## January 3, 2008

### Comparative Smootheology

#### Posted by Urs Schreiber

Here in the $n$-Café we happen to talk about the various notions of generalized smooth spaces every now and then (last time starting here).

I was dreaming of having, at one point, a survey of the various definitions and their relations in our non-existent wiki. Luckily, while I was just dreaming, Andrew Stacey did it.

Andrew is an expert on the index theorem for Dirac operators on loop spaces (see his list of research articles), and for that work he needs to deal with generalized smooth structures that render loop space a smooth space.

Last time I visited Nils Baas in Trondheim I had the pleasure of talking quite a bit with Andrew. Ever since then I had planned to post something about the intriguing things about loop space Dirac operators he taught me, but never found the time (but see this comment).

Now recently he sent me a link to his new article, which gives a detailed survey of the various definitions of generalized smooth spaces, and a careful and detailed comparison between them:

A. Stacey
Comparative Smootheology
(pdf, arXiv)

Abstract. We compare the different definitions of “the category of smooth objects”.

Of the four or five different definitions he considers, Andrew favors Frölicher spaces, and he explains why.

I am reading this in particular with an eye towards our recent discussion in Transgression of $n$-Transport and $n$-Connections, where I am falling in love with a general definition that Andrew does not discuss explicitly: smooth spaces as general presheaves over the site of manifolds or open subsets of $\mathbb{R} \cup \mathbb{R}^2 \cup \mathbb{R}^3 \cup \cdots$.

The set such a presheaf assigns to any object $U$ of the domain category (manifold or open subset of sorts) is to be thought of as the collection of smooth maps from $U$ into the smooth space thus defined.

The slogan here is

A generalized smooth space (in the sense of presheaves on manifolds) is a space which need not locally look like a manifold, but which may be probed by manifolds.

Chen smooth spaces and/or diffeological spaces are a special case of such presheaves, namely presheaves which are quasi-representable: while not in general representable, these are presheaves $X$ for which there exists a set $X_s$ such that for $U$ any test domain, we have $X(U) \subset \mathrm{Hom}_{Set}(U,X_s)$. Moreover, morphisms of diffeological or Chen-smooth spaces are morphisms of presheaves $X \to Y$ induced by maps $X_s \to X_y$ of these sets.

On the other hand, Frölicher spaces and “differentiable spaces”, as in Mostow’s article, are defined not just by specified smooth maps into them, but also by specified smooth maps out of them.

This is extremely useful for instance for having such a standard concept as the chain rule available for generalized smooth spaces. The chain rule for mere presheaves, as above, is, in contrast, a headache.

Moreover, the slick thing about Frölicher’s definition is that he realized that with maps in and out, it is already sufficient to consider all maps from just $\mathbb{R}^1$ into the generalized smooth space. Hence a Frölicher smooth space is a set together with a specified collection of smooth curves in it, and a specified collection of smooth functions on it, satisfying some compatibility conditions.

I like that. But personally I haven’t quite made up my mind yet.

I am thinking that maybe this is telling us that we eventually may want to consider things that are pairs consisting of a presheaf and a co-presheaf on manifolds, compatible in some way.

Notice that for presheaves on manifolds it is natural to define all contravariant differential geometric constructions, notably differential forms.

While for co-presheaves on manifolds, it is natural to define all covariant differential geometric constructions, notably vector fields.

Then recall one of the points emphasized in Transgression of $n$-transport and $n$-connections: every non-negatively differential graded commutative algebra sits inside the dg-algebra of differential forms on some presheaf-like generalized smooth space.

And then recall from On BV-quantization, Part VIII that we are really looking for a way to generalize this statement to dg-algebras with no restriction on the grading.

So possibly one way to realize this is to consider things that are both presheaves and co-presheaves on manifolds. “Frölicher presheaves”, in a way.

Hm…

Posted at January 3, 2008 10:21 PM UTC

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### Re: Comparative Smootheology

I didn’t put in the “presheaves” version because I didn’t know of it. If someone sends me a precise definition then I’ll add it in (providing I understand it, of course!). Now that, thanks to Urs, I have a copy of Mostow’s paper I’ll add a section on that (again, subject to the proviso of comprehension).

