to the page infinitesimal path ∞-groupoid, but there needs to be some fixing of typos first.

It says

$(Red \dashv \Pi_{inf}) \colon \mathbf{H}_{th} \stackrel{\overset{i_*}{\leftarrow}}{\underset{i^*}{\to}} \mathbf{H} \stackrel{\overset{i_!}{\leftarrow}}{\underset{i_*}{\to}} \mathbf{H} \,.$and should be

$(Red \dashv \Pi_{inf}) \colon \mathbf{H}_{th} \stackrel{\overset{i_!}{\leftarrow}}{\underset{i^*}{\to}} \mathbf{H} \stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\to}} \mathbf{H}_{th} \, ,$I think.

Note that $\Pi_{inf}$ is now written $\Im$, and $Red$ is written $\Re$.

]]>Oh, I see that my interpretation is basically written in de Rham space, under Properties, As a quotient. But then the comment “inclusion of constant paths into all infinitesimal paths” below $i \colon X \longrightarrow \Im X$ in jet bundle is still somewhat confusing for me. Shouldn’t it read: “collapsing infinitesimal paths to constant paths”?

]]>Re 7&8: Yes, thanks! I woke up noticing that in your first post you wrote $Jet(Jet(E_2))$ instead of $Jet(Jet(E_1))$, thats a typo right?

Thanks for the references of Marvan, I had not seen them.

Strictly speaking, what is still missing is a precise proof that the co-product c∞,∞ used there is really the coproduct of the Jet comonad as defined more abstractly.

I see your point. I first need to catch up with your writings on differential cohesion etc. to say more. I keep wanting to do that. Maybe you could help me gain some intuition starting with

$i \colon X \longrightarrow \Im X$For concreteness, suppose $X$ is the affine line $R$ from SDG, can I think of $\Im R$ as a quotient of $R$, where infinitesimally close points have been identified? Although that does not make sense with what is written in jet bundle (“inclusion of constant paths into all infinitesimal paths”).

]]>So we now have two similar constructions

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

$\Box_W: \mathbf{H}_{/W} \to \mathbf{H}_{/W} \,.$Both arise as dependent product then context extension for the units of modalities, $\ast$ and $\Im$, in the Aufhebung table.

Would it be worth looking at this for the other modalities, so $\sharp$, ʃ, $R$ and $\rightrightarrows$? Then also possibilty, reader, writer, randomness.

Then what about these for the counits of comodalities, e.g., along $\Re X \to X$?

]]>in Marvan 89 at least the statement about the Jet operation itself being a comonad is made in passing. It seems that Marvan says that he introduced the terminology “jet comonad”.

]]>the statement is explicit in Marvan 93, section 1.1

]]>I have turned that into a statement in the entry on *differential operator*, see *this defnition/proposition* with consecutive proof.

Let me know what you think.

Strictly speaking, what is still missing is a precise proof that the co-product $c^{\infty,\infty}$ used there is really the coproduct of the Jet comonad as defined more abstractly.

]]>Here the key point, for the purpose of #1, is the middle rectangle (in #7). Regarding the full middle rectangle identifies the bottom middle morphism as $\psi_\infty^{\Delta_1}$, while factoring the middle rectangle, as shown, through the $J^\infty$-image of the square defining $\psi^{\Delta_1}$ identifies the bottom middle morphism as $J^\infty(\psi^{\Delta_1})$.

]]>You mean this commuting diagram here, I suppose:

$\array{ P_1 &\stackrel{id}{\longrightarrow}& P_1 &\stackrel{\Delta_1}{\longrightarrow}& P_2 &\stackrel{\Delta_2}{\longrightarrow}& P_3 \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{j_\infty}} && \downarrow^{\mathrlap{j_\infty}} && \downarrow^{\mathrlap{id}} \\ P_1 &\stackrel{j_\infty}{\longrightarrow}& J^\infty(P_1) &\stackrel{J^\infty(\Delta_1)}{\longrightarrow}& J^\infty(P_2) \\ \downarrow^{\mathrlap{j_\infty}} && \downarrow^{\mathrlap{j_\infty}} && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}} \\ J^\infty (P_1) &\stackrel{c^{\infty,\infty}}{\longrightarrow}& J^\infty(J^\infty(P_1)) &\underoverset{= J^\infty (\psi^{\Delta_1})}{\psi^{\Delta_1}_\infty}{\longrightarrow}& J^\infty(P_2) &\stackrel{\psi^{\Delta_2}}{\longrightarrow}& P_3 }$ ]]>I don’t see that they state it explicitly.

But it also seems to me, that it follows from what they say, by writing out a commutative diagram (which I don’t know how to display here).

]]>Thanks again!

Now, do they ever say explicitly that the “Jet-associated” (as they call it) map corresponding to the composite of two differential operators is given by (in whatever notation) the formula I gave at the bottom of #1 above?

I think it follows from what they say, but for citation purposes I wonder if they (or anyone else) says it explicitly.

]]>