Comparative Smootheology, II

April 17, 2008

Posted by John Baez

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A while back, Urs blogged about Andrew Stacey’s paper comparing various flavors of ‘smooth space’ that generalize the concept of manifold:

Andrew Stacey, Comparative Smootheology.

My student Alex Hoffnung and I are writing a paper on two of these flavors: Chen’s ‘differentiable spaces’ and Souriau’s ‘diffeological spaces’. So, I found Andrew’s detailed comparison to be very helpful, and I decided to ask him a question that had been bugging me: could Chen’s spaces be equivalent to Souriau’s?

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Chen spaces and diffeological spaces are formally very similar. The key difference is that a Chen space is equipped with a bunch of ‘plots’ that are maps into it from convex subsets of $ℝ^{n}$ , while a diffeological space has plots that are maps into it from open subsets of $ℝ^{n}$ . It seemed unlikely that the resulting notions were equivalent, but I didn’t have a proof — and it would be embarrassing to write a paper about two different kinds of smooth space and only later realize they were the same! My first hoped-for counterexample, manifolds with boundary, fell through quite a while ago. So, I wanted to ask Andrew about this.

In the process of getting ready to ask this question, I reread his paper to see precisely which definition of Chen spaces he was using. In the process, I came up with some other questions that were so detailed and technical that I didn’t want to bring them up here — when you ask nitpicky questions in public it’s easy to seem like you’re trying to score rhetorical points.

So, I sent him a couple of emails… but then he suggested talking about this stuff on the blog, which seems like a great idea.

So, here are my emails. I’ll post Andrew’s reply as a ‘comment’. His reply gives all 4 definitions of Chen spaces that I vaguely allude to here.

Dear Andrew -

Hi! I’m really enjoying your paper Comparative Smootheology, especially now that my student Alex Hoffnung are writing a paper about Chen spaces and diffeological spaces, so that all sorts of detailed issues are on my mind.

Here are two questions / comments:

1) Chen defined “differentiable spaces” in 3 different ways in the 3 papers I have access to right now:

K.-T. Chen, Iterated integrals of differential forms and loop space homology, Ann. Math. 97 (1973), 217–246.

K.-T. Chen, Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206, (1975), 83–98.

K.-T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83, (1977), 831–879.

I need to look at his fourth paper again:

K.-T. Chen, On differentiable spaces, Categories in Continuum Physics, Lecture Notes in Math. 1174, Springer, Berlin, (1986), 38–42

Does use the 1977 definition or yet another one? (That’s not a question to you, mainly — I’m just wondering. But, if you know I’d be interested, since I can’t get that paper here in Shanghai.)

Anyway: I think your “early Chen spaces” are not precisely the differentiable spaces from Chen’s 1973 paper. In this paper he requires that the space be, not just a topological space, but a Hausdorff space.

Also: I think your “Chen spaces” are not precisely the differentiable spaces from Chen’s 1977 paper. In this paper he takes the domain of a plot to be any convex subset of $ℝ^{n}$ , while you require that this domain be closed.

Is there some reason you made these changes?

I don’t know how important these issues are, but it might be helpful, for people trying to straighten out this tangled tale, to note that you’re adding two new definitions to Chen’s three. (Unless, of course, I’m making a mistake!)

2) Are your categories “Chen” and “Souriau” equivalent or not? It seems like probably not. You don’t seem to prove this: instead, you construct some adjunctions between them and show they’re not equivalences. But, maybe you understand the situation well enough to easily figure this out! Maybe it’s easier to show there’s no equivalence that acts as the identity on the underlying sets and functions. In principle there could be some sneakier equivalence.

I’m actually interested in showing that Chen’s 1977 category is not equivalent to “Souriau”, but I’ll take whatever words of advice you can offer!

Best,
jb

And then:

Dear Andrew -

Here’s another niggly little remark. I really like the theorem of Kriegl and Michor that you cite:

Let $K$ be a convex subset of $ℝ^{n}$ Let $f : K \to ℝ^{m}$ . Then $f$ maps smooth curves in $K$ to smooth curves in $ℝ^{m}$ iff $f$ is smooth on the interior of $K$ and all derivatives (on the interior) extend continuously to the whole of $K$ .

I hadn’t known it!

But, it seems hard to understand, and probably even false, in the case when the interior of $K$ is empty - e.g. a line sitting in the plane. Then ANY function f is smooth on the interior of $K$ (the empty set), and god knows whether its derivatives extend continuously from the interior to the whole of $K$ . I guess they do: any continuous function counts as a continuous extension of a function defined on the empty set!

So, maybe we should add the assumption that $K$ has nonempty interior. Pondering this, I reread Chen’s paper in that 1986 Springer volume (kindly forwarded to me by my student Alex Hoffnung) and found that he added an assumption: the domains of his plots must be convex subsets of $ℝ^{n}$ with nonempty interior.

Maybe Kriegl and Michor build this in somehow.

Rereading Chen’s 1986 paper, I then noticed he cleverly starts by taking his convex sets to be “abstract”, not embedded in any particular $ℝ^{n}$ Then he gets them embedded in $ℝ^{n}$ in such a way that they have nonempty interior. He’s quite quick and sketchy about this.

So: my Chen spaces will henceforth have plots whose domains are convex subsets of Euclidean spaces, with nonempty interior.

Best,
jb

Andrew’s reply follows — I’ll use my superpowers to pretend he posted it as a comment here.

Posted at April 17, 2008 12:26 AM UTC

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Re: Comparative Smootheology, II

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Hi John,

Thanks for your emails. I’m delighted that you’re looking at the paper and welcome your comments; particularly as I know that you actually want to use this stuff!

Okay, so on to your comments …

Firstly, Chen’s definitions. Yes, he does define “differentiable spaces” in several different ways. Let me see if I can hunt them all down. I’ll repeat them all here since bytes are cheap and it’ll make it easier to comment on them.

1973

By a convex $n$ -region (or, simply a convex region), we mean a closed convex region in $ℝ^{n}$ . A convex $0$ -region consists of a single point.

Definition. A differentiable space $X$ is a Hausdorff space equipped with a family of maps called plots which satisfy the following conditions:

(a) Every plot is a continuous map of the type $ϕ : U \to X$ , where $U$ is a convex region.

(b) If $U^{'}$ is also a convex region (not necessarily of the same dimension as $U$ ) and if $θ : U^{'} \to U$ is a $C^{∞}$ map, then $ϕ θ$ is also a plot.

(c) Each map ${0} \to X$ is a plot.

So you’re right: I missed the Hausdorff condition here. That’s extremely annoying on two counts: firstly that I missed it, and secondly because it mucks up the functors. My functor to Early Chen Spaces used the indiscrete topology (essentially to make it irrelevant). I can’t do that any more. I’ll have to think about how much of a difference that makes. Of course, to a certain extent then it doesn’t make any difference since no one actually uses these spaces as anyone who is aware of Chen’s original definition is almost certainly aware of his later definitions and would prefer to use those.

1975

By a convex region we mean a closed convex set in $ℝ^{n}$ for some finite $n$ .

Definition. A predifferentiable space $X$ is a topological space equipped with a family of maps called plots which satisfy the following conditions:

(a) Every plot is a continuous map of the type $ϕ : U \to X$ , where $U$ is a convex region.

(b) If $U^{'}$ is also a convex region (not necessarily of the same dimension as $U$ ) and if $θ : U^{'} \to U$ is a $C^{∞}$ map, then $ϕ θ$ is also a plot.

(c) Each map ${0} \to X$ is a plot.

Remark. in [1973], a predifferentiable space is called a “differentiable space”. We propose to amend the definition of a differentiable space by adding the following condition:

(d) Let $ϕ : U \to X$ be a continuous map and let ${θ_{i} : U_{i} \to U}$ be a family of $C^{∞}$ maps, $U$ , $U_{i}$ being convex regions, such that a function $f$ on $U$ is $C^{∞}$ if and only if each $f \circ θ_{i}$ is $C^{∞}$ on $U_{i}$ . If each $ϕ \circ θ_{i}$ is a plot of $X$ , then $ϕ$ itself is a plot of $X$ .

I had not come across this paper before and it is extremely interesting. First, in his recollection of what is now a predifferentiable space Chen drops the Hausdorff condition. Thus what I called “Early Chen Spaces” are these predifferentiable spaces. Secondly, and much much more importantly, is his introduction of condition d. This appears to be a sheaf condition but it is not; it is much stronger. By Kriegl and Michor’s result on curves in convex regions (see later for more on this), we could take the family of functions $θ_{i}$ to be the family of smooth curves in $U$ . Thus condition d is saying, “any continuous map which is a plot when restricted to smooth curves is a plot”.

Interestingly, Chen retains the assumption of an underlying topology.

1977

The symbols $U$ , $U^{'}$ , $U_{i}$ , $\dots$ will denote convex sets. All convex sets will be finite dimensional. They will serve as models, i.e. sets whose differentiable structure is known.

…

Definition 1.2.1 A differentiable space $M$ is a set equipped with a family of set maps called plots, which satisfy the following conditions:

(a) Every plot is a map of the type $U \to M$ , where $\dim U$ can be arbitrary.

(b) If $ϕ : U \to M$ is a plot and if $θ : U^{'} \to U$ is a $C^{∞}$ map, then $ϕ \circ θ$ is a plot.

(c) Every constant map from a convex set to $M$ is a plot.

(d) Let $ϕ : U \to M$ be a set map. If ${U_{i}}$ is an open covering of $U$ and if each restriction $ϕ ∣ U_{i}$ is a plot, then $ϕ$ is itself a plot.