Urs wrote (and I emphasised):

I am thinking that maybe this is telling us that we eventually may want to consider things that are pairs consisting of a presheaf and a co-presheaf on manifolds, compatible in some way.

It’s the compatibility relationship that leads one to Frölicher spaces. I suspect that you would end up with a non-set-based version of Frölicher spaces which I don’t think anyone would have any qualms about.

As I see it, Frölicher’s main insight was this compatibility relation. The fact that he only used curves and functionals (rather than maps into or out of more general spaces) is more of a technicality. Due to useful results such as Boman’s, one can always reduce a more complicated definition to one using only curves and functionals without losing any information (just “bloat” as I refer to it).

The compatibility relationship can be thought of as “If it looks like a duck and quacks like a duck, then it is a duck.”. Without wishing to start a flame war, the other approaches are more in the line of “It is only a duck if it appears in my list of ‘approved ducks’.”.

Andrew

PS And without wishing to start a subthread on a completely irrelevant issue, a chemist I know (rather well) told me of the following version: “If it looks like a duck and quacks like a duck but doesn’t have the NMR of a duck, it ain’t a duck.”

In categorical terms, that would translate to: “If it looks like a duck and quacks like a duck but doesn’t transform like a duck, it ain’t a duck.” though, of course, it may be a 2-duck.

Posted by: Andrew Stacey on January 4, 2008 9:52 AM | Permalink | Reply to this

### Re: Comparative Smootheology

Andrew, the presheaf version Urs is referring to is but one of many of a class of sheaf models for doing smooth analysis. You may have seen some of this already, so I’ll try to keep this brief.

The specific version Urs is looking at is the category of functors

$S^{op} \to Set$

where $S$ is the category of finite-dimensional smooth manifolds and smooth maps between them. He denotes this category of functors by $S^{\infty}$.

Besides simplicity, there are a number of pleasant features of this category. For one thing, this category is a topos, which has many wonderful properties: (local) cartesian closure, completeness and cocompleteness, and nice global exactness conditions. Also, the category of ordinary smooth manifolds $S$ embeds fully and faithfully in $S^{\infty}$ via the Yoneda embedding, viz. the functor sending $M$ to $hom(-, M)$. So in some respects it is a paradisal extension of ordinary manifolds.

But not all respects. Notably, the Yoneda embedding fails to preserve some colimits one would very much like to consider in practice, in particular colimits which arise from gluing patches of an open covering:

$\sum_{i, j} U_i \cap U_j \stackrel{\to}{\to} \sum_i U_i \to M.$

More sophisticated sheaf models would repair such defects by considering not all presheaves on $S$, but sheaves with respect to a Grothendieck topology on $S$ which takes into account such coverings.

The full development of this kind of approach was initiated by Lawvere in the late 60’s. Actually, Lawvere had in mind a development which would incorporate objects of nilpotent infinitesimals (the kind used by Grothendieck in algebraic geometry). The way this works is that one first expands $S$ to a larger category $C$ = $C^{\infty}$-Alg^{op} in which $S$ fully embeds. That is, there is a full and faithful functor

$S \to C^{\infty}-Alg^{op}$

which sends $M$ to the $C^{\infty}$-algebra of smooth functions on $M$. But $C^{\infty}$-Alg, the category of finitely presented (commutative) $C^{\infty}$-algebras, contains other more exotic objects, for example $C^{\infty}(\mathbb{R})/I$ where $I$ is the $C^{\infty}$-ideal generated by $x^2$. The corresponding “smooth affine scheme” in $C^{\infty}-Alg^{op}$ may be regarded as an infinitesimal object consisting of formal elements of $\mathbb{R}$ of square zero. It is the “generic tangent vector” $T$, and one of the beauties is that the tangent bundle of an object $X$ may be constructed as as internal hom $X^T$, which we have available by cartesian closure.

In a nutshell, a typical sheaf model will consider the presheaf topos consisting of functors

$C^{\infty}-Alg \to Set$

and then cut back to sheaves with respect to some topology on $C^{\infty}-Alg^{op}$ to force various other nice things to occur. Some of the fancier versions also faithfully reflect the use of invertible infinitesimals, à la Abraham Robinson in his nonstandard analysis. (I interpret Urs’s choice as just sticking to one of the simplest types of sheaf models, possibly with an option of switching to something fancier once things are a little more settled.)

