### Comparative Smootheology, II

#### Posted by John Baez

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?

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 $\mathbb{R}^n$, while a diffeological space has plots that are maps into it from *open* subsets of $\mathbb{R}^n$. 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 $\mathbb{R}^n$, 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 $\mathbb{R}^n$ Let $f: K \to \mathbb{R}^m$. Then $f$ maps smooth curves in $K$ to smooth curves in $\mathbb{R}^m$ 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 $\mathbb{R}^n$

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 $\mathbb{R}^n$ Then he gets them embedded in $\mathbb{R}^n$ 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.

## Re: Comparative Smootheology, II

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

usethis 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.

1973So 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.

1975I 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 $\theta_i$ 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.

1977Again, 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.

1986Up to trivial rephrasing, this is the same as the 1977 definition.

You ask:

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

fourdefinitions with certain characteristics: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:

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

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

$|S| \cong \mathbf{S}(\{*\}, S) \cong \mathbf{C}(G(\{*\}), G(S)) \cong |G(S)|$

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 $\mathbf{S}$ and $\mathbf{C}$ 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 $\mathbf{C}(C_1, C_2)$ and $\mathbf{S}(F(C_1), F(C_2))$ are

the samesubset of $Set(|C_1|, |C_2|)$ and similarly for $G$. This means that the compositions $G F$ and $F G$ areexactlythe 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 $\mathbf{C}(U,C)$ 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 convexsubset 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$\mathbf{S}(U, F G(U)), \quad \mathbf{S}(F G(U), U); \quad \mathbf{C}(U, G F(U)), \mathbf{C}(G F(U), U)$

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

convexsets. We therefore have$\mathbf{S}(U,S) = \mathbf{C}(G(U), G(S)) = \mathbf{C}(U, G(S))$

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

$\mathbf{S}(V, S) = \mathbf{C}(G(V), G(S)) = \mathbf{C}(V, G(S))$

where $V$ is given the canonical Chen structure wherein all

inclusionsof convex subsets are plots.Hence the functor $F : \mathbf{C} \to \mathbf{S}$ 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

abstractinterior, not the interior as embedded in some arbitrary $\mathbb{R}^n$. 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