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stub for jet bundle
There seems to be a bit of a contradiction between what is said at jet bundle and at jet space about the meaning of “jet bundle”. At jet bundle it is said that a jet bundle consists of sections of a given map $p:P\to X$, with the fiber over $x\in X$ being the jets of germs of sections defined in neighborhoods of $x$; so that in particular, the jet bundle is a bundle over the domain of the jets. But at jet space it is said that jet spaces and bundles generalize tangent spaces and bundles by replacing order-1 equivalence of curves by order-k equivalence of maps with an arbitrary domain, which would suggest that a “jet bundle” would be, like a tangent bundle, a bundle over the codomain of the jets.
Observe that a jet bundle is also a bundle over the codomain $P$ of the sections of $p$. The page on jet space does not give a precise definition, but sometimes I have heard the terminology “jet space” used for the case of jets of maps from some domain $X$ to a codomain $Y$, which can be considered a special case of the jet bundle of the trivial (non dependent) bundle $p: X\times Y \to X$.
By the way, I think your observation (from the cojet differential forms thread), that the infinite order jets are the cofree T-coalgebra, generalizes straightforwardly to jet bundles. One only needs to replace the tangent functor with the functor $J^1$. It might be nice to add that to the nlab page. Here is a suggested stub, feel free to expand or modify to fit the nlab style:
The infinite order jet bundle as the cofree $J^1$-coalgebra: Let $J^1$ be the functor on bundles over $X$ sending a bundle $p:P\to X$ to the bundle $J^1 p: J^1 P \to X$ of first order jets of sections of $p$. This functor is co-pointed: the natural projection $J^1 P \to P$ forgets first order information. A $J^1$-coalgebra is then by definition a bundle $p:P\to X$ with a bundle map $P\to J^1 P$, which is the same as a connection on $P$. Whit this, the infinite order jet bundle of sections of $p$ may be defined as the cofree $J^1$-coalgebra on $P$. By definition it comes with a bundle map $p_\infty: J^\infty P \to P$ (“forgetting all higher order information”) and a universal connection $C: J^\infty P \to J^1( J^\infty P)$ sometimes called the Cartan connection or infinite order contact structure on $J^\infty P$. The universal property of this cofree $J^1$-coalgebra states that for any bundle $q:Q\to X$ with connection $K: Q \to J^1 Q$ and bundle map $\varphi: Q\to P$, there exists a unique prolongation of $\varphi$ denoted with $j^\infty \varphi: Q \to J^\infty P$ commuting with all given maps: $p_\infty \circ j^\infty \varphi =\varphi$ and $J^1(j^\infty \varphi)\circ K= C \circ j^\infty \varphi$.
In the particular case when $q:Q\to X$ is the identity $X\to X$ with its canonical connection, then a morphisms of bundles $\varphi$ from $q$ to $p$ is the same as a section of $p$ and $j^\infty \varphi$ is what is traditionally called the infinite jet of the section $\varphi$.
re #2 (since it’s likely that I wrote this): I entirely agree that this is stated suboptimally and would deserve to be improved.
Do you have time to edit it? I might find time only later…
re #4: you should feel invited to add this to the $n$Lab right away
Is there anything new in #4 for the ’abstract general’ perspective of jet bundle?
While on that page, in
$Jet : \mathbf{H}/X \stackrel{\overset{i^*}{\leftarrow}}{\underset{Jet := i_*}{\to}} \mathbf{H}_{\mathbf{\Pi}_{inf}}(X)$is $\mathbf{H}_{\mathbf{\Pi}_{inf}}(X)$ a notational variant of $\mathbf{H}/\mathbf{\Pi}_{inf}(X)$?
Then one can see the right hand side as a cofree coalgebra of sorts?
Ah, sorry for the notation mismatch, I have fixed it now. Yes, as the text did say, this was meant to denote the base change between the slice $\infty$-toposes.
So $i \colon X \to \mathbf{\Pi}_{inf}(X)$ is meant to denote the unit of the infinitesimal shape modality and $(i_! \dashv i^\ast \dashv i_\ast) \colon \mathbf{H}_{/X}\to \mathbf{H}_{/\mathbf{\Pi}_{inf}(X)}$ is the induced base change (dependent sum $\dashv$ context extension $\dashv$ dependent product).