There are a number of texts which treat this development. The one I am most familiar with is the one by Moerdijk and Reyes, reviewed here.

Posted by: Todd Trimble on January 4, 2008 12:24 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Hi Andrew,

thanks for your comment!

You write:

I didn’t put in the “presheaves” version because I didn’t know of it. If someone sends me a precise definition then I’ll add it in

I had written the following long response, when I saw that Todd had pipped me. I’ll post it nevertherless. Here goes.

A presheaf is not much more than a contravariant functor. We are usually compelled to think of a contravariant functor as a presheaf if

- its domain is a category with the structure of a site: it’s objects have the essential properties which make them behave roughly like subsets of some topological space

- its codomain is the category of sets, or a category of sets with extra structure.

Sometimes people call any contravariant functor a presheaf. And since contravariant functors are in bijection with covariant functors, that pretty much amounts to calling every functor a presheaf.

But the point of presheaves is, as the name suggests, that some of them are in fact sheaves. And that is a condition which only makes sense when the domain is a site.

And another important point of presheaves is that some of them are representable, and this only makes sense when the codomain is a category that the domain is enriched over.

Let me see, I am starting explaining this from the wrong end now, it seems. But the point I am making so far is, to get started, that:

Generally speaking a presheaf is just any functor $f : S^{\mathrm{op}} \to T$, but usually we want the objects of $S$ to behave like open subsets and, unless we feel a little more sophisticated, want $T$ to be the category of sets.

Then the category of all presheaves on $S$ is simply the category of all contravariant functors from $S$ to $Set$. We write this category of presheaves on $S$ as $Set^{S^{op}} \,.$

One important point of this is that the category $S$ itself sits inside the category of all presheaves on $S$:

$Yon : S \hookrightarrow Set^{S^{op}}$

This is called the Yoneda embedding.

This is one of those almost completely tautological and yet immensely deep things that make abstract nonsense such a delight.

For understanding what presheaves mean, let’s quickly look at how Yoneda’s embedding works:

for every object $U \in Obj(S)$ we obtain a presheaf on $S$ by sending any other object $V$ of $S$ to the set of morphisms from $V$ into $U$:

$V \mapsto \mathrm{Hom}_{S}(V,U) \,.$

Notice that this is indeed a contravariant functor, since the Hom-functor is contravariant in its first argument.

The presheaves isomorphic to one of this form are called the representable presheaves. They are represented by the given object $U$ of $S$.

What does this tell us? That’s important. It tells us that we should think of the set that any given presheaf $X : S^{op} \to Set$ assigns to any given object $U \in Obj(S)$ as the set of morphisms from $V$ into $X$. Only that, for presheaves that don’t come from the Yoneda embedding, there is not really anything like a morphism from $V$ into $X$.

So you should think of the presheaf $X : S^{op} \to Set$ as a rule that assigns to any open set $U$ the set of plots from $U$ to $X$.

The fact that $X$ is a contravariant functor on $S$ then says that these plots behave sensibly under pullback along morphisms of objects of $S$.

Moreover, if the presheaf is actually a sheaf, it has the special property that if a bunch of objects $U_1, U_2, \cdots$ of $S$ “cover” an object $V$ in the way open subsets may cover one another, then the plots that $X$ assigns to $V$ are already fixed by the plots it assigns to the $U_1, U_2, \cdots$.

So you can see now how Chen-smooth spaces and diffeological spaces are examples of certain presheaves:

As I have said, presheaves on any category $S$ are “generalized objects of $S$”. More precisely: they are “things that may be probed by objects of $S$”.

So for Smootheology, we take one of the following choices:

- $S$ the category whose objects are ordinary (smooth) manifolds, and whose morphisms are ordinary morphisms of (smooth) manifolds.

or

- $S$ the category of open subsets of $\mathbb{R} \cup \mathbb{R}^2 \cup \cdots$ and smooth maps between these

or

- $S$ the category of open convex subsets of $\mathbb{R} \cup \mathbb{R}^2 \cup \cdots$ and smooth maps between these

Or some variant of that.