Again, you are right. I did not spot the fact that he has here dropped the requirement that the convex sets be closed. They are just arbitrary convex sets of finite dimension, and not necessarily embedded in Euclidean space (not that that matters). Comparing with the 1975 definition, we see that the fourth condition is now a sheaf condition.

I do have the 1986 paper in front of me. Here’s the definition from that.

1986

We take as the model category the one whose ojects are convex subsets with nonempty interiour in $ℝ^{n}$ , $n = 0, 1, \dots$ , and whose morphisms are $C^{∞}$ maps.

…

Definition 1.1. A $C^{∞}$ space $M$ is a set equipped with a family of set maps called plots, which satisfy the following conditions:

(a) Every plot is a map of the type $U \to M$ where $U$ is a convex set.

(b) If $ϕ : U \to M$ is a plot and if $U^{'}$ is also a convex set (not necessarily of the same dimension as $U$ ), then, for every $C^{∞}$ map $θ : U^{'} \to U$ , $ϕ θ$ is also a plot.

(c) Every constant map from a convex set to $M$ is a plot.

(d) Let ${U_{i}}$ be an open convex covering of a convex set $U$ , and let $ϕ : U \to M$ be a set map. If each restriction $ϕ ∣ U_{i}$ is a plot, then $ϕ$ itself is a plot.

Up to trivial rephrasing, this is the same as the 1977 definition.

You ask:

Is there some reason you made these changes?

Yes. Sheer ignorance! Stupidity cannot be ruled out either. I simply did not spot the myriad of changes. In my defence, I would say that rather than simply copying the definitions in the paper I was trying to standardise the language.

We appear to have four definitions with certain characteristics:

1973, no sheaf-like condition, topology, Hausdorff, closed domains.
1975a, no sheaf-like condition, topology, not necessarily Hausdorff, closed domains.
1975b, strong sheaf-like condition, topology, not necessarily Hausdorff, closed domains.
1977 (and 1986), sheaf condition, no topology, arbitrary domains.

By “arbitrary” I mean still convex, but not assumed to be closed.

Phew!

I got one of these, at least. My “Early Chen Spaces” are the 1975a definition. But you’re right, my “Chen Spaces” are not on the list. Whoops. However, I think that one can simply delete the word “closed” from my definition of a Chen space to get the 1977 definition and this would not require any other changes to the mathematics. I’ll have to check that, of course, but I’m reasonably confident. The other definitions will require a little thought.

I’d certainly consider putting them all in my paper but I think it warrants a little reorganisation. Perhaps in the main flow of the paper it would be best to concentrate on the last definition and then have a separate section comparing all the different variants of Chen space.

Does that go some way to answering your question on definitions?

On to the equivalence (or not) of Chen and Souriau spaces. You ask:

Are your categories “Chen” and “Souriau” equivalent or not? It seems like probably not. You don’t seem to prove this: instead, you construct some adjunctions between them and show they’re not equivalences. But, maybe you understand the situation well enough to easily figure this out! Maybe it’s easier to show there’s no equivalence that acts as the identity on the underlying sets and functions. In principle there could be some sneakier equivalence.

I’m actually interested in showing that Chen’s 1977 category is not equivalent to “Souriau”, but I’ll take whatever words of advice you can offer!

I think that they are not equivalent. Let’s see if we can prove this. To shorten the notation, let $C$ be the category of Chen spaces (1977 definition) and $S$ the category of Souriau spaces (diffeological spaces).

The first thing to do is to rule out your “sneaky equivalence”. Suppose we have functors $F : C \to S$ and $G : S \to C$ . Suppose that these define an equivalence of categories. Then in particular, they take terminal objects to terminal objects. We therefore have natural isomorphisms

∣ S ∣ ≅ S ({*}, S) ≅ C (G ({*}), G (S)) ≅ ∣ G (S) ∣

and vice versa, and this works on morphisms. Therefore up to natural isomorphism, $G$ and $F$ are set-preserving. We can make this strictly true if we want essentially by regarding $S$ and $C$ as lying over two copies of $Set$ and using $F$ and $G$ to identify the two copies in a (possibly) non-standard fashion.

So any equivalence has to define a set-preserving one. Let us now assume that our functors are set-preserving. This means that $C (C_{1}, C_{2})$ and $S (F (C_{1}), F (C_{2}))$ are the same subset of $Set (∣ C_{1} ∣, ∣ C_{2} ∣)$ and similarly for $G$ . This means that the compositions $G F$ and $F G$ are exactly the identity functors on their respective categories.

Now, I think, we can show that the functor from Chen spaces to Souriau spaces is the one that I describe in my paper. In fact, this is easier with the assumption of closedness dropped.

The set of plots of a Chen space, $C$ , is precisely the union of the sets $C (U, C)$ where $U$ runs over the family of convex regions with their standard Chen structure. A similar statement for Souriau spaces holds only with $U$ the family of open sets (in Euclidean spaces).

Let $U$ be an open convex subset of some Euclidean space. We can give this a canonical Chen structure and a canonical Souriau structure; both of which are characterised by the fact that they contain the identity map. As $G F$ and $F G$ are the identity functors, we see that the identity map $∣ U ∣ \to ∣ U ∣$ is contained in all of

S (U, F G (U)), S (F G (U), U); C (U, G F (U)), C (G F (U), U)

so we deduce that, with absolutely horrendous notation, $G (U) = U$ and $F (U) = U$ .

Now as Souriau spaces satisfy the sheaf condition, the Souriau plots are completely determined by the subfamily where $U$ runs over the family of open convex sets. We therefore have

S (U, S) = C (G (U), G (S)) = C (U, G (S))

More generally, we see that if $V$ is an open subset of some Euclidean space then using the sheaf conditions

S (V, S) = C (G (V), G (S)) = C (V, G (S))

where $V$ is given the canonical Chen structure wherein all inclusions of convex subsets are plots.

Hence the functor $F : C \to S$ is the functor that I describe in my paper.

Now we arrive at a contradiction. I’m pretty sure that even with the modified definition of Chen spaces, my example of two distinct Chen spaces with the same underlying Souriau space remains valid. Thus the functor $F$ cannot be part of an equivalence of categories and so the categories of Chen spaces and Souriau spaces are not equivalent.

(Insert end-of-proof symbol here)

Right, I worked that out more or less as I wrote it so there’s probably bits that I’ve overlooked. It’ll probably look a bit neater when run through iTeX (you can do that without posting it on the cafe). Let me know if you’re convinced!

Now, on to Kriegl and Michor’s theorem. I simplified the statement of the theorem in their book since that deals with convex subsets in arbitrary convenient vector spaces. In doing so, perhaps I lost a little precision. What I was not careful about was defining the interior of a convex set. What I ought to have said was that this was the abstract interior, not the interior as embedded in some arbitrary $ℝ^{n}$ . If one embeds the abstract convex set in its “natural” affine space, then this abstract interior is the interior that you inherit from the topology on the affine space. I guess that this is what Chen had in mind in the 1986 paper.

So you were right to pick up on that, but it was my error in being imprecise and misquoting Kriegl and Michor.

Right, that’s probably enough to be going on with for now. It’s getting near lunchtime here and I’m getting hungry.

Best, Andrew

Posted by: Andrew Stacey on April 17, 2008 3:53 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Concerning the sheaf-like condition from 33 years ago (the one from 1975, that is): if you take Chen by the letter here I suppose you are right that this is stronger than the ordinary sheaf condition.

But is there any indication that Chen meant to anderstand and to use it that way? The notation suggests (at least from 33 year hindsight) that he did simply have the ordinary sheaf condition in mind.

Since he never seems to actually mention the very word “sheaf” it might be that he wasn’t aware of the concept (could that be?) and gradually “discovered” it himself.

Posted by: Urs Schreiber on April 17, 2008 4:46 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

It’s impossible to know what Chen had in mind in his various definitions, but I would hazard a guess that he knew that his third definition was stronger than the sheaf condition he eventually settled on. Boman’s paper preceded Chen’s by eight years and was known to Frölicher. The two certainly knew each other, at least by 1982. Mostow makes an interesting point in his paper. He uses Sikorski spaces; but he mentions Smith spaces and explains why he doesn’t use them. His reason is that the closure condition is difficult to check. It may be that Chen decided that his condition was difficult to check and the sheaf condition would be adequate for what he wanted to do.

Posted by: Andrew Stacey on May 6, 2008 9:16 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

A small point of terminology: at least among those who regularly work with convex sets in Euclidean spaces, the notion of interior Andrew describes is usually called the “relative interior”.

Posted by: Mark Meckes on April 17, 2008 2:38 PM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Thanks! That would disambiguate things nicely, at least after it’s explained.

Posted by: John Baez on April 18, 2008 5:06 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

Thanks for that, Mark. I’ll add that in to the next version to make it clear.

Posted by: Andrew Stacey on April 22, 2008 9:50 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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I think that they are not equivalent.

Maybe you could help me here:

for $S$ a site with a forgtful functor $f$ to $Set$ , we are looking at categories of quasi-representable sheaves over $S$ : those sheaves $X$ on $S$ for which there exists a set $X_{s}$ such that for each $U \in S$ we have $X (U) \subset Set (f (U), X_{s})$ and for each $ϕ : U \to V$ we have $X (ϕ) = ϕ^{*} .$ Moreover, a morphism between quasi-representable sheaves $X \to Y$ is a morphism of sheaves which comes from a map between the underlying sets $X_{s} \to Y_{s}$ .