As implicit in Beilinson-Drinfeld and as amplified more abstractly on p. 6 of Lurie’s “Notes on crystals and D-modules” (pdf) the jet bundle construction is $i_\ast$ if one regards the jet bundle with its crystal structure, and it is the comonad construction $i^\ast i_\ast$ if regarded as a bundle over the original base. So $Jet(P) \simeq i^\ast i_\ast(P)$ and in this sense this is, too, a cofree coalgebra construction.
re #4, that’s also interesting, but it doesn’t seem to me to be actually a generalization of the observation in the other thread, because of the mismatch mentioned in #2. You’re talking about the jet bundle as a bundle over the domain, whereas the jets I was talking about are bundles over the codomain. Or am I confused?
as to how to fix #2, I’m not sure — it depends on how the term “jet bundle” is most often used in the literature. And if we call one of these the “jet bundle” then what is left to call the other one? (Also, should we maybe merge the pages jet bundle and jet space?)
I’m confused. Re #2
…at jet space it is said that jet spaces and bundles generalize tangent spaces and bundles by replacing order-1 equivalence of curves by order-k equivalence of maps with an arbitrary domain, which would suggest that a “jet bundle” would be, like a tangent bundle, a bundle over the codomain of the jets.
But tangent bundle has
The tangent bundle $T X \to X$ of a space $X$ is a bundle over $X$ whose fiber over a point $x \in X$ is a collection of infinitesimal curves in $X$ emanating at $x$
Doesn’t that suggest a bundle over the domain?
re Urs #6: I went ahead and added it, expanding it a bit with the sequential construction discussed in the other thread. I also moved what used to be in the idea section to the concrete definition and added a few lines to the idea section. I hope that is ok. I feel somewhat uncomfortable for adding those things since I can’t relate them yet to the general abstract definition. Feel free to revert any changes.
re Mike #9: concerning the codomain domain question, maybe it’s me who has not understood the issue. I think I can put the jets from the other thread into the setting of jets of bundles: you considered jets of maps from the line $\mathbb{R}$ to some space $X$ at $0\in \mathbb{R}$. If you take the trivial bundle $p: \mathbb{R}\times X\to \mathbb{R}$ this should correspond to the fiber of what I called $J^\infty P \to \mathbb{R}$ over the base point $0 \in \mathbb{R}$. So you are right that it’s strictly not a special case of a jet bundle (unless maybe we replace $\mathbb{R}$ with its infinitesimal nbhd around $0$).
Mike is right, the usual way to speak is to regard jet bundles as bundles over the domain of the maps whose derivatives are considered. I.e. the jet bundle of a bundle $P \to X$ is itself a bundle over $X$. Those paragraphs comparing to the tangent bundle are not well formulated. I can try to have a go at improving it sometime later. But feel invited to have a go yourself (any one of you) if you feel you have the energy. (Spare time, where art thou?)
Doesn’t that suggest a bundle over the domain?
A curve in $X$ is a curve whose codomain is $X$. “Emanating at $x$” is perhaps a bit confusing, but it means we consider only such curves that map a fixed point in their domain (usually $0$) to $x$.
So you are right that it’s strictly not a special case of a jet bundle
Good, I’m not confused then. But I also see your point that the “most general” thing to say is that the jet bundle of a bundle $E\to X$ is a bundle over $E$, since then the “usual” (as Urs says) bundle would be the composite $J E \to E \to X$, while in the case of a trivial bundle $E=X\times Y$ the “other” bundle would be $J(X\times Y)\to X\times Y\to Y$.
I feel somewhat uncomfortable for adding those things since I can’t relate them yet to the general abstract definition.
You should not! For one thing, the onus should rather be on the general abstract definition to be related to the traditional concrete definition. But more importantly, the inability to write more should never get in the way of writing something; the nLab is always a work in progress.