Chen-smooth spaces and diffeological spaces are what, it seems, should be called “quasi representable” presheaves on one of these sites, as I indicated in the above entry.

One nice aspect of realizing that Chen-smooth spaces and diffeological spaces are special kinds of presheaves is that we can make use of the immense amount of knowledge about presheaf categories.

Presheaf categories have a bunch of nice properties. They are cartesian closed, for instance. They are even topoi.

Useful for our purposes is the cartesian closedness. It says that the thing of morphisms between any two presheaves on $S$ is itself again a presheaf on $S$.

For $X$ and $Y$ any two presheaves, the presheaf $hom(X,Y)$ is the one whose assignment of sets works as $hom(X,Y) : U \mapsto Hom_{Set^{S^{op}}}( U \times X, Y) \,,$

where on the right $U$ denotes the presheaf represented by $U$, and where the cartesian product $\times$ of presheaves is the componentwise one

$X \times Y : U \mapsto X(U) \times_{Set} Y(U) \,.$

You can easily check that this general – powerfully elegant – notion of internal homs in presheaves restricts to the ordinary internal hom for Chen-smooth spaces and diffeological spaces. I discuss that in a little more detail in my notes.

Posted by: Urs Schreiber on January 4, 2008 1:06 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Thanks, Todd and Urs, for that; someone has finally given me a reason to be interested in (pre)sheaves!

There’s a lot to absorb there and the semester’s starting next week so it’ll take me a while to ponder all of it, and a bit longer to incorporate it into what I wrote.

One thing that strikes me immediately is that in saying why presheaves are so wonderful, you both start listing fantastic properties of the category, starting with completeness, cocompleteness, and closed. If I have an arbitrary category, in this case the category of smooth manifolds, there may be lots of ways to embed this in a category with lots of nice properties, such as those listed above. What makes one such enlargement better than another? This is perhaps not a good mathematical question but I would be interested your answers.

The category of Frölicher spaces is complete, cocomplete, and closed and I suspect that it is the “smallest” such category containing the category of smooth manifolds. Certainly it embeds in all of the set-based categories of smooth objects that I’ve encountered so far.

Perhaps a more leading question would be: why haven’t I managed to convince any of you that Frölicher spaces are worth considering? (I detest smileys, but if I didn’t then I would put one there which indicated a wry smile at that point (is there such a smiley? I don’t think I’ve ever seen one) just to show that I didn’t really expect to convince anyone of anything)

Anyway, as I said, the semester starts next week and I’m teaching functional analysis. Perfect opportunity to promote my point of view before they encounter any other!

Andrew

Posted by: Andrew Stacey on January 4, 2008 2:27 PM | Permalink | Reply to this

### Re: Comparative Smootheology

why haven’t I managed to convince any of you

I wouldn’t quite put it that way. As I said, I haven’t quite made up my mind yet. I do appreciate the the Frölicher definition has its charms.

To me, it seems, the most striking aspect of Frölicher spaces is that for them we have the chain rule.

For presheaves the chain rule holds only under very specific circumstances, as far as I am aware.

Posted by: Urs Schreiber on January 4, 2008 2:46 PM | Permalink | Reply to this

### Re: Comparative Smootheology

why haven’t I managed to convince any of you

I wasn’t actually trying to promote sheaf models over other possibilities; I was just putting out there some facts I know or have read. To be honest, I haven’t had a chance to study the Frölicher spaces approach.

If you look at the category of Frölicher spaces over a Frölicher space, Fröl/$U$, does that have good categorical properties too (in particular, is it cartesian closed)?

Since we’re not salesmen here, one should come right out and issue some warnings about the sheaf models approach. Maybe the worst is that it’s pretty hard to get one’s head wrapped around what these sheaves are really like, concretely. A lot of the experts on sheaf models seem to cheerfully counter with something like, “Don’t worry about the analytic complications of these models; it’s easier just to work with and prove theorems in the axiomatic theories which they model. You just have to be careful that your reasoning is intuitionistic: the law of excluded middle does not hold internally in these toposes.” I expect most potential customers would start to get a little nervous around talk like that; it’s sort of like saying, “Look, this is a wonderful car, you’re gonna love it, but whatever you do, don’t push this red button here.”