Let me write $QSh (S)$ for such a category of quasi-representable sheaves.

Then I’d like to know:

is $QSh (open subsets) ≃ QSh (open convex subsets)$ ?

I did once think that this should be true. But maybe I was wrong. So is $QSh (open subsets) ≃ Chen spaces$ and/or $QSh (abc convex subsets) ≃ Soriau spaces,$ where “abc” is your favorite among “general”, “open”, “closed”. ?

Generally, I’d think that $QSh (S)$ is the “right” thing to look at here. But please correct me if I am wrong about this.

Posted by: Urs Schreiber on April 19, 2008 2:22 PM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Urs wrote:

Generally, I’d think that $QSh (S)$ is the “right” thing to look at here. But please correct me if I am wrong about this.

I think you’re right! Indeed, the way Alex Hoffnung and I are investigating Chen spaces and diffeological spaces is by considering them as examples of quasirepresentable sheaves.

Actually we consider a slight variation on (and I believe improvement of) the definition you mention, and we speak of a category of ‘concrete sheaves’ over a ‘concrete site’ — ideas explained to us by James Dolan. But, I think you’re very much on the right track here.

I don’t have the energy to explain the details, since the paper will be done in a couple of weeks, and everything will be explained very nicely there!

So is

$QSh (open subsets) ≃ Chen spaces$

and/or

$QSh (abc convex subsets) ≃ Soriau spaces$

where “abc” is your favorite among “general”, “open”, “closed”?

I think you made a typo here. Here’s the true story:

$QSh (open subsets) ≃ Souriau spaces$

$QSh (convex subsets) ≃ Chen spaces$

where ‘subsets’ means ‘subsets of Euclidean spaces of arbitrary finite dimension’.

The difficulty with Chen spaces is that Chen gave several definitions leading up to the final one which I’m using here. Andrew reviews the earlier definitions very nicely above But, for the purposes of doing elegant mathematics (as opposed to history) we should ignore all the earlier definitions and focus on the final one.

It can be useful to take any convex subset of a Euclidean space and embed it into the lowest-dimensional Euclidean space in which it fits. We get a convex subset with nonempty interior, which makes derivatives of functions on this set a bit easier to explain. We can do this without any loss of generality, so Chen’s final definition can equivalently be phrased this way:

$QSh (convex subsets with nonempty interior) ≃ Chen spaces$

This is what I do in my paper with Alex. (Hmm, maybe I need to explain this trick better in the paper.)

Anyway, Andrew’s argument above has convinced me that

$Chen spaces ≃ Souriau spaces$

is false, even though both turn out to be equally good for handling manifolds with boundary and manifolds with corners. And, to answer another question of yours, I also believe that

$QSh [convex subsets] ≃ QSh [convex open subsets]$

is false.

(I can’t figure out how to draw a “not \simeq” symbol.)

Posted by: John Baez on April 20, 2008 3:38 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Thanks a lot for this reply! I am looking forward to your article with Alex Hoffnung. (I did already begin a while ago to cite it in my various notes :-)

I am hoping that the issue about non-equivalences here is all in whether or not we use open subsets. Is that right?

I’d be puzzled if $QSh (open subsets) ≃ QSh (open convex subsets)$ (with open sets in both cases) were false (which possibly just shows that I didn’t follow Andrew’s argument).

Or it is the $Q$ ? We should have $Sh (open subsets) ≃ Sh (open convex subsets)$ at least.

Posted by: Urs Schreiber on April 20, 2008 4:11 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Urs almost wrote:

I’d be puzzled if

$QSh (open subsets) ≃ QSh (open convex subsets)$

(with open sets in both cases) were false (which possibly just shows that I didn’t follow Andrew’s argument).

I haven’t carefully checked, but I’ve been spending the weekend redoing a lot of proofs in Andrew’s paper, developing more intuition for this stuff… and I’m willing to bet that this is true:

$QSh (open subsets) ≃ QSh (open convex subsets)$

(If you didn’t follow Andrew’s argument, perhaps it’s because his argument here just showed that if there were any equivalence between $QSh (open subsets)$ and $QSh (convex subsets)$ , it would have to be a certain functor called $So$ that he’d already studied in his paper… and there, in Section 5, he had shown that functor was not an equivalence. So, the truly central issue is not to be found here, but in his paper! A summary will appear in the paper Alex and I are writing.)

By the way, I believe every open convex subset of $ℝ^{n}$ is diffeomorphic to $ℝ^{n}$ , so I also believe this:

$QSh (open convex subsets) ≃ QSh (Euclidean spaces)$

So, all these

$QSh (open subsets), QSh (open convex subsets), QSh (Euclidean spaces)$

should be various equivalent ways of defining diffeological spaces! And of course the last is the most succinct.

I’m much happier with diffeological spaces now that I’ve shown they correctly handle manifolds with corners. Unfortunately, it requires a rather amazing technical result by Kriegl and Michor, Theorem 24.5 in their book. You don’t need this result if you use Chen spaces! But, if you’re willing to use Kriegl and Michor’s result (which is free), you can sail ahead quite nicely studying cobordism $n$ -categories and path $n$ -groupoids using diffeological spaces.

I like your idea that quasirepresentable sheaves on the site of superEuclidean spaces give a notion of super smooth space.

Also by the way: I think the reason Andrew is so quiet right now is that he has limited access to the web. He said he’d be back.

Posted by: John Baez on April 21, 2008 5:06 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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John boggled:

By the way, I believe every open convex subset of $ℝ^{n}$ is diffeomorphic to $ℝ^{n}$ , so I also believe this:

QSh(openconvexsubsets)≃QSh(Euclideanspaces)

Can you expand on that diffeomorphism a little? Certainly they are homeomorphic but the obvious homeomorphism isn’t always a diffeomorphism.

Nonetheless, you are right about the categories of quasi-representable sheaves since an open convex subset is locally diffeomorphic to a Euclidean space and that’s all you need - as I’m sure you already know! (Grandma, these things are called “eggs”.)

Urs scrabbled:

I think among all the various choices of test domains which we are discussing, the “right” one is simply: Euclidean spaces.

I came to this conviction for two reasons:

first when Todd pointed me to Moerdijk&Reyes: for precisely this choice of S (but none of the other choices which involve subsets) do we have a beautiful theory of duality between “spaces” (contravariant functors on S) and “quantities” (covariant functors on S) with the latter behaving very (very) nicely as algebras of functions (if monoidal).

It sounds here as if you’re willing to drop the sheaf condition. Otherwise you get the same category of smooth things for quite a range of sites. Are you trying to drop the sheaf condition?

It maybe that you want to replace it by the monoidal condition. Doesn’t this mean that everything is determined by its action on simply $ℝ$ ? So why not go the whole hog and use just $ℝ$ ?

This ends you back at the category suggested by looking at the definition of Frölicher spaces, but by a rather circuitous route.

On the other hand, if you retain the sheaf condition then the choice of site seems a little related to the notion of an adequate subcategory, as defined by Isbell (see the review of MR175954, or the original paper if you have access). It would be useful to determine a smallest (left) adequate subcategory, and Euclidean spaces certainly seems to be such, but it doesn’t change the properties of the actual category.

The question of the site is of secondary importance to me, despite the impression I may have given elsewhere; though I like the notion of a super-site (is it a bird, is it a … sorry, already done that joke). For any of our candidate sites we have a hierarchy of categories:

Presheaves on $𝒮$
Sheaves on $𝒮$
Quasi-representable sheaves on $𝒮$
Isbell-stable presheaves on $𝒮$

(have I missed any?)

By the argument I gave in the Space and Quantity post, every Isbell-stable presheaf is actually a quasi-representable sheaf so this is a total ordering.

If we consider two comparable sites, one can ask at what stage the two hierarchies become equivalent, or isomorphic, or set-preservingly isomorphic. For the cases we’re studying, the sites are all families of subsets of Euclidean spaces and we can rephrase this question as to whether the intersection family defines a left adequate subcategory for any of the four levels (assuming I grok what adequacy means, here). This turns the question around a little so that the focus is on the resulting category and not on the site used in its definition.

A second question, and one I’m much more interested in, is to ask what level on the hierarchy we want to work in. I suspect I’ve made it fairly obvious where I stand on this issue so I’ll not bore you with more on this.

A point I’ve made before, but which I think is worth making again, is that these two questions: the site and the hierarchy, are independent.

Posted by: Andrew Stacey on April 22, 2008 10:29 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Are you trying to drop the sheaf condition?

My attitude is the one expressed above: spaces=presheaves, rather nice spaces=sheaves, pretty nice spaces=self-dual wrt $Ω^{•}$ , extremely nice spaces=Frölicher ones, where the last two items are thanks to discussion with you.

(And you give essentially the same list now.)

So, I don’t want to drop the sheaf condition in general. But I am saying that even with the sheaf condition, the choice of site matters, since it’s only with nice enough sites that very nice co-presheaves make sense.

Also, when we move away from ordinary smooth test domains, things that look like entire Euclidean spaces tend to be more naturally present than any notion of subsets of them.

It maybe that you want to replace it by the monoidal condition.

I don’t want to replace it by that. The monoidal condition is a condition on nice co-presheaves.

But it is only with test domains full Euclidean spaces that we naturally have the monoidal condition, since it makes use of the vector space structure on the Euclidean space. That fails for most subsets of Euclidean spaces.

Doesn’t this mean that everything is determined by its action on simply $ℝ$ ?