It’s nice to nail down the standard meaning of “jet bundle”, but it would also be nice to have a name for the sort of “jet thingy” in the other thread, i.e. the fiber over $0$ of the jet bundle of the trivial bundle $\mathbb{R}\times X\to \mathbb{R}$, regarded as a bundle over $X$. Is this by any chance what a “jet space” is (or could be)?
Oh, I see.
So there’s really quite a generalization going on from tangent bundle to jet bundle in the sense here: from (1) tangent bundle of manifold to (2) jet bundle of manifold and (3) jet spaces for general maps between manifolds, and then finally to (4) jet bundle of bundle.
As it stands jet bundle seems to cover (3) (in the case of submersions) and (4).
Hmm, couldn’t there also be a jet construction to complete an analogous move from (1) tangent $(\infty, 1)$-category to (2) jet $(\infty, 1)$-category to (3) some jet thing for a mapping between $(\infty, 1)$-categories and then to (4) a jet thing for an $(\infty, 1)$-module bundle?
re Mike 14#: the people I’ve heard talk about jets (Vinogradov and collaborators) used the terms jet bundles and jet spaces almost interchangeably. They might call the total space of the jet bundle $J^k P$ a jet space when the bundle structure was of no relevance. And I would certainly call that jet thingy from the other thread a jet space (or even jet bundle over $X$). But maybe in other parts of mathematics people use the terminology more restrictively. I think in algebraic geometry people mean jets of arcs (maps from some one dimensional thing to some space $X$) when they say jet space, which seems to agree with your suggested use of the term.
re David #15: currently jet bundle is not explicitly covering (3) (jet spaces for maps between manifolds). I’ve added the following below the concrete definition:
In the case when $p$ is a trivial bundle $p:X\times Y \to X$ its sections are canonically in bijection with maps from $X$ to $Y$ and two sections have the same partial derivatives iff the partial derivatives of the corresponding maps from $X$ to $Y$ agree. So in this case the jet space $J^k P$ is the space of jets of maps from $X$ to $Y$ and commonly denoted with $J^k(X,Y)$.
Thanks Michael. Do you have any suggestions for terminology that would refer specifically to the jet thingy from the other thread? The “tangent jet bundle of $X$”? “Arc jet bundle”? “Curve jet bundle”?
I’m not 100% sure if that arc jet bundle I once read about was really the same, and otherwise I had not explicitly encountered that jet thingy before. Might it be reasonable to call it the space of infinitesimal paths (of higher oder) in $X$? It seems to be related to things at infinitesimal interval object.
A note along the lines of #15: there is yet another generalization of jets (at least in the setting of differentiable manifolds): jets of submanifolds of dimension $n$ of some given space $P$. Think of them of infinitesimal germs of submanifolds of $P$. Since (local) sections of some bundle $p:P\to X$ may be regarded as special submanifolds of $P$ (those transversal to the fibers) the jet bundle of $p$ is an (open dense) subset of the space of jets of $\dim X$ dimensional submanifolds of $P$.
These jets of submanifolds seem to be a natural thing to consider for example when talking about the PDE of minimal surfaces in some Riemannian manifold $P$.
A note concerning terminology. One name that already has at least some usage in the literature on jets [1] is higher (order) velocity space or velocities of order $r$. Even more generally, one can consider higher order multi-velocities or $n$-velocities of order $r$. These are $r$-jets of maps of $R^n$ into a manifold $M$, as a bundle over $M$. The case $n=1$, $r=1$ gives the usual tangent vectors and $n>1$, $r=1$ gives Grassmannian elements. Generally, these describe $r$-jets of $n$-dimensional submanifolds in $M$. Formally, take the trivial bundle $\mathbb{R}^n \times M \to \mathbb{R}^n$ and its jet bundle $J^r(\mathbb{R}^n,M) \to \mathbb{R}^n \times M \to \mathbb{R}^n$. Then naturally project its fiber at $0$ onto $M$, the fiber at $0$ of $\mathbb{R}^n \times M \to \mathbb{R}^n$.