I should have a look at your paper.

Posted by: Todd Trimble on January 4, 2008 7:08 PM | Permalink | Reply to this

### Re: Comparative Smootheology

So you should think of the presheaf $X:S^{op}\to Set$ as a rule that assigns to any open set $U$ the set of plots from $U$ to $X$.

Sorry, what’s a plot?

Posted by: Mike Stay on January 4, 2008 5:29 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Sorry, what’s a plot?

Just a name for the elements in the set $X(U)$ which such a presheaf assigns to an object $U$.

The point is that we think of these elements as “smooth maps from $U$ to $X$” only that there is a priori no smooth map, instead we are defining what a smooth map is by decree. To emphasize this, we say “plot” from $U$ to $X$ instead of “smooth map from $U$ to $X$”.

Or, in fact, Chen did so, back then, for the special case of his quasi-representable presheaves, where each plot is in fact a map of sets.

Posted by: Urs Schreiber on January 4, 2008 6:01 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Okay, here goes.

Define the Fundamental Frölicher Category as the category with one object whose morphisms are $C^\infty(\mathbb{R}, \mathbb{R})$.

A pre-Frölicher object is a triple $(C,F,ev)$ where $C$ is a covariant functor from $FFC$ to the category of sets, $F$ is a contravariant functor from $FFC$ to the category of sets, and $ev$ is a natural transformation of functors $FFC^{op} \times FFC \to Set$ from $F \times C$ to the hom functor. A morphism of pre-Frölicher objects is a pair of natural transformations making the obvious diagram commute.

A refinement of a pre-Frölicher object, $X_1$, is a pre-Frölicher object $X_2$ and a morphism $X_1 \to X_2$ for which both of the maps in the natural transformation are injective.

A Frölicher object is a pre-Frölicher object with no non-trivial refinements. A morphism of Frölicher objects is simply a morphism of them as pre-Frölicher objects.

Don’t know what properties this category would have because I just invented it while collecting the boxes for the Christmas decorations (I think I might be a troll - my brain works better at subzero temperatures (the boxes were in the cellar)). I’ll happily think more about it on Monday morning but just thought I’d share the idea with you all to see whether it’s worth thinking about some more.

Andrew

PS This was actually my second idea and I like it much more than my first. However if no one likes this idea I’ll try out the other on you before I climb back under my bridge.

Posted by: Andrew Stacey on January 4, 2008 9:45 PM | Permalink | Reply to this

### Sheaves and Smootheology

I’ve been thinking a bit about the sheaf versions and here are some thoughts. Hopefully they glue together to give something coherent.

I think that the inclusion of quasi-representable presheaves in the category of all presheaves has both a left and right adjoint. This should work for any of the suggested source categories (smooth manifolds, open subsets of Euclidean spaces, convex subsets of Eulidean spaces, or my FFC category described elsewhere).

The left adjoint is fairly simple. For each of the source categories we can define “constant plots”. The only one where we don’t have an object which is a singleton set is the FFC but we can still define constant curves here. Given an arbitrary contravariant functor, $P: S \to Set$, we define the underlying set of $P$, say $P_c$, to be the set of all constant plots. Any element in $P(s)$ can be viewed as a map $s \to P_c$ in the obvious way. This defines an associated “quasi-representable” presheaf. Putting this together in the obvious way yields the left adjoint.

The right adjoint is more complicated. For the natural morphism $P \to P_c$ to fail to be an isomorphism, it must fail to be injective and so there must be two distinct plots in $P$ which define the same constant plots. What we need to do is add in extra constant plots so that they no longer do so. To be the adjoint, we need to ensure that we add in the minimum possible extra points.

Let $x$ be a constant plot. Consider the set of plots $\phi$ such that $x$ factors through $\phi$. Write this as $P_x$. Define a quasi-ordering on $P_x$ by $\phi \preceq \psi$ if the germ of $\phi$ at $x$ factors through $\psi$. That is to say, there is a neighbourhood of $x$ in the domain of $\phi$ such that the restriction of $\phi$ to this neighbourhood factors through $\psi$. We then take the union over the constant plots of the sets of maximal disjoint unions of maximal chains of the $P_x$’s.

