Pointer regarding jet bundles of $\mathbb{Z}$-graded manifolds and their vector bundles:

- Jan Vysoky.
*Graded Jet Geometry*. (2023). (arXiv:2311.15754)

added this pointer:

- Jean-François Pommaret, Chapter II in:
*Partial Differential Equations and Group Theory*, Springer (1994) [doi:10.1007/978-94-017-2539-2&rback;

(here and in related entries)

]]>I changed the action in the $E$ component so that it corresponds to the one on $E$.

Mauro Mantegazza

]]>Added a small section about the role of the first jet bundle in the Atiyah exact sequence (which also means I should probably get around to creating an entry for the Atiyah class and its role in Chern-Weil theory).

]]>Thanks for the prodding! Have fixed it now, finally.

There is now also a pdf-version (here), kindly produced by the FOMUS organizers from the nLab source

]]>Wasn’t I right in #53 about the ordering of $\Im_n$? As $n$ gets larger, points further away are being identified. So it’s still wrong at Modern Physics formalized in Modal Homotopy Type Theory?

]]>Right, Dennis kept telling me about this when we were close by a while back. It always sounded intriguing and like the right idea. Unfortunately I haven’t found the time to look into it yet.

]]>I’ve only had a glance, but maybe if we return to the germ/jet issue, there may be something in

- Dennis Borisov, Kobi Kremnizer,
*Beyond perturbation 1: de Rham spaces*, ( arXiv:1701.06278 )

which uses ∞-nilpotent elements due to Moerdijk and Reyes instead of ordinary nilpotent elements.

]]>Something completely different happens when we apply de Rham space formalism to contracting ∞-nilpotent neighbourhoods. Instead of quotients by actions of formal neighbourhoods of diagonals we get quotients by actions of germs of diagonals… Since in differential geometry there are many more orders of vanishing than just the finite ones, the infinitesimal theory we get here is much richer.

Good questions, somebody should look into these!

I try to find time to fix that typo with the sequence of modalities. Thanks for alerting me!

]]>Ok so a puzzle for me. What is

$\mathbf{H}_{/X} \stackrel{\overset{i^\ast}{\longleftarrow}}{\underset{i_\ast}{\longrightarrow}} \mathbf{H}_{/\Im_{germ}(X)} \,?$Can we call it the ’germ comonad’? Presumably coalgebras are locally integrable PDEs.

It seems that in some situations $\Im_{germ}$ and $\Im$ coincide (complex and real analytic, some parts of algebraic geometry). Is it anywhere other than real functions, where Taylor expansions needn’t converge, that we see this distinction?

And would that part of the Aufhebung story of the rise of (co)monads need to be told differently. There it just says

Continuing the process, we posit a furrther opposition of moments lifting the previous ones,

and notes that Aufhebung is automatic. So does $\Im_{germ}$ have a right adjoint, something like ’locally étale’? The expression does exist.

By the way, just after there is

$X \to \cdots \to \Im_{(3)}X \to \Im_{(2)}X \to \Im_{(1)}X \to \Im X \to ʃ X \,.$which as I suggested elsewhere is wrong. Higher order infinitesimal neighbourhoods are larger.

]]>Yes, that’s right, there is the infinitesimal neighbourhoods to any finite order, then the formal neighbourhood, being of unbounded infinitesimal order, and then the germ, in increasing order.

My “infinitesimal germ” was meant for “formal neighbourhood”, if one wants to keep the word “germ”, but we don’t have to do that.

]]>Did your addition clarify the situation?

Am I right in thinking there’s

the spaces corresponding to nilpotent infinitesimals, right up to the union of all of these, for which jet spaces are relevant.

the larger space supporting germs.

Both occur at infinitesimal interval.

Confusion is possible, because of our using ’infinitesimal germ’ in (1) (e.g., at jet bundle), and using ’infinitesimal neighbourhood in (2), (e.g., at infinitesimal object).

What I was worrying about in #49, was that the shift from the previous sections on (1) to this section on (2) wasn’t clear. Now I think it’s mixing up (1) and (2).

Bunge and Marta speak of the space of kind (2) about $0$ in the real line as an ’infinitesimal object’ (but including more than nilpotent infinitesimals) as the intersection of all opens containing $0$.

]]>Is that confusing at infinitesimal object, if it speaks of the ’infinitesimal neighbourhood’ as supporting the germ, beyond the jet?