[1] Jets and contact elements, D. Krupka and M. Krupka Proceedings of the Seminar on Differential Geometry Mathematical Publications Volume 2 Silesian University in Opava, Opava, 2000, 39–85
@igor, thanks! It does seem a little strange to me to say “velocities” — I think of “velocity” as a concept from physics, which mathematically is an instance of a tangent vector. So “higher order tangents” would make more sense as a name for the mathematical concept (a “higher order velocity” would be a physical concept like acceleration or jerk). Has anyone used a similar phrase that you know of?
@Mike, I’ve also seen D.J. Saunders (of the Geometry of Jet Bundles renown) use the same terminology [1]. If Krupka and Saunders are not sufficient authorities for you, I don’t know who would be. :-)
[1] Homogeneous variational complexes and bicomplexes, D. J. Saunders (2009) JGP 59 727-739
Another remark, this one is about coalgebras. Unfortunately, the talk of coalgebras in the preceding comments is black magic to me, so I cannot say anything about that directly. However, the very excellent book by Seiler [1] on the algebraic and geometric aspects of the formal theory of PDEs talks about coalgebras in the context of jets as well. The idea is quite simple. There is a commutative algebra that is formed by differential operators with constant coefficients over $\mathbb{R}^n$, where multiplication is just composition. Thinking of the differentiation and evaluation at 0 as a dual pairing, $\langle \partial_1^{i_1} \partial_2^{i_2} \cdots, (x^1)^{j_1} (x^2)^{j_2} \cdots \rangle = (\partial_1^{i_1} \partial_2^{i_2} \cdots)((x^1)^{j_1} (x^2)^{j_2} \cdots)|_{x=0}$, it is not hard to see that the space of formal power series is dual to the algebra of differential operators in the sense of vector spaces. So, the space of formal power series is naturally endowed with a coalgebra structure.
In terms of jets, it is well known that formal power series are essentially $\infty$-jets of real valued functions on a manifold $M$, say of dimension $n$ to fit in with the above discussion. It is not hard to see that the algebra of differential operators with constant coefficients is the symmetric algebra generated by the tangent space $T M$ (so there’s a copy of this algebra for every point of $M$). The interest in this identification of a coalgebra structure on $\infty$-jets is that the sometimes mysterious Spencer cohomology that plays an important role in the formal theory of PDEs is none other than the linear dual of the so-called Koszul homology of the symmetric algebra over $T M$.
[1] Seiler, W. M. Involution Algorithms and Computation in Mathematics 24 (Springer, 2010).
Its black magic to me in the sense that I can’t see how Mike noticed that, but once you have the statement it is quite straightforward to check. Just a remark @Igor: the coalgebra you mention is a coalgebra in the more classical sense and is, as far as I can see, not related to the $J^1$-coalgebra.
Alright then, can anyone suggest an application of this “$J^1$-coalgebra” structure? For instance, the other “classical” coalgebra structure that I mentioned has application in the interpretation of Spencer cohomology. Obviously, this is not a competition. But I’m curious whether this “$J^1$-coalgebra” structure is just an observation or does it have some immediate utility.
Currently I think of it as a characterization of the infinite (nonlinear) jet space by a universal property. I don’t remember seeing a universal property of $J^\infty$ with the Cartan distribution before, but I’d be interested in learning about others if you know one. (Probably there is one in the general abstract definition on that page but I don’t have enough background on higher topoi and cohesion). I don’t know of any new applications, but it certainly reduces some arguments in the jet literature to “one-liners”.
It sounds like we should have something on Spencer cohomology and Koszul homology. Does the latter arise from a Koszul complex?
Igor,
to clarify: in what Michael is talking about, the word “algebra” and “coalgebra” is used in a sense vastly more general than what you probably imagine these terms might ever mean. There is a field called “universal algebra” or “categorical algebra” or similar in which essentially any kind of mathematical structure for which there is a theory in a formal (but again vastly general) sense is an “algebra” of sorts. For instance the theory might be exhibited by a monad/comonad and then one would speak of an algebra over a monad. For instance something entirely non-algebraic (in the usual sense) like a graph is still an algebra in this sense of universal algebra (over the free graph monad, in this case).
What Michael is talking about here is even more general than algebras over monads: he is talking about algebras over endofunctors/coalgebras over endofunctors.
