The n-Category Café

p. 4, 4th full paragraph: one of “definitions notions” probably should not be there

p. 6: the itemize list is referred to by numbers of items. Probably you want to make it enumerate

p.6-7: concerning exposition, why not discuss diffeological spaces first, then Chen spaces? The diffeological spaces convey the idea and the problem with convex subsets is then clearly seen to be a technical subtlety that is not the heart of the matter.

p. 7, last item (6): is the embedding $$\mathrm{Top}\hookrightarrow C^{\mathrm{\infty }}$$ full and faithful? How does that functor combine with the one $C^{\mathrm{\infty }}\to \mathrm{Top}$ on p. 8?

p. 9, first full paragraph: “functor that is the identity on maps” = full and faithful (?)

p. 18: concerning the terminology “sections” versus “plots” for the set $X(D)$ assigned by a sheaf to a domain $D$, I think one wants to think “sections” iff the site is one of (open) subsets of some fixed space, while for sites such as those of all open subsets in the world, one thinks “plot”.

There must be a good abstract nonsense way to think of this difference, which I wish I knew. It’s a relevant source of confusing in the context of comparative smootheology: for instance when talking about locally ringed spaces, these are conceived in terms of sheaves on open subsets of a fixed underlying (topological) space. While a sheaf, this is morally the thing of out-maps, of course, and plays an opposite (dual, maybe) role to the kind of sheaves of in-maps that arise in Chen spaces and in diffeology.

p. 19, last sentence of first paragraph: probably one superfluous “underlying”

t8m8r wrote:

It seems that the paper is not accessible by public.

I thank you and Daniel for pointing out this problem, and Urs for cleverly getting around it! I’ve fixed the problem and taken the liberty of deleting your comments (since now they’ll only confuse the poor readers of this blog). Please don’t take offense.

Why, on UNIX, when I copy a file with permissions set to 744 (readable by the whole world), do the permissions get reset to 700 (not readable by the whole world)?

I guess it’s a way to prevent people from accidentally exposing secret information to the world at large… but for me it only means I have to carefully remember to reset the permissions to 744 — which I often forget, on this blog.

I just fixed all the problems Urs has noted so far. Thanks, Urs!

Urs wrote:

p. 9, first full paragraph: “functor that is the identity on maps” = full and faithful (?)

No, these concepts are very different. Fullness and faithfulness are useful properties of functors, invariant under natural isomorphism. ‘Acting as the identity on morphisms’, on the other hand, is a crazy, evil property:

$$F(f)=f$$

It just so happens to be the quickest way to describe what the functor we’re talking about does to morphisms: it leaves them alone!

We’re setting up a functor from Chen spaces to diffeological spaces. So, we need to turn a smooth map between Chen spaces into a smooth map between diffeological spaces. And, we’re doing this by not changing this map at all! Of course this only makes sense because this map is a function between sets — and we’re not changing those sets, either.

I agree that this is a bit confusing to someone who has fully incorporated category theory into their thinking. It’s an old fashioned, set theoretic thing to say.

Anyway, $F(f)=f$ implies the functor $F$ is faithful — but not necessarily full, since there could be new smooth maps between diffeological spaces that didn’t come from maps between Chen spaces.

But, later we prove that this functor is actually full, too!

In fact there are two full and faithful functors

$${\mathrm{Ch}}^{♯},{\mathrm{Ch}}^{♭}:\mathrm{DiffeologicalSpace}\to \mathrm{ChenSpace}$$

that are right and left adjoint, respectively, to a certain functor

$$\mathrm{So}:\mathrm{ChenSpace}\to \mathrm{DiffeologicalSpace}$$

The point of this section is to review Andrew Stacey’s work on these functors. I want it to be an easy place to find everything you might ever need to know about the relation between Chen spaces and diffeological spaces! It’s a bit technical, but it’s the sort of thing we comparative smootheologists wonder about.

Here’s a question I’ve asked before around here but which I don’t think has been answered:

Are any of these kinds of smooth spaces the result of gluing certain local models together along certain equivalence relations?

In other words, is there a category $M$ (=”local models”) with a Grothendieck topology such that our category $S$ of spaces is equivalent to the full subcategory of sheaves on $M$ containing $M$ and closed under quotients by equivalence relations of a certain type $T$. I’m not exactly sure what the key axiomatic properties of $T$ should be, but if it’s imporant, I could probably take a stab at it. Maybe I should add that $M$ should be simpler in some sense than our category $C$ of smooth spaces. Otherwise, you could probably just take $M$ to be $C$ itself!