Not sure how to answer this. I guess I see what you mean, but am not sure if that’s the way to think about it. What I know is that assuming “nice” algebras of functions to be monoidal functors on the $ℝ^{n}$ s site makes them have all the nice properties that Moerdijk&Reyes describe.

(have I missed any?)

As I mentioned above and before in some discussion we had elsewhere, I am thinking that an important class in between your level 2 and 3 are those things which I put at level 3: those that are self-dual under duality induced not by $C^{∞} (-)$ but by $Ω^{•} (-)$ .

what level on the hierarchy we want to work in. I suspect I’ve made it fairly obvious where I stand on this issue

I think there is no general best answer where in the hierarchy to work in. You will want to work as high up as possible, generally, but how high is possible depends on the application.

For instance, I have an important (for me, at lest :-) application which forces me to work with things at level 3, but mix them with things at mere level 1.

Posted by: Urs Schreiber on April 22, 2008 1:20 PM | Permalink | Reply to this

Re: Comparative Smootheology, II

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That makes things a little clearer, Urs. Thanks for the explanation.

I’m still of the opinion that the site should be secondary to the theory. What I’d rather say is: here are the various categories of types of smooth spaces, ranked by how “nice” they are. Let us fix an object in one of them. For any suitable choice of site (i.e. family of subsets of Euclidean spaces), we get a resulting family of plots from the objects of that site.

Am I making myself clear? I mean that the categories (I’m now resigned to having several!) should come first and then we can evaluate them on any given site to get things of the form that you want.

If I take a Chen space and look at the Chen-morphisms from that space to Euclidean spaces then I get $C^{∞}$ algebras. That doesn’t depend on the fact that I used convex sets to define Chen spaces.

We still, however, have to define the categories. For this, it seems that we should strive for the smallest site possible. My argument for this is that this approach promotes clarity. If the site is large (non-technically speaking) then it is harder to separate out those properties of the resulting categories that are inherent from those which arise from the specific choice of site.

At this point, you (Urs) usually start going on about $Ω^{•}$ . This is the hom functor if we work with superspaces. I see no difficulties moving from ordinary spaces to super spaces and I quite like the idea. The argument I gave over in Space and Quantity about “ever so extremely nice spaces” being quasi-representable carries over to the super situation as it depended only on there being a separator in the category. In this case, the separator would be the superline, $ℝ^{0 ∣ 1}$ .

However, sometimes it seems that you want to use $Ω^{•}$ for ordinary spaces. If so, I’d like to separate the two uses in some fashion as it sometimes makes my head spin (or superspin) trying to keep up! Also if so, this may mean that we need Euclidean spaces of arbitrary dimension but I have a sneaky suspicion that it all gets determined by its effect on lines anyway.

Posted by: Andrew Stacey on April 23, 2008 8:48 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Hi Andrew,

I agree of course that if sheaves on $S$ and $S'$ are equivalent, it does not really matter much wether we think of using $S$ or $S'$ . I just said that if we talk about $S$ but not $S'$ also for other reasons, it is somehow less conceptual overhead to think of $S$ instead of $S'$ .

it seems that we should strive for the smallest site possible.

Sure. But what is “possible” depends on the application. In your application a “smaller” site is possible than in mine. In fact, the site of Euclidean space is the smallest – for my application. :-)

At this point, you (Urs) usually start going on about $Ω^{•}$ .

At which point you (Andrew) usually start to stubbornly refuse following me ;-)

This is the hom functor if we work with superspaces.

Right, this is the argument I made a while ago. I was trying hard to fit my example into your philosophy and noticed that maybe that’s the solution. But now I think it isn’t the solution. While it is of course true that forms are superfunctions of the odd tangent bundle, the “problem” is that

a) this regards forms as $ℤ_{2}$ instead of a $ℤ$ -graded (at least with the usual meaning of super, which one should stick to), which is not what I need

b) in the generalization of my application to superspaces, I actually need superforms on superspaces.

So this proposal is not in fact resolving the tension between my application and your philosophy.

sometimes it seems that you want to use $Ω^{•}$ for ordinary spaces.

Always! In all the latest notes that I kept posting, in particular.

The idea of using the interpretation of forms as superfunctions to reconcile them with your point of view I mentioned in a single blog comment. (Should have told you that I discarded it shortly afterwards.)

this may mean that we need Euclidean spaces of arbitrary dimension but I have a sneaky suspicion that it all gets determined by its effect on lines anyway.

But that can’t be. The classifying space of 2-forms has a single point and a single curve, but infinitely many surfaces.

The classifying space of 4634-forms has a single $k$ -simplex for all $k < 4634$ , even! :-)

Posted by: Urs Schreiber on April 24, 2008 2:20 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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I besiqued:

this may mean that we need Euclidean spaces of arbitrary dimension but I have a sneaky suspicion that it all gets determined by its effect on lines anyway.

Urs risked:

But that can’t be. The classifying space of $2$ -forms has a single point and a single curve, but infinitely many surfaces.

The classifying space of $4634$ -forms has a single $k$ -simplex for all $k < 4634$ , even! :-)

But that’s precisely my point! You talk of a classifying space. A space consists of two things: the underlying set and the smooth structure. The former is completely determined by all maps from points to it and the latter is completely determined by all maps from lines to it. There’s a difference between a space and its simplicial decomposition.

The difficulty lies in the fact that when we talk of $B$ being a classifying space for something then we don’t mean that there is a bijection between “things classified by $B$ ” and “morphisms into $B$ ” but rather we divide each side by an equivalence relation. One could ask: what are the things classified by actual morphisms into $B$ ? but that’s a side issue.

So when we say “the sheaf $X$ is represented by the classifying space $B X$ ” then what we actually mean is that the sheaf $X$ , defined on the homotopy category of smooth manifolds, lifts to a quasi-representable sheaf on the original category of smooth manifolds. This means that we can extend the definition of the original sheaf to all smooth spaces.

An example may illustrate my point. We think of $K$ -theory as being constructed from equivalence classes of differences of vector bundles. That is actually only true for compact spaces. For non-compact spaces, we define $K$ -theory as the set of homotopy classes of maps into $(ℤ \times) B U$ . That is, knowing that there is a classifying space allows us to extend the definition of $K$ -theory beyond the context in which the naïve definition is valid.

There’s still a part of the story that I’m missing, I think. Why do you want to treat $Ω^{•}$ as if it were a smooth manifold? I can see why one might wish to extend $Ω^{•}$ to smooth spaces rather than smooth manifolds. Depending on how one sets up the categories of smooth spaces this may or may not be easy. But in what way do you want to treat it itself as a smooth manifold?

I still see a distinction between “smooth spaces” and “things one might do to smooth spaces” but I get the feeling that for others then this distinction is at best blurred and at worst non-existent.

That’s why I want to say that Frölicher spaces are smooth spaces and that all the rest are things that can be done to smooth spaces.

Let me illustrate again with an example. Take Chen’s definition of forms on a Chen space. It is a very neat definition and easily extends to more the more general sheaf setting. But in the case of a quasi-representable sheaf, i.e. a Chen space, one really wants to know that one is computing something of the underlying topological space (yes, I know that a topology is not part of the data of a Chen space but nevertheless one can always topologise afterwards). Chen wanted to show that

H (Ω^{•} (X)) ≅ H^{•} (X)

I don’t know too much about the technical details of his proof of this; I’ve looked at the one championed by Jones, Getzler, and Petrak using spectral sequences. But one way one could attempt to prove this would be to argue as follows:

The functor $H (Ω^{•} (-))$ is a cohomology theory on the category of smooth spaces.
It is representable therein.
Its representing spaces are equivalent to those for ordinary cohomology (more precisely, they map down to those for ordinary cohomology under the functor from smooth spaces to topological spaces).

And now, in other news,

I reversied:

At this point, you (Urs) usually start going on about $Ω^{•}$ .

to which Urs othelloed:

At which point you (Andrew) usually start to stubbornly refuse following me ;-)

Yes, and I feel mildly apologetic for that. It’s not that I don’t want to follow; just that I’m a bit hesitant about going off on a journey without a map. I sometimes feel as though I’m following a breadcrumb trail that’s already been eaten by the crows.

I’m afraid I’m still learning some of the language that you all use and so get easily confused (imagine, trying to learn category theory and norwegian. Hmm, maybe I should combine the two. Let’s see, “adjungert” is fairly easy, “omegn” is a little more obscure – though that’s more of a topological notion that categorical – while I wonder if anyone would like to tell me the properties of the category of “mengder”?)

Then I goed:

This is the hom functor if we work with superspaces.

To which Urs chessed:

Right, this is the argument I made a while ago. I was trying hard to fit my example into your philosophy and noticed that maybe that’s the solution. But now I think it isn’t the solution.

Ah, right. Okay, I’ll ignore that.

Don’t get me wrong. I like the idea of saturating with respect to something other than the hom functor and I think that this is an interesting generalisation of Frölicher spaces. However, my argument given over in Space and Quantity still applies so that if one has a separator for the hom-like functor then there is still a sort of “underlying set” controlling the behaviour.

Interesting technical note: I tried cut-and-pasting Urs’ comments which, as they contained a bit of mathematics, didn’t work perfectly. For example, $Ω^{•}$ copied over as Ω $•$ . Rather than write out \Omega^\bullet again, I just inserted the dollar and sup signs. This didn’t work! The \bullet character came back as an ‘Unknown character’. I guess the markdown+iTeX filter is not idempotent.