Yes, that was wrong. I have made it read “infinitesimal germ”, but of course it would be good to expand on this…

]]>Since we’re thinking about germs in relation to jets at the moment, perhaps the opening at jet bundle needs a word. Since ’infinitesimal’ and ’germ’ weren’t linked in

A jet can be thought of as the infinitesimal germ of a section of some bundle,

I put in such links. But maybe we should say what work ’infinitesimal’ is doing in the combined term.

Is that confusing at infinitesimal object, if it speaks of the ’infinitesimal neighbourhood’ as supporting the germ, beyond the jet?

]]>44: the contents, preface and the bibliography of the book are on the researchgate, one small file with many hyperlinks, 21 pages, https://www.researchgate.net/publication/286776030_Geometry_in_a_Frechet_Context_A_Projective_Limit_Approach

]]>Thanks for the sanity check; and thanks for the pointer to that MO comment! For the moment I have added that here. (No time for more right now.)

]]>That’s all there is, right, there is no subtlety hidden here?

Correct. At the time I think I was still processing and so hedging my bets, but that’s the argument my subconscious was trying to supply.

In fact this kind of argument also works for ILB spaces, or I suppose ILB manifolds more generally, as is treated in the book you mention in #44 (in fact it seems that every Fréchet space can be written in *some* way as an inverse limit of Banach spaces, see http://math.stackexchange.com/a/53020/3835). I was disappointed to find that despite my library generally having access to LMS lecture notes, we don’t (yet) have access to that volume. I’m inquiring to see if we can buy it, given the amount of infinite-dimensional geometry we do in Adelaide.

Hi David,

just to be sure, allow me to get back to this paragraph of yours above:

As far as I can tell, the Fréchet topology on the sequential limit $\mathbb{R}^{\leftarrow \infty}$ (so denoted to distinguish it from the colimit) of the $\mathbb{R}^n$’s (with arrows $\mathbb{R}^n \to \mathbb{R}^m$ if $n \geq m$) is induced as for an ILH space. One takes the limit in the category of topological vector spaces to be the coarsest such that the projections are continuous, but I believe this is induced by the countable sequence of seminorms $||-||_n \colon \mathbb{R}^{\leftarrow \infty} \xrightarrow{pr_n} \mathbb{R}^n \xrightarrow{||-||} \mathbb{R}$.

Since you say “believe”, let me make this explicit to see if there is any pitfall hiding here:

So the topology induced by the seminorms is such that for every point $x$ then the open balls

$B_\epsilon^n(x) \coloneqq \left\{ y \;|\; \Vert y-x \Vert_n \lt \epsilon \right\}$form a base of neighbourhoods of $x$. For fixed $n$ of course these open balls are the preimages of a base of neighbourhoods of $\mathbb{R}^n$, hence they form the coarsest base of neighborhoods that makes all the $pr_{n}$ continuous. Hence also the topology induced by that base of neighbourhoods is the coarsest one that makes all the $pr_n$ be continuous, hence is the projective limit topology.

That’s all there is, right, there is no subtlety hidden here?

]]>I am starting to make some notes on this at *Frechet manifold – Projective limits of finite-dimensional manifolds*. Please feel invited to expand further!

There is this new book

- C. T. J. Dodson, George Galanis, Efstathios Vassiliou,
*Geometry in a Fréchet Context: A Projective Limit Approach*, Cambridge University Press (2015)

but I have not seen much of its inside yet (GoogleBooks is shy about showing it).

]]>Thanks, David!

So we need to show that given a compatible sequence of smooth maps $g_n\colon \mathbb{R}^k \to \mathbb{R}^n$ the resulting continuous map $g\colon \mathbb{R}^k \to \mathbb{R}^{\leftarrow\infty}$ is smooth. But this is just showing that all the partial derivatives exist and are continuous, and I believe that this reduces to showing that the partial derivatives of $g$ are the induced maps from all the partial derivatives of the $g_n$ (and note that the transition maps $\mathbb{R}^n\to \mathbb{R}^m$ have rather trivial partial derivatives so that the chain rule doesn’t do anything strange), and then continuity is assured.

Yes, this is what I found spelled out in Saunders’s “The geometry of jet bundles”, chapter 7. There it is Lemma 7.1.8.