For example, the category of schemes satisfies this property. So does the category of algebraic spaces.

Another small typo: on p.23, in the last paragraph before section 5, the description of going from an ‘$F$-space’ to an abstract simplicial complexes doesn’t seem quite right.

Given an a.s.c. $(X,K)$, the associated $F$-space should have $$X(n)=\{f:n\to X\mid \mathrm{im}(f)\in K\}$$ (not just $\{$sets in $K$ of size $n\}$), I think—essentially, since $F(m,n)$ includes all maps $m\to n$, not just monos, we have to throw degenerate simplices into $X(n)$.

Hi John and Alex,

I’ve now had time to read through your paper in some detail. Obviously, my main interest is in where it directly relates to my paper (though all of it is interesting) but as I read I wielded my dread red pen and so you may as well reap the benefit.

Let me make it absolutely clear from the outset that I think it is a good paper and easy to read. My comments, particularly the nit-picky ones, are not to be taken as criticisms of the paper overall. Rather, as I read it then whenever I thought “I don’t immediately see what you’re getting at there”, then I marked it. The key is the ‘immediately’! It usually was the case that I figured it out after a moment or two, but for the majority of the paper then I didn’t have to do that so when I did have to pause then it was worth marking. Also, most of my comments are ‘judgement calls’ and I’m being at my most nit-picky because it’s your job to write the paper and if I don’t tell you something that seemed a little odd to me then you don’t have a choice as to whether or not to change it.

Some general comments first.

1. In sections 3 and 5 you do a lot on categorical properties of your categories – section 3 descriptive and section 5 provides the proofs. In the abstract you make it clear that all the category stuff is there for “differential geometers” who, by implication, do not know much category theory. It therefore seems a little odd that you never explicitly say:

   We do all these constructions by doing the construction to the underlying set and then “imposing” a suitable smooth structure on it.

   In section 3 this would just make a little clearer what you are actually doing. However, in section 5 this would simplify things a lot, I think. Basically, “concrete sheaves on a concrete site” is an example of a “topological category over Set” in which case all of these “standard” constructions are formed by doing them for the underlying sets and then imposing a suitable smooth structure.

   I think this approach would save a lot of time and make the categorical stuff easier to read for non-categorists (after all, anyone familiar enough with basic category theory is going to skip most of section 5. The only part of that section that I really paid attention for was 5.2 – mainly since I have yet to answer Todd’s question on local cartesian closedness of Frölicher spaces posed over here and I was looking for inspiration.) Where it would also save a fair bit of effort is with colimits. After all, it is fairly intuitive to see that one can make colimits by forming the colimit of the underlying set and then imposing a suitable smooth structure. On the other hand, the concretisation of sheaffification (how many ‘f’s in that?) is obscure in the least!

   Section 5 is also a little oddly balanced in that you are very detailed in the early sections: Subspaces and Mapping Spaces, but very abstract in the Colimit section.

2. The inclusion of simplicial complexes is a little strange. It’s main role, it seems, is to provide a further example of a category of “concrete sheaves over a concrete site”. Certainly it appears to have little to do with categories of “Smooth Spaces”. However, this is never actually said and, moreover, it is given quite high prominence in the early stages – even getting a mention in the abstract.

3. In relation to my paper, there are a few hang-overs from my mistaken assumption that the convex sets be closed. In particular, the “locally smoothly extendible” functor has changed. Full details will appear in the next draft, but the basic idea is to factor the plots of the Chen space through plots of the original Souriau space. If the convex sets were closed, Whitney’s theorem would give the “locally extendible” description but without the closed assumption, we can no longer describe them in such a fashion. The problem with “locally extendible” (which, after all, makes sense for any choice of subset of ${ℝ}^n$) is that the composition of a locally extendible map with a smooth map need not be locally extendible again.

Now some slightly more focussed comments.

1. Kriegl and Michor’s work. The references to their work are not quite as they should be. In the introduction you imply that they construct a cartesian closed category of manifolds. In fact, a careful read of the introduction of Chapter IX of their book reveals that this is precisely what they don’t do. They have done it, they say, elsewhere but

   Unfortunately they [Kriegl and Michor] found no applications, and even the authors [Kriegl and Michor again] were not courageous enough to pursue them further and to include them in this book.