Posted by: Andrew Stacey on April 25, 2008 9:45 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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A space consists of two things: the underlying set and the smooth structure.

Wait. We agreed that a “space” does not have to have an underlying set. Only an “extremely nice space” has!

We think of K-theory

No – we think of differential K-theory! :-)

The space that I am denoting $S (CE (b^{n - 1} u (1)))$ is the classifying space for $n$ -forms, in that a map from $U$ into is is precisely an $n$ -form on $U$ .

This is the classifying space of ordinary differential cohomology in degree $(n + 1)$ , in the sector where the corresponding integral class is trivial.

In this smooth setting, classifying maps are not defined up to homotopy, but up to thin homotopy.

Why do you want to treat $Ω^{•}$ as if it were a smooth manifold? I can see why one might wish to extend $Ω^{•}$ to smooth spaces rather than smooth manifolds.

I am not sure what comment of mine this is referring to. I am not thinking of $Ω^{•}$ as a manifold. It is a sheaf, and hence a “pretty nice space”, but not a manifold.

(Though the main work of Getzler and Henriques, for instance, was concerned with cutting down on $Ω^{•}$ such that it does become a manifold, or a Banach space.)

a journey without a map

Okay, you’d do me a favor if you expanded on in which respect the stuff I kept writing fails to serve as a map. Then I can try to improve on it.

The essential part of the map was laid out in On Lie $N$ -tegration and rational homotopy theory.

The journey along the road on that map is then described, for instance, from section 6.3 on and then through section 7 in On nonabelian differential cohomology.

As every text I’ll ever write, this is in a state of imperfection. But if you could give me more details on where you find too many bredcrumbs removed by crows, I’d might have a better chance replacing them (by stones, maybe).

Posted by: Urs Schreiber on April 25, 2008 12:42 PM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Urs jengaed:

Wait. We agreed that a “space” does not have to have an underlying set. Only an “extremely nice space” has!

and then later battleshiped:

I am not thinking of $Ω^{•}$ as a manifold. It is a sheaf, and hence a “pretty nice space”, but not a manifold.

Here’s where we have to be careful on terminology. By talking of “spaces”, “nice spaces”, “extremely nice spaces” and so forth then I want to treat them all as fundamentally the same type of thing. A bit like talking of “topological spaces”, “Hausdorff spaces”, and “normal spaces”. When saying that $Ω^{•}$ is a “pretty nice space” then that says to me that you want to see how far you can treat it as an “extremely nice space”; you know that you might not be able to do so completely but you’d like to see how far you can get.

That’s why I want to separate out the “things” from the “things you do to things”. May be it is clear to everyone else, but at the moment this is still confusing me. Perhaps also different people here have different agenda for these objects and I’m picking up on those differences. This is not intended as a criticism! Simply that I sometimes find it hard to keep up.

Urs buckarooed:

But if you could give me more details on where you find too many bre[a]dcrumbs removed by crows, I’d might have a better chance replacing them (by stones, maybe).

It is probably more my failing than yours that I feel as though I am on a journey without a map. To a certain extent, I am mainly interested in one reasonably small part of your journey. You have maps of the whole thing, but they are covered with strange symbols and runes that – for the moment – I have trouble decoding. I was hoping that I could manage this part without needing to study the whole map. Perhaps that was overoptimistic of me. It would probably be useful to both of us if I try to understand the whole map so I shall make the attempt.

That’s not to say that I’m not interested in the whole journey, but this one area is one where I thought I could make a contribution so my original intention was simply to sort out the smooth spaces and then sit back and watch the rest from the sidelines – having plenty of other things to do! But “the best laid plans” and all that!

So, when I have a moment, I’ll take a much more careful look at the two documents you mention above and see what I can make of it.

By the way, some animals eat stones too, aids their digestion it seems, so perhaps small lumps of arsenic? Which reminds me of my fascinating norwegian discovery of this week: the word for “poison” is the same as that for “married”!

Posted by: Andrew Stacey on April 25, 2008 2:59 PM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Andrew,

I should probably pave the road for our interaction a bit better by going more into how I am headed towards Dirac operators on loop spaces.

The last sections of my notes are indicating how we construct String-2-bundles with connection from lifts of Spin-bundles when a certain obstructoin vanishes.

Transgressing these to loop space yields Spin bundles on loop space. Before long, I want to understand the 2-Dirac operators down on base space and the loop space Dirac operators up on the transgressed thing on loop space better.

Then I need your help! I’ll draw a map for you for where to find me.

But now I need to hurry up to check in and then get my flight back to the Old World.

Posted by: Urs Schreiber on April 25, 2008 7:51 PM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Andrew wrote:

By the way, I believe every open convex subset of $ℝ^{n}$ is diffeomorphic to $ℝ^{n}$ …

Can you expand on that diffeomorphism a little? Certainly they are homeomorphic but the obvious homeomorphism isn’t always a diffeomorphism.

This is one of those things where coming up with explicit formulas can be rather tricky.. the basic idea is that while the boundary of a convex open set can be very jagged and un-smooth, it’s not actually in the open set, so we can indefinitely ‘postpone’ the onset of un-smoothness when we start building a diffeomorphism between this set and an open ball, starting from the inside and working our way out.

If that’s too vague: pick a point $p$ in your convex open set $C$ and let $d (x)$ be the distance you can march in the $x$ direction ( $x$ any unit vector) starting from $p$ until you hit the boundary of $C$ . I hope your ‘obvious’ homeomorphism between $C$ and the open unit ball is the one that maps $p$ to the origin and sends $q \in C$ to something like

$(q - p) / d (x)$

where $x$ is the unit vector in the direction $q - p$ . But this involves dividing by $d (x)$ , which isn’t a smooth function of $x$ , so as you note, this ‘obvious’ homeomorphism isn’t smooth.

However, instead of dividing by $d (x)$ , you can do something similar but subtler, which is smooth as a function of $q$ , but approaches ‘dividing by $d (x)$ ’ as $q$ approaches the boundary of $C$ .

I think I could come up with a formula that does the job, if I were being paid a bit more…

However, as you point out, this whole business is not really necessary for getting

$QSh (open convex subsets) ≃ QSh (Euclidean spaces)$

Posted by: John Baez on April 22, 2008 3:00 PM | Permalink | Reply to this

Re: Comparative Smootheology, II

Yes, of course. I should have thought of this since it basically relies on the fact that we can approximate a continuous function by a strict monotone sequence of smooth functions. In other words, we can find a sequence of “smooth shells” which converge to the boundary of our convex set in such a way that each is properly contained in the hull of the next. As they converge, every point in the interior of the convex set is eventually within a shell. We then define a diffeomorphism of the convex set and the ambient Euclidean space by sending the nth shell to the sphere of radius n, and smoothly interpolating between the shells.

One could call this the “infinite onion” method.

Posted by: Andrew Stacey on April 23, 2008 8:26 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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I have another prejudice about what is “right” here:

I think among all the various choices of test domains which we are discussing, the “right” one is simply: Euclidean spaces.

I mean: not subsets of those, but complete Euclidean spaces, such that our site is $Obj (S) = ℕ$ $S (n, m) = SmoothManifolds (ℝ^{n}, ℝ^{m}) .$

I came to this conviction for two reasons:

first when Todd pointed me to Moerdijk&Reyes: for precisely this choice of $S$ (but none of the other choices which involve subsets) do we have a beautiful theory of duality between “spaces” (contravariant functors on $S$ ) and “quantities” (covariant functors on $S$ ) with the latter behaving very (very) nicely as algebras of functions (if monoidal).

second: I very much have super on my mind these days. I claim that all you ever do with Chen-like smooth spaces generalizes in a very smooth, very seamless, very autopilot, very satisfactory way if you replace in the above Euclidean spaces simply with super-Euclidean spaces $Obj (S) = ℕ \times ℕ$ $S (n ∣ n', m ∣ m') = SmoothSuperManifolds (ℝ^{n ∣ n'}, ℝ^{m ∣ m'}) .$

(Well, point 2 was in effect also pointed out to me by Todd, of course.)

My picture of the world of smootheology, at the moment, is this realization of Lawvere’s picture in “Taking categories seriously”:

Spaces are presheaves over $S$ , with $S$ as above.

Rather nice spaces are sheaves over $S$ .

Pretty nice spaces are sheaves over $S$ which are weakly self-dual with respect to dualization with respect to $Ω^{•}$ .

Very nice spaces are quasi-representable sheaves in $S$ .

Extremely nice spaces are Isbell-self dual sheaves on $S$ (Frölicher sheaves).

(Terminology roughly as in Space and Quantity.)

Posted by: Urs Schreiber on April 20, 2008 4:26 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Andrew wrote:

I got one of these, at least. My “Early Chen Spaces” are the 1975a definition. But you’re right, my “Chen Spaces” are not on the list. Whoops. However, I think that one can simply delete the word “closed” from my definition of a Chen space to get the 1977 definition and this would not require any other changes to the mathematics. I’ll have to check that, of course, but I’m reasonably confident. The other definitions will require a little thought.

I’m glad we’re straightening things out!

For me, at least, it’s much less important that you discuss all Chen’s early definitions than that you treat a definition that precisely matches — in substance, if not in presentation — the final polished definition given in his 1977 and 1986 papers. The reason is that I want to use results from your paper in my own work on Chen spaces! And, I’d hate to need a footnote saying “even though his definition of Chen spaces is different from Chen’s, the proofs carry over.”

So, if it doesn’t cause trouble, I’d love for you to delete the word ‘closed’ from your definition of Chen spaces.