A maximal chain in $P_x$ corresponds to a constant plot with controversy removed; i.e., wherever there was a choice for an ambient plot we made a decision. Two such chains are disjoint if the germs don’t interact but we regard the points as the same. Such chains are independent in that controversy in one chain has no effect on the other. An example where we need this is where the point in question is the origin and the ambient space is the union of the $x$ and $y$ axes.

I think that this gives the right adjoint.

The adjoints are, respectively, ignoring controversy and resolving controversy.

As I said above, a presheaf is not quasi-representable if are two (or more) plots which look the same when probed with constant plots but which are (declared) different. As an example, consider the following presheaf. Start with two copies of $\mathbb{R}$. The corresponding presheaf assigns to an object $s$ in our source category (whatever it is) the set $C^\infty(s,\mathbb{R}) \amalg C^\infty(s,\mathbb{R})$. Now identify all constant plots in one part with their corresponding plot in the other part (depending on one’s definition of “constant plot” this may necessitate identification of some other plots; i.e., any that factor through a constant plot). The resulting presheaf is not quasi-representable.

So far, so good. The above applies to any of the theories. Now let us add in the Frolicher requirement that input and output should be somehow linked. I gave a suggestion elsewhere in this discussion as to how one might do this (that suggestion doesn’t work, but no one has pointed out the flaw so I’m guessing no one has looked at it in great detail; that’s not important here, though) but the actual implementation isn’t important. What we need is the general philosophy that “functionals and curves determine each other in some fashion”.

We certainly need to start with a presheaf $C$ (“plots”) and a copresheaf $F$ (“coplots”) and a composition $C \times F \to C^\infty(-,-)$ (I’m not going to assume a particular source category here for maximum generality). One aspect of the Frolicher viewpoint says that this composition is my only source of pertinent information.

Suppose I have two plots, $\phi$ and $\psi$, on the same domain, say $s$, which yield the same constant plots. Let $f$ be a coplot with codomain $t$. The compositions $f \circ \phi$ and $f \circ \psi$ are both in $C^\infty(s,t)$. Now as this is simply regular smooth maps between two standard smooth objects, such a map is completely specified by its restrictions to points, i.e. constant maps. Hence $f \circ \phi$ and $f \circ \psi$ are the same map. Thus $\phi$ and $\psi$ are Frolicher indistinguishable so why do we formally distinguish between them?

Notice that we haven’t used the “saturation” part of Frolicher’s philosophy. If we did, things would get even more bizarre! For then we could take an arbitrary subset of the domain of $\phi$, say $a$, and define a new plot $\theta$ by declaring $\theta$ to be $\phi$ on $a$ and $\psi$ on $s \setminus a$. This plot would be Frolicher-indistinguishable from $\phi$ and $\psi$ and so, by saturation (however that is interpreted), would have to be a plot.

I don’t know about you lot but that seems to be a little bit strange.

Strange or not, what it does say is that if one takes the map from an arbitrary presheaf to the nearest quasi-representable one and looks at the fibres of this then those fibres have no interesting structure whatsoever. So one may as well just consider quasi-representable presheaves (i.e., Frolicher spaces) and be done with it.

Of course, without the saturation requirement then there may be “interesting” structure on the fibres but in effect it is only interesting because it has been declared to be interesting and not because it is intrinsically interesting.

Andrew

Posted by: Andrew Stacey on January 11, 2008 10:09 AM | Permalink | Reply to this

### Re: Sheaves and Smootheology

Andrew Stacey wrote:

what it does say is that if one takes the map from an arbitrary presheaf to the nearest quasi-representable one and looks at the fibres of this then those fibres have no interesting structure whatsoever. So one may as well just consider quasi-representable presheaves (i.e., Frolicher spaces) and be done with it.

Hm, that’s interesting.

I was distracted in the middle of this discussion and am now coming back to it. I want to understand this argument by Andrew better.

One remark: don’t these conclusions all follow from the assumption that I can put the composition operation $C \times F \to C^\infty(-,-)$ on my pair (presheaf,copresheaf)?

In any case, I would like to see a concrete example. Can we do that?