   What they do do in their book is find a cartesian closed category of smooth linear spaces. The best non-linear ambient cartesian closed category would be that of Frölicher spaces.

   Incidentally, at the end of the first part of section 2 you mention their Theorem 24.5 as part of why life is simpler when the domains are closed. Their theorem makes no assumption on closedness and so applies to all convex sets. This is probably another symptom of my error in thinking that Chen’s domains were closed. What makes things easier in the closed setting is Whitney’s theorem on extensions of smooth maps.

   Incidentally, I don’t know whether Kriegl and Michor are the originators of this theorem or not. They do not attribute it to anyone else in their book, but they may view it as minor enough not to require attribution.

2. Mostow’s contribution. Mostow did not invent his version of “differentiable space”, although he does not make this clear when he gives the definition. On p266 he writes

   The definition which we have given of differentiable space appeared in the work of R. Sikorski [29], but he did not define differential forms in the way we did.

   Incidentally, I have not come across any definition of “differentiable space” which used coplots and which had target spaces other than $ℝ$ for those coplots.

3. Wherefore art thou concrete? After definition 15 you say

   Finally, axiom 3 implies that any Chen space X gives a special kind of sheaf on the site Chen: a ‘concrete’ sheaf, …

   ‘Fraid not. Axiom 3 just says that the ‘underlying set’ functor agrees with the ‘set of points’ functor; in other words, the plots ‘see’ the whole of the set. It says nothing about concreteness. The concreteness is sort-of built in to the definition so is more of an underlying assumption. I don’t think you can separate it out into an axiom without making the whole definition much more abstract.

4. Terminal objects. Is there a reason why the definition of a concrete site requires the terminal object? Couldn’t I just take any separator? I want it to be a terminal object for looking at equivalences of categories of smooth objects, but I’m not sure I want it to be an assumption. For example, if I had an algebraic theory then the underlying set functor is represented by the free gadget on one element whereas the terminal object has underlying set the terminal object. These are rarely the same thing. So what if my site were built out of algebras in some fashion? Could I still have concrete sheaves over it?

Finally, some even more picky points.

1. Introduction.

   Abstract. Bit on the waffly side, I thought. If I were just scanning this on the arXiv I’d’ve probably missed the main point: nice objects versus nice categories.

   Paragraph beginning “Now consider differential geometry”, phrase “$C^{\mathrm{\infty }}(X,Y)$ usually is not”. That’s a bit weak isn’t it! Unless $X$ is $0$-dimensional then it most certainly is not.

2. Smooth Spaces.

   “Souriau’s notion of a ‘diffeological space’”

   Axiom 3 of diffeological space: points in ${ℝ}^n$ are not usually (!) open.

   On Chen spaces: “Here we use his final, most refined approach”. In retrospect, was it really the most refined? That’s a little debatable since his first forcing condition was actually stronger than the sheaf condition.

   On Chen’s evolution. Maybe emphasise the changes a little more, particularly where the topology is concerned. For example, the 1975 definition did indeed drop the Hausdorff condition but it still retained the topology.

   “Pythagorean triples” is this really relevant?

   Paragraph “Situations like this”. You are using the fact that a map into a convex region is smooth if and only if it is smooth into its ambient vector space. As you are going into quite some detail about smoothness between convex sets, this is quite an important point to make. It explains why all the focus is on the domain and not the codomain.

   2.1 Examples. I get a little lost in distinguishing between “smooth” in the usual sense and “smooth” as a morphism between Chen (or diffeological) spaces.

   “with better formal properties”. Would “categorical” be better here? Also, much of the rest of the paper is to justify this sentence so stating it here as a fact downplays the rest of the paper a little.

   With regard to the functor $C^{\mathrm{\infty }}\to \mathrm{Top}$, there is another natural functor given by taking the projective topology for all smooth functions on the Chen space (i.e. morphisms to $ℝ$ with its standard Chen structure). I think that these provide left and right adjunctions to the functor $\mathrm{Top}\to C^{\mathrm{\infty }}$ which assigns to a topological space all its continuous plots.

   2.2 Comparison

   “identity on maps” issue, I guess the point is that the underlying set functor is faithful and so both $\mathrm{So}(f)$ and $f$ have the same underlying set map.