I’ve checked that your two adjunctions between Chen spaces and diffeological spaces still work when I do this… but don’t trust me, please check for yourself! (I’ve also checked that ${Ch}^{♯}$ is a one-sided inverse to $So$ , but I haven’t yet checked ${Ch}^{♭}$ .)

Perhaps in the main flow of the paper it would be best to concentrate on the last definition and then have a separate section comparing all the different variants of Chen space.

That sounds good! For all but the most extreme fans of deviant Chen spaces, this will be clearer and more useful.

I’ve got more to say but my wife and I can’t use the internet at the same time in this apartment in Shanghai, and I’m getting in trouble for hogging it right now…

Posted by: John Baez on April 19, 2008 5:46 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

John monopolied:

So, if it doesn’t cause trouble, I’d love for you to delete the word ‘closed’ from your definition of Chen spaces.

No trouble at all!

bzr checkout bm:papers/smthcat/main smthcat
cd smthcat
emacs smthcat.tex
M-x replace-regexp
closed,? ?

C-x C-s
C-x C-c
bzr commit -m "Removed word 'closed' as requested by John Baez"

Ta-da!

Seriously, though, I’d like this paper to be both accurate and useful. Getting Chen’s definition wrong is a demerit on both accounts and one I’m happy to correct. This paper is still definitely in the beta stage and so as you (John) have said that you want to use the results of this paper, that gives you the privileged status of a beta tester and ignoring suggestions from beta testers is a bit like … ow, who put that hole in my foot?

Ho hum, off to learn some norwegian now. I’ll return to this tomorrow.

Posted by: Andrew Stacey on April 22, 2008 10:54 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Okay, she let me back online.

Andrew wrote:

John wrote:

I’m actually interested in showing that Chen’s 1977 category is not equivalent to “Souriau”, but I’ll take whatever words of advice you can offer!

I think that they are not equivalent. Let’s see if we can prove this.

It looks like you proved it! I’m confused about this step:

Let $U$ be an open convex subset of some Euclidean space. We can give this a canonical Chen structure and a canonical Souriau structure; both of which are characterised by the fact that they contain the identity map. As $G F$ and $F G$ are the identity functors, we see that the identity map $∣ U ∣ \to ∣ U ∣$ is contained in all of

$S (U, F G (U)), S (F G (U), U); C (U, G F (U)), C (G F (U), U)$

so we deduce that, with absolutely horrendous notation, $G (U) = U$ and $F (U) = U$ .

but a few days ago I thought I was able to carry out this step in a slightly different way. (Now I forget how.)

Anyway, for the same sort of self-interested reason as above, I’d be delighted if you included this result in your paper.

(Indeed, in the unlikely event that you’re short of things to do, it would be nice to know that all the categories of smooth spaces you study are inequivalent!)

Posted by: John Baez on April 19, 2008 10:22 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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John, this has been bugging me too. I haven’t commented on it until now as I haven’t had anything to add. Perhaps I ought to have mentioned that I was thinking about it, but I abhor spurious comments.

You are right. It is not so simple as I thought. But I think I now know how to do it. In brief, the key is to note the special properties of $ℝ$ in each category. That all “interesting” functors were set-preserving came from the fact that the underlying-set functor was related to a categorically determinable object – a terminal object. To show that all “interesting” functors preserve higher structure, we need to show that $ℝ$ can be categorically determined.

(One of the problems of this method of communication is that I cannot see whether you see what I’m getting at. I want to add a “If you see what I’m getting at, turn to page 183; otherwise, turn to page 23.”, a bit like those daft make-your-own-story books that were around when I were a young’un.)

So, if you see what I’m getting at skip the next paragraph or two; otherwise, read on.

What I mean is that suppose a manical alien hands you one of these categories, possibly in a slightly warped form, and challenges you to find $ℝ$ using only categorical tools. Can you find it? (Yes, we can! Err, I think so.)

We can certainly identify all those objects whose underlying set is (isomorphic to) $ℝ$ since the underlying set functor is equivalent to evaluating the hom-functor on a terminal object. Amongst these, can we find $ℝ$ ?

The one thing you cannot use is the set of plots (or coplots if using Smith or Sikorski spaces). In terms of the category, these are given by evaluating the hom-functor on – you guessed it – $ℝ$ (and similar). I enjoy a good circular argument as much as the next gibbon, but as not everyone is as loopy as me, I’d better use more traditional techniques.

What we can use is that any object with underlying set $ℝ$ determines a submonoid of $Map (ℝ, ℝ)$ via the natural inclusion

Hom (X, X) \to Set (∣ X ∣, ∣ X ∣)

I think that $ℝ$ is characterised by the fact that the image of this map is precisely $C^{∞} (ℝ, ℝ)$ . There might be a few other conditions as well, involving limits or colimits (depending on whether we are mapping in or mapping out of our test spaces), but the above is the key property.

As I said in another comment, I’m currently revising the paper and will include the details of this in there. I’ll treat the general case as this won’t involve much more than the specific of Chen-Souriau.

Posted by: Andrew Stacey on May 6, 2008 9:08 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Here’s another question. Some of you category mavens might want to take a crack at this.

Andrew considers Souriau’s category of diffeological spaces, which we’re calling $S$ , and Chen’s category of differentiable spaces, which we’re calling $C$ . He constructs a functor

$So : C \to S$

and shows this has a right adjoint

${Ch}^{♯} : S \to C$

He also shows that the composite

$S \overset{{Ch}^{♯}}{\to} C \overset{So}{\to} S$

is equal to the identity.

He later has a corollary saying that ${Ch}^{♯}$ embeds isomorphically $S$ as a reflective full subcategory of $C$ .

I only see how this follows if I also assume that the identity natural transformation

$1 : So {Ch}^{♯} \to 1$

is the counit of the adjunction between $So$ and ${Ch}^{♯}$ . Is this extra assumption really necessary to complete the argument, or not?

This extra assumption is probably true and easy to check; I’m just wondering. It’s only in my relatively old age that I’ve warmed to reflective and coreflective subcategories, and I could still be missing plenty of tricks.

By the way, $So$ also has a left adjoint ${Ch}^{♭}$ for which

$S \overset{{Ch}^{♭}}{\to} C \overset{So}{\to} S$

is the identity, and Andrew says this embeds $S$ isomorphically into $C$ as a coreflective full subcategory. My formal question applies here, too, dually — but in this case I’m more loath to figure out the unit

$1 \to So {Ch}^{♭}$

because the functor ${Ch}^{♭}$ is a bit annoying.

(Actually, it’s just been a long day — I’ve been trying to reprove everything Andrew showed about Chen spaces and diffeological spaces, making sure I understand what’s going on, and it was fun at first but now I’m getting tired.)

Posted by: John Baez on April 19, 2008 10:51 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Hi John! I think you can pretty much do this with abstract nonsense.

There are two little lemmas about adjunctions that work well together in situations like this. The first is Lemma A.1.1 in Peter Johnstone’s elephant book. I was surprised when I heard about it:

Lemma 1. Let $F : C \to D$ be a functor having a right adjoint $G$ . If there is any natural isomorphism between the composite $F G$ and the identity functor on $D$ , then the counit of the adjunction is an isomorphism.

Proof: One can transport the comonad structure on $F G$ across the isomorphism, to obtain a comonad structure on $1_{D}$ . But the monoid of natural endomorphisms of the identity functor on any category is commutative, so the counit and comultiplication of this comonad must be inverse isomorphisms. Transporting back again, the counit of $(F ⊣ G)$ is an isomorphism.

I bet you already know the second one. It’s Theorem IV.3.1 in Mac Lane’s book.

Lemma 2. Let $G : D \to C$ be a functor having a left adjoint $F$ . The counit of this adjunction is invertible if and only if the functor $G$ is full and faithful.

Proof: Consider the composite

(1)

D (a, b) \overset{G}{\to} C (G a, G b) ≅ D (F G a, b),

where the isomorphism comes from the adjunction. This natural transformation is the image, under the Yoneda embedding, of the counit of the adjunction; and clearly it’s invertible just when the first part is invertible, i.e. when $G$ is full and faithful!

So, the end result is that, given an adjunction $F ⊣ G$ , we have $F G ≅ 1$ iff the counit is invertible iff $G$ is full and faithful. Of course everything dualises too, so $G F ≅ 1$ iff the unit is invertible iff $F$ is full and faithful.

Posted by: Robin on April 20, 2008 4:20 PM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Robin wrote:

Lemma 1. Let $F : C \to D$ be a functor having a right adjoint $G$ . If there is any natural isomorphism between the composite $F G$ and the identity functor on $D$ , then the counit of the adjunction is an isomorphism.

Wow — that’s incredible: it seems too much too hope for! Just because somebody is an isomorphism, the guy we care about is an isomorphism. How often does that happen?

But now I think I’d vaguely remembered someone discussing this — either here or on the category theory mailing list. Maybe you. Maybe that’s why I had the gall to hope for such a wonderful result.

Thanks a million!

Posted by: John Baez on April 21, 2008 4:42 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

Urs wrote:

Since he never seems to actually mention the very word “sheaf” it might be that he wasn’t aware of the concept (could that be?) and gradually “discovered” it himself.

That’s my impression too. It would be very interesting to know if someone ever told Chen he was studying sheaves, and if so, what his reaction was.