As usual in category theory, the impact of identifying anything already well-known as an instance of such-and-such general abstract formalism is often not so much useful in further understanding the already well-known thing itself. It is instead usually useful for putting that already well-known thing into broader context, understanding its similarity to other possibly superficially very different-looking things and knowing what the sensible generalizations of the well-known thing to new and more exotic contexts might be.
That is the case here: in itself the observation in #4 above doesn’t tell you anything about jet bundles as such which you didn’t already know. It is in particular not a new structure on top of the jet bundle, such as the coalgebra structure (the co-associative co-algebra structure) that you are thinking of above. Instead it is a way of pointing a magic wand at the mundane concept of jet bundle and saying: “Hereby though art a concrete particular incarnation of something general abstract.”
David,
yes, Koszul homology is the chain homology of a Koszul complex.
Re #26, putting the $J_1$-coalgebra story together with the ’abstract general’ one, presumably the $Jet$ of #7 is the $J^{\infty}$ of #4. $Jet$ is idempotent presumably. Is there no need in this set up for a $J_1$?
But we’re pointed in #6 to p. 6 of Lurie’s “Notes on crystals and D-modules” (pdf), where $Jet$ or $J$ is constructed as an inverse limit.
Hmm, does this work at the level of objects in $\mathbf{H}$, giving a sequence of slices? I mean is ${\Pi}_{inf}(X)$ some kind of limit, from a sequence of $n$-th order infinitesimal coreductions, so that ${\Pi}_{inf}(X)$ is a fixed point for some $J^1$ like operator? And that induces something on $\mathbf{H}_{/\mathbf{\Pi}_{inf}(X)}$ as fixed point?
I added Igor’s remark about Spencer cohomology to Koszul homology.
Hmm, I now think my generalisation in #4 was not as straightforward as I thought, and is probably wrong as stated. Presumably the universal property of $J^\infty P$ should be that there exists a lifting for of any bundle morphism from $q$ to $p$ when $q$ is supplied with a flat connection. Sorry. I’ll try to think about it, although currently I don’t see how one would formulate that universal property it in terms of coalgebras of some functor. Mikes original observation might still be right, since a vector field is automatically flat (in involution with itself).
I erased that section from the nlab page. What I was constructing there was not $J^\infty$ but the “infinite iteration of $J^1$”.
David, I certainly wouldn’t expect a jet construction to be idempotent. Is it really so in the infinitesimal cohesion case?
I don’t think it’s idempotent.
And to amplify again: the “differential cohesion case” subsumes the ordinary construction in algebraic geometry (“D-geometry”). This was first observed by Simpson-Teleman.
Whoops, yes. I was thinking about $\mathbf{\Pi}_{inf}$ being idempotent.
So, that would give the jet of a bundle over $\mathbf{\Pi}_{inf}(X)$ as just the bundle itself, induced by $i \colon \mathbf{\Pi}_{inf}(X) \to \mathbf{\Pi}_{inf}(\mathbf{\Pi}_{inf}(X))$, I guess.
David, the jet bundle construction is push followed by pull along that single unit map. This pull-push operation has no reason to be idempotent.
Of course one could also consider what you write here, to push twice alobg the iterated unit map. That then would be, for an idempotent monad, the same as pushing once, as you say.
expanded at jet bundle the remark on precursors in the literature of the general abstract definition, by adding pointer to Krasilshchik-Verbovetsky 98, p. 13, p. 17, which almost shows that the jet bundle construction is a comonad (it seems they don’t explicitly check the counit property, or do they?)
I have also streamlined the entire section definition – general abstract just a little bit more
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
$Hom_{FrechetMfd}(X, J^\infty E) \simeq \underset{\longleftarrow}{\lim}_k Hom_{SmthMfd}(X, J^k E)$?
Ah, never mind, it seems that chapter 7 of Saunders’ The geometry of jet bundles has everything I need.
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?
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.
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.
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
but I have not seen much of its inside yet (GoogleBooks is shy about showing it).
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?
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.
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.)
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
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?
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…
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 $0$.
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.
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.
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!
I’ve only had a glance, but maybe if we return to the germ/jet issue, there may be something in
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.
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.
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?
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
added this pointer:
(here and in related entries)
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