   Last line of paragraph beginning “The embedding ${\mathrm{Ch}}^{♯}$ lacks this defect”. The ${\mathrm{Ch}}^{♯}$ is typeset wrong (italic instead of roman).

3. Convenient Properties.

   Is “strong” the more usual terminology? I’ve been using “extremal” but I’m newish to this game so I don’t know.

4. Smooth Spaces as Generalised Spaces

   “Axiom 1 in the definition of a Chen space”

   Definition 16. The displayed maths in this definition should really be taking place in $\mathrm{Set}$ so there should be lots of $\mathrm{hom}(1,?)$s in it.

   Definition 18. Any chance of modifying the “D space” notation a little? I don’t like the actual space (rather than the word “space”!) in it. Putting in a hyphen or something would make it clearer that I should read “D-space” as one thing.

   Lemma 21. A $C_j\to C$ has snuck in here instead of $U_j\to U$.

   Proposition 25. Err… check axiom 1? (Sorry, couldn’t resist).

5. Convenient Properties

   Proposition 27. In the proof, the “underlying set” functor didn’t typeset so well (bit on the short side).

   Proposition 28. Careful of circular arguments (or being seen to be circular). It would be clearer if, in the proof, you make it clear that the $\Omega$ that you start with is the functor $D\mapsto 2^{underlineD}$ and that you will go on to show that it is a concrete sheaf.

   Definition 31. “We say a morphism of”

   Generally, in this section, the typesetting of your diagrams goes a little skew-if when the arrows have tails or strange heads. Are you using xy.sty? If so, you can add a little spacing here and there to sort this out. I don’t know about other packages.

   After Definition 41. “We now prepare to show”. Yuk! I know you guys are American, but really! You’re not related to Humphrey Appleby are you?

   Grothendieck plus construction. Is there a smallness assumption needed here? Perhaps this is implicit somewhere.

   Chen’s predifferentiable spaces. It’s a bit strong to say that he “described a systematic process”. He gives a one line description of the subsequent plots. On the other hand, his one line description is a lot easier to “see” than the plus construction.

Well, I think that’s enough to be going on with! I ought to get back to improving Comparative Smootheology. Feel free to rip that apart in revenge!

After reading this paper, I pondered Chen’s different definitions again and realised that his middle definition is a sheaf condition except that the site is not what one would expect. I haven’t checked the details (i.e. that this does genuinely give a site). The idea is to take the category of convex subsets (closed if you like) with $C^{\mathrm{\infty }}$ maps as morphisms and declare a family of maps into an object, say $C$, to be a covering if it is deterministic for smooth maps. That is, a map $f:C\to C\prime$ is smooth if and only if $f\theta$ is smooth for all $\theta :C″\to C$ in the family. This ought to define a site. Then the Chen spaces in the sense of the 1975 definition are the quasi-representable (that is, concrete) sheaves on this site.

So Urs’ comment somewhere else was exactly wrong. It wasn’t that Chen was gradually converging on the idea of a sheaf. The concept of a sheaf is there as soon as he realised the need for some sort of forcing condition. Rather, he was unsure as to the correct site. Thus in going from the 1975 to 1977 definitions, he only changed the site – albeit in two ways.

This doesn’t, of course, mean that Chen knew that the original definition was sheafy.

I don’t know whether or not it’s worth pursuing the idea of the “smooth site” any further (for example, to check that it is actually a site). But it’s certainly worth noting that there is an intermediate step in between “q-r sheaves on the usual site of convex sets” and Frölicher spaces which will generalise to a non-set-based theory.

These then lie in between “very nice spaces” and “extremely nice spaces”. “Pretty nice spaces”?

I guess it’s the time-zone that means that I’m the first to notice this paper going on the arXiv. Great!

I’ll read it properly and thoroughly when I next get close to a printer.

One thing I pondered on a casual look-through was your Cantor set example of how bad Chen spaces can be. Actually, I’m not sure that this is a particularly bad space. Presumably we give it the smooth structure as a subset of the real line, whereupon its plots are those plots of the real line which factor through the Cantor set. Seems to me as if all of these should be constant, and thus the Cantor set gets a discrete smooth structure which, apart from its sheer size, is not particularly scary.

Am I missing something here?

Added in proof: The “Great!” in the first paragraph was linked to the appearance of the paper on the arXiv, not to the fact that I was the first to comment on it.