I imagine someone must have told him when he gave his talk at Lawvere and Schanuel’s 1982 conference on continuum mechanics and synthetic differential geometry at SUNY Buffalo, the talk that was written up here:

K.-T. Chen, On differentiable spaces, Categories in Continuum Physics, Lecture Notes in Math. 1174, Springer, Berlin, (1986), 38–42

After all, in this paper he considers generalized Chen spaces lacking the property that makes them have an underlying set. I believe these are simply sheaves, not ‘concrete’ or ‘quasirepresentable’ sheaves.

Alas, as Jim Stasheff told me, “sadly we lost Chen back in 1987 when he was just beginning to be appreciated.”

Posted by: John Baez on April 20, 2008 3:58 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

I’m back online again and with a minute or two to spare so I’ll try to respond to the various points raised in this post. I intend to stick to one point per comment so I may post a few different comments; this is so that the ensuing discussion is properly threaded and is certainly not so that I can get the combined comment number for the two Comparative Smootheology posts past the sesquicentury (TeXnichal issues is currently on 151).

However, there’s a couple of minor points that I’d like to make before launching into mathematics and I’ll make them here.

First off, the daft comment (with abject apologies for stooping so low).

John wrote:

Andrew’s reply follows — I’ll use my superpowers to pretend he posted it as a comment here.

Is it a bird? Is it a plane? No, it’s n-man! Faster than an adjoint functor, more powerful than a coequaliser, and able to leap large categories in a single bound.

Secondly, and slightly more seriously, John started this discussion by email because he felt that some of his comments were nitpicky and he didn’t want to offend by airing them in public. To coin a phrase, the only thing worse than being blogged about is not being blogged about.

It reminded me a little of the various referee’s reports that I’ve received on other papers. Of course, there’s always the initial reaction: “What do you mean, the paper isn’t perfect as it is?” but after that, I’d far rather have a detailed report that, at the least, shows that the referee actually read the paper! I’m early enough in this game that I know I don’t know how to write a brilliant paper and so any help anyone can give me is welcome. I’m even up for a spat over the Oxford Comma.

So, nitpick away! Of course, spelling mistakes are probably better pointed out by email than clogging the blogging (and please do email me; Bruce pointed out several such mistakes on the first draft which helped a lot - thanks Bruce), but anything mathematical that there’s a chance someone else might be confused about or interested in is worth blogging about.

(Hmmm, the hosts of this blog may wish to add their own riders to that since although bytes are cheap, bandwidth has a habit of not being so.)

Having gotten that off my chest, I’m ready to weigh in on the mathematics.

Posted by: Andrew Stacey on April 22, 2008 9:48 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Here’s another question that I’m dying to know the answer to, Andrew:

You describe a functor from Souriau’s diffeological spaces to Chen spaces

${Ch}^{♯} : So \to Ch$

and you show that this is not an equivalence of categories, even though it’s full and faithful. So, it must not be essentially surjective.

So: what’s a Chen space that’s not isomorphic to the image of any diffeological space?

(I think I know the answer for your other functor, ${Ch}^{♭}$ : the closed unit interval with its usual Chen structure. But, precisely for this reason, I think ${Ch}^{♯}$ is more interesting! I’m guessing the answer here is: the closed unit interval with a certain goofy Chen structure, where the plots are ‘locally smoothly extendible maps’.)

Posted by: John Baez on April 23, 2008 5:21 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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I almost wrote:

So: what’s a Chen space that’s not isomorphic to the image of any diffeological space under the functor ${Ch}^{♯}$ ?

Oh, never mind! As I guessed, the closed unit interval with ‘locally smoothly extendible maps’ as plots will do. And, the proof is just abstract fiddling, not requiring any substantial thought. I’ll include it in my paper with Alex, as part of a little story about ‘Chen spaces versus diffeological spaces’.

The darn paper is really close to being done…

Posted by: John Baez on April 24, 2008 5:06 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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John wrote that he almost wrote:

So: what’s a Chen space that’s not isomorphic to the image of any diffeological space under the functor ${Ch}^{♯}$ ?

To which he rhetorically replied:

Oh, never mind! As I guessed, the closed unit interval with ‘locally smoothly extendible maps’ as plots will do. And, the proof is just abstract fiddling, not requiring any substantial thought. I’ll include it in my paper with Alex, as part of a little story about ‘Chen spaces versus diffeological spaces’.

Yes, that was the example I would have given had you not gazumped me. Although the proof does not require any deep thought, I think that the example is illustrative. It demonstrates that when comparing Chen and diffeological spaces then there is really no difference in what “smooth” means inside a space but that they have slightly different meanings on the boundary.

Essentially, Chen spaces say that I am allowed to test smoothness on the boundary by testing “half-smoothness”, namely testing with functions that approach the boundary at speed (the identity function on $[0, 1]$ being a prime example). The diffeological approach says that I am only allowed to test “full-smoothness” on the boundary, so I can only test with functions that “rebound” from the boundary (the function $(- 1, 1) \to [0, 1]$ , $t \mapsto t^{2}$ being a prime example). By shrinking the number of allowable test functions you get that certain functions which are not “half-smooth” are “full-smooth”, such as the identity on $[0, 1]$ regarded as a morphism from the usual Chen structure to the “locally extendible” Chen structure.

The darn paper is really close to being done…

Great. I look forward to reading it. Incidentally, I’ll rewrite CS as I’ve indicated elsewhere in this discussion to include the results that you want. Are there any more on your wishlist that fit into the remit of CS?

Posted by: Andrew Stacey on April 25, 2008 10:29 AM | Permalink | Reply to this

Read the post Convenient Categories of Smooth Spaces
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Excerpt: Chen spaces and Souriau's diffeological spaces are two great contexts for differential geometry. Alex Hoffnung and his thesis advisor just wrote a paper studying these in detail.
Tracked: May 17, 2008 4:27 AM

Read the post Spivak on Derived Manifolds
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Excerpt: David Spivak has an interesting thesis on 'derived differential geometry'.
Tracked: August 19, 2008 7:17 PM

Re: Comparative Smootheology, II

From the arXiv today:

0808.2996

has a definition of a “differentiable space” that may be of mild interest. In terms of the definitions that we know, it’s closest to Sikorski spaces; that is, it takes the “maps out” point of view, the spaces have to be topological spaces, and the functions have to satisfy a sheaf condition. However, it doesn’t have the algebra condition or the post-composition by smooth functions. What it does have is local models for the space: every point in the space must have an open neighbourhood isomorphic to one of a family of local models.

These local models are “locally closed subspaces of Euclidean spaces”. I’d never thought about these before (had to look up the definition on Wikipedia!) and it seems to me as if one can get quite nasty looking sets this way so I’m curious as to what benefit these local models supply.

Looking at the references, the authors of this paper have written a SLN on their notion of differentiable spaces (no. 1824, year 2003). I guess someone should take a look at this and see what they say.

Given that they’d written an SLN I’m a bit surprised that no one has mentioned them before. Anyone willing to own up to having heard of this notion of differentiable space?

Posted by: Andrew Stacey on August 25, 2008 8:43 AM | Permalink | Reply to this

Re: Comparative Smootheology, II

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Hi Andrew,

thanks for the pointer. I feel too busy at the moment to have a close look at this but am certainly interested.

Similarly concerning higher versions of this, as mentioned here, where function algebras are replaced by “higher function algebras” of sorts.

It seems that the important question to find the answer to is this:

what is the relation between

- the homotopy category of concrete presheaves of simplicial sets/ $∞$ -categories

on the one hand, generalizing Chen-like diffeological spaces, and

- topological spaces locally equipped with objects in a homotopy category of $∞$ -function algebras

generalizing Sikorski’s and other’s locally ringed spaces?

As I said, we need comparative $∞$ -smootheology. Eventually.

Posted by: Urs Schreiber on August 25, 2008 1:35 PM | Permalink | Reply to this

Read the post Comparative Smootheology, III
Weblog: The n-Category Café
Excerpt: The third episode in our continuing comparison of various frameworks for differential geometry.
Tracked: September 3, 2008 8:46 PM

Read the post Bär on Fiber Integration in Differential Cohomology
Weblog: The n-Category Café
Excerpt: On fiber integration in differential cohomology and the notion of generalized smooth spaces used for that.
Tracked: November 26, 2008 7:59 AM

Re: Comparative Smootheology, II

Anders Kock has just made available a ‘beta version’ of his new book, Synthetic Geometry of Manifolds, a successor to his classic Synthetic Differential Geometry (also available via that link). Inspiring stuff!

Posted by: Tom Leinster on March 2, 2009 9:28 AM | Permalink | Reply to this

Cross Modules and Non-Abelian 2-Forms: Cubes vs Simplices

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Thanks for posting the link.

I’ve always wished I could understand the relation between Urs’ stuff and Kock’s stuff and ultimately relate this to our (Urs and my) stuff. My gut has always told me they should all be part of a big pretty picture.

The first thing I did was to search for “Schreiber” in Kock’s book, which took me to Section 3.9 (page 130). There, he seems to say that there are two possible approaches to describing crossed modules (whatever those are): one utilizing cubical forms and another utilizing simplicial forms. He says:

The notion of crossed module may seem somewhat ad hoc, but the category of crossed modules is equivalent to some other categories, whose description are conceptually simpler: the category of group objects in the category of groupoids; the category of groupoid objects in the category of groups; the category of 2-groupoids with only one object (a 2-groupoid is a 2-category where all arrows and also all 2-cells are invertible); or the category of “edge symmetric double groupoids with connections” [8], [9]. The latter description is particular well suited for being lifted to higher dimensions, and for the theory of cubical differential forms, and higher connections, cf. [55] and [56]; however, for the purpose of describing a theory of non-abelian 2-forms, the crossed module description is sufficient, and the one most readily adapted for concrete calculations. So we shall adopt this version (following in this respect [2] and [103]); we shall consider differential forms in their simplicial manifestation.