]]>As far as I can tell, the Fréchet topology on the sequential limit $\mathbb{R}^{\leftarrow \infty}$ (so denoted to distinguish it from the colimit) of the $\mathbb{R}^n$’s (with arrows $\mathbb{R}^n \to \mathbb{R}^m$ if $n \geq m$) is induced as for an ILH space. One takes the limit in the category of topological vector spaces to be the coarsest such that the projections are continuous, but I believe this is induced by the countable sequence of seminorms $||-||_n \colon \mathbb{R}^{\leftarrow \infty} \xrightarrow{pr_n} \mathbb{R}^n \xrightarrow{||-||} \mathbb{R}$. In particular, given a Fréchet manifold $M$ and compatible maps $f_n\colon M \to \mathbb{R}^n$ forming a cone, we get a *continuous* map $f\colon M \to \mathbb{R}^{\leftarrow\infty}$. So there can be *at most one* map of the kind required by the universal property of the limit in the category of Fréchet spaces.

To check this map is smooth, we can use the fact Fréchet manifolds form a full subcategory of diffeological spaces, and so check what happens on plots. Thus for a smooth map $p\colon \mathbb{R}^k \to M$, we need to check that $f p\colon \mathbb{R}^k \to \mathbb{R}^{\leftarrow \infty}$ is smooth. But such a map is precisely determined by its compositions with the projections, so we might as well assume wlog that our Fréchet manifold is in fact $\mathbb{R}^k$.

So we need to show that given a compatible sequence of smooth maps $g_n\colon \mathbb{R}^k \to \mathbb{R}^n$ the resulting continuous map $g\colon \mathbb{R}^k \to \mathbb{R}^{\leftarrow\infty}$ is smooth. But this is just showing that all the partial derivatives exist and are continuous, and I believe that this reduces to showing that the partial derivatives of $g$ are the induced maps from all the partial derivatives of the $g_n$ (and note that the transition maps $\mathbb{R}^n\to \mathbb{R}^m$ have rather trivial partial derivatives so that the chain rule doesn’t do anything strange), and then continuity is assured.

(There is a nice collision here between the two senses of the word ’limit’: we want to take the limit of functions to a limit, and so take the limit of the functions in the cone :-)

All in all, I think the argument works out.

Something like this should also be true for sequential ILH manifolds, in all likelihood, say for those where the limit is over a diagram of submersions.

]]>I still have a question, on the relation between the following three levels of finite order-ness.

Let $X = \lim_k X_k$ be a sequential projective limit of manifolds (formed in some bigger ambient category). For a function

$f \;\colon\; X \to \mathbb{R}$say that

$f$ is

**globally of finite order**if it comes from a function $f_k \colon X_k \longrightarrow \mathbb{R}$ for some $k \in \mathbb{N}$;$f$ is

**locally of finite order**if for every point of $X$ there is an open subset $U_k$ around its projection in $X_k$, for some $k$, such that restricted to the pre-image of $U_k$ in $X$, $f$ comes from a function $f_{U,k} \colon U_k \longrightarrow \mathbb{R}$;$f$ is

**formally of finite order**if at each point of $X$ the partial derivatives of $f$ are non-vanishing only along a finite-dimensional subspace of the tangent space at that point.

**Question:** How are points 2 and 3 related? Are they equivalent?

Ah, never mind, it seems that chapter 7 of Saunders’ *The geometry of jet bundles* has everything I need.

Various authors discuss the realization of infinite jet bundles $J^\infty E$ as Frechet manifolds (here). It’s the projective limit $\underset{\longleftarrow}{\lim}_k J^k E$ of the underlying sets, or even the projective limit of the underlying Banach spaces, and then equipped with Frechet manifold structure.

Takens 79 puts it succinctly like so (p. 3):

The underlying set is the projective limit $J^\infty E := \underset{\longleftarrow}{\lim}_k J^k E$ of underlying sets, and the smooth structure is determined by requiring that a function $J^\infty E \to \mathbb{R}$ is smooth if for each point $s \in J^\infty E$ there exists $k \in \mathbb{N}$ and a neighbourhood $U_k$ of the component of $s$ in $J^k E$ and a smooth function $f_k \colon U_k \to \mathbb{R}$ such that $f$ restricted to the preimage of $U_k$ in $J^\infty E$ is given by $f_k$.

**Question**:

Does this definition make $J^\infty E$ also be the projective limit $\underset{\longleftarrow}{\lim}_k J^k E$ *formed in Frechet manifolds*?

Hence:

For $X$ a finite dimensional smooth manifold, regarded as a Frechet manifold, is it true that

$Hom_{FrechetMfd}(X, J^\infty E) \simeq \underset{\longleftarrow}{\lim}_k Hom_{SmthMfd}(X, J^k E)$?

]]>