[55] A. Kock, Infinitesimal cubical structure, and higher connections, arXiv:0705.4406[math.CT]

[56] A. Kock, Combinatorial differential forms - cubical formulation, Applied Categorical Structures 2008

[2] J. Baez and U. Schreiber, Higher Gauge Theory, arXiv:math/0511710v2 [math.DG], 2006.

[103] U. Schreiber and K. Waldorf, Smooth Functors vs. Differential Forms, arXiv:0802.0663v2[mathDG]

Then the story gets more interesting. In the intro to [55] he says:

This research was partly triggered by some questions which Urs Schreiber posed me in 2005; for n = 1, an attempt of an answer was provided in my [14]. I want to thank him for the impetus. I also want to thank Ronnie Brown for having for many years persuaded me to think strictly and cubically. Finally, I want to thank Marco Grandis for useful conversations on cubical and other issues.

Four people that I wish I could understand appearing in the same paragraph! :)

One of the things I learned from Urs as we worked through our stuff was that cubes (diamonds actually) are more appropriate for modeling physical spacetimes. I came away convinced that simplices were somehow less desirable for physics. I still think so. If you don’t care about physics and are interested in the pure mathematics, then the choice between simplices or cubes/diamonds may be moot (or maybe not), but I was hoping someone could help me understand what the choice entails.

I’d guess that simplicices and cubes are in many ways “equivalent” as far as their mathematical properties, but I can also guess that each would represent certain computational advantages depending on what you are interested in. In what cases is it better to work with simplices? In what cases is it better to work with cubes? In what cases does it not matter?

Thanks for any words of wisdom that might help get a high level understanding.

Posted by: Eric on March 2, 2009 4:30 PM | Permalink | Reply to this

Re: Cross Modules and Non-Abelian 2-Forms: Cubes vs Simplices

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[…] they should all be part of a big pretty picture.

Here is what I understand of the big picture:

The general topic is that of “generalized smooth spaces” in the sense that these are objects $X$

a) which can be probed by throwing “smooth test spaces” into them;

b) such that there are smooth homotopies between different ways of throwing a given smooth test space $U$ into $X$ , so that the collection of all ways of throwing $U$ into $X$ forms a higher groupoid.

These two conditions are formalized by saying

1) $X$ is a presheaf on smooth test spaces;

2) this sheaf takes values in $∞$ -groupoids.

together with a consistency condition:

c) so that the interpretation of $X$ as a generalized space probed by $U$ s is consisten

which in turn is formalized by saying

3) throwing equivalent objects into $X$ must yield equivalent results

or in more esoteric language designed to hide a simple idea:

3) $X$ satisfies descent and hence is a smooth $∞$ -stack.

Within this general picture there are various variations possible, notably concerning the choice of the collection of “smooth test spaces”.

The focus of “synthetic differential geometry” is on such choices of “smooth test objects” which contain “infinitesimally extended spaces”. This pretty much always boils down to regarding certain algebras as dual incarnations of smooth test spaces, and regarding the algebra free on a single generator that squares to 0 as a dual model for the smooth test space that looks like the infinitesimally extended interval.

What Anders Kock does in his book and in a long series of articles that he published is to develop a language carefully (but naturally) designed such, roughly, all of the intuitive statements in differential geometry which involve infinitesimal objects can be stated in the way that you’d expect them to be stated intuitively, while still making fully rigorous sense as statements about presheaves on “smooth test objects”.

It’s a bit like having a very intuitive graphical user interface to a huge and highly complex supercomputer.

On the other hand, much of the stuff concerning generalized smooth spaces that we have been discussing around the Café invokes a less high-powered machine in the background, notably in that it does not require that there are concrete infinitesimal smooth test spaces.

So far this concerns choices regarding points a) and 1) above. In principle the choice of technical realization of points b) and 2) above is pretty much independent of the choice for a) and 1). But then, when concretely implementing all these things, some choices seem to pair more naturally than others.

Usually the central construction in this context which pairs the choice of realization of item 1) with that of item 2) is the assigment to each smooth test space of its higher groupoid of paths:

$Π : U \mapsto Π (U) .$

If one wants to talk about higher bundles with connection, higher nonabelian differential cohomology, etc, much goes through by abstract nonsense, but the realization of this functor $Π$ is sensitive to the concrete technical implementation.

So then it matters what you regard as a convenient and useful way to draw higher-dimensional smooth paths onto your smooth test space $U$ . If you think this is most naturally done cubically, by drawing higher dimensional cubes on $U$ , then chances are that you’ll find it convenient take a cubical model for $∞$ -groupoids. If you think that when working with higher structures there is no reason ever not to use simplicies, you’ll use those.

There need not be any absolute preference here. On the other hand, things may change when we pass from considering just generalized smooth spaces to generalized smooth spaces equipped additionally with some extra structure. Such as that of having a lightcone structure.

Posted by: Urs Schreiber on March 2, 2009 5:58 PM | Permalink | Reply to this

Re: Cross Modules and Non-Abelian 2-Forms: Cubes vs Simplices

Does Kock really say:

The notion of crossed module may seem somewhat ad hoc,

Nothing like ignoring the history and the fact that crossed modules occurred quite naturally in a specific problem!

Posted by: jim stasheff on March 2, 2009 9:50 PM | Permalink | Reply to this

Re: Cross Modules and Non-Abelian 2-Forms: Cubes vs Simplices

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Here is a more complete quote:

3.9 Crossed modules and non-abelian 2-forms

Recall that a crossed module $𝒢$ consists of two groups $H, G$ , together with a group homomorphism $\partial : H \to G$ , and an action (right action $⊢$ , say) of $G$ on $H$ by group homomorphisms, s.t.

1) $\partial : H \to G$ is $G$ -equivariant (takes the $G$ -action $⊢$ on $H$ to the conjugation $G$ -action on $G$ ),

$\partial (h ⊢ g) = g^{- 1} . \partial (h) . g$

for all $h \in H$ and $g \in G$ ;

2) the Peiffer identity

$h^{- 1} . k . h = k ⊢ \partial (h)$

holds for all $h$ and $k$ in $H$ .

A homomorphism of crossed modules is a pair of group homomorphisms, compatible with the $\partial$ s and the actions.

The notion of crossed module may seem somewhat ad hoc…

I interpreted his remark to mean that the fairly convoluted definition he just gave may not seem very inspired, but it is equivalent to other things that are more intuitive. In other words, I don’t think he was saying crossed modules themselves are ad hoc, but rather the definition he just gave might not seem very motivated at first. At least until you see how it relates to other things.

I don’t know enough to judge, but that is what I thought.

Posted by: Eric on March 2, 2009 10:40 PM | Permalink | Reply to this

Re: Cross Modules and Non-Abelian 2-Forms: Cubes vs Simplices

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Jim wrote:

Does Kock really say:

“The notion of crossed module may seem somewhat ad hoc”?

Nothing like ignoring the history and the fact that crossed modules occurred quite naturally in a specific problem!

I agree with Eric — Kock is quite right to say what he said.

He didn’t say the notion was ad hoc. He said it may seem ad hoc. Indeed, the notion does seem ad hoc if you happen to randomly stumble upon the definition in a textbook: two groups, a homomorphism from one to the other, an action of the other on the one, and two mysterious equations.

Of course crossed modules turn out to be very important and fundamental, which becomes utterly obvious when you either study their uses or realize that they’re equivalent to something that can be defined using three words: groups in $Cat$ .

Posted by: John Baez on March 3, 2009 2:55 AM | Permalink | Reply to this

Re: Cross Modules and Non-Abelian 2-Forms: Cubes vs Simplices

I ought to explain that in the period 1965-74 I was examining the potential role of double groupoids in homotopy theory, suggested by examining the proof I had written down of the van Kampen theorem for the fundamental groupoid on a set of base points.

One question was whether there were good examples of double groupoids, and Chris Spencer and I in 1971-3 or so found you could construct them first from normal subgroups of a group and then from crossed modules. The latter were defined by Henry Whitehead in 1946 following on from his earlier work on second relative homotopy groups.

Another key step in 1974 was with Philip Higgins, defining a homotopy double groupoid of a based pair (more symmetric than the traditional relative homotopy group) and allowing multiple compositions in 2 directions, essential for doing “algebraic inverses to subdivision”, a useful step in local-to-global problems, which I learned from Dick Swan in 1958 are central in mathematics.

My preprint web page has a recent entry developing some double examples, algebraically and homotopically.

I have been unable to understand why in general algebraic topologists seem to refuse to admit the existence of such cubical higher homotopy groupoids, and indeed are generally tied down to a one base point approach! Mind you, if you have just two base points in a space X, then the two loop spaces at these get embedded in a bigger object of paths, which I suppose is scary.

I have not been able to recover analogous local-to-global arguments using simplices or globes, and using cubes we could also develop tensor product, internal hom, and higher homotopies. (Although the relation between cubes, globes, simplices and in the groupoid case, discs, is useful in the theory.)

Nobody seems to have tried `quasi-categories’ in a cubical context.

Cubical methods are also used in the concurrency area, as I learned to my surprise at an Aalborg meeting in 1999.

Posted by: Ronnie Brown on March 23, 2009 11:02 PM | Permalink | Reply to this

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