I would like to start with the apparently not quasi-representable presheaf

$X_g$ for any Lie algebra $g$, which assigns to each test domain $U$ the collection of flat $g$-valued 1-forms on $U$:

$X_g : U \mapsto \mathrm{Hom}_{dg-algebras}(CE(g), \Omega^\bullet(U))$

i.e. the set of all elements $A \in \Omega^1(U) \otimes g$ which satisfy $d A + [A \wedge A] = 0 \,,$ where $[.,.]$ is the Lie bracket on $g$.

This presheaf looks like it is not quasi-representable.

Can you find me a

nearest quasi-representable presheaf and look at the fibers of this [showing that] those fibers have no interesting structure whatsoever

?

That would be helpful.

Posted by: Urs Schreiber on January 23, 2008 5:39 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

I think that there are two themes to explore from Urs’ comment above (is it worth splitting them into two threads?).

Firstly,

One remark: don’t these conclusions all follow from the assumption that I can put the composition operation $C \times F \to C^\infty(−,−)$ on my pair (presheaf,copresheaf)?

Yes. However, earlier Urs said:

I am thinking that maybe this is telling us that we eventually may want to consider things that are pairs consisting of a presheaf and a co-presheaf on manifolds, compatible in some way.

My natural transformation was an attempt to make sense of the compatibility requirement. The idea is that if $C$ is meant to represent maps into the object and $F$ maps out then at the very least one ought to be able to “compose” a map in with a map out. Moreover, as we are trying to pin-point the notion of smoothness, the composition of a map in with a map out ought to be a smooth map between our test spaces.

(Incidentally, to return to my original point, merely specifying the maps in allows one to consider things strictly weaker than smoothness. For example, if I take the family of all continuous maps from my source objects into some manifold then I get a Chen space or diffeological space or whatever space. But by doing this I have somehow smothered the smooth structure. By having both the maps in and the maps out then I ensure that I am examining smooth, the whole smooth smooth, and nothing but the smooth.)

Given just a presheaf then I can define a compatible copresheaf simply by taking the set of all natural transformations from this presheaf to the presheaf $C^\infty(-,\mathbb{R})$.

So my point is that this composition should be considered to be part of the initial structure and not something tacked on afterwards. At least, for a Frölicher-like definition.

Okay, so to Urs’ next point. I would very much like to understand this example. I’m not so bothered about where it came from (at least, as far as this discussion goes) but I think it would be useful to know how Urs thinks of this as a “smooth object”.

In my mind, when someone says “Here’s a presheaf, say $C$, on the category of stuff. This is a ‘smooth object’.” then I want to think of $C(X)$ as being ‘maps from $X$ into … something’. The ‘something’ may not even be a set, but nonetheless we can probe it with $C(X)$. Even if $C(X)$ has some other meaning, when someone says, “This is a smooth object.”, I want that to mean, “These are (generalised) plots.”.

So I think of an element of $C(X)$ as a map $c : X \to ?$. I can maybe reconstruct $?$ by patching together the images of these maps. Namely, by starting with

(1)$\coprod_X \coprod_{c \in C(X)} X$

and then quotienting out by some suitable equivalence relation.

I guess the idea is that I have an invisible object in the room and I want to see what it is. I can shove bits of paper up against it and trace its outline. Problem is, it’s quite big, so I need to use lots of bits of paper and these will overlap. But if I use enough paper I will eventually see the shape of the whole thing.

So my question regarding Urs’ example is this: how am I to think of elements of $Hom_{dg-algebras} (CE(\mathfrak{g}), \Omega^\bullet(U))$ as maps from $U$?

I want the definition of a “smooth object” to be like a cheap airline flight: no baggage allowed.

I’m going to think a little more about this example, this is just my initial thoughts and question.

Here’s another random thought. For each object in my test category I get an obvious presheaf: $C^\infty(-,X)$. For a presheaf $C$ to be a “smooth object” then the ‘plots’ in $C(X)$ should be precisely the natural transformations from $C^\infty(-,X)$ to $C$. I suspect that this is not automatic and so would need to be an additional assumption.

Finally, here’s a confession: I am not a fully paid up member of the category party (despite Bruce’s efforts over the last couple of years). I accept that it may well be a good thing to say, “Let’s find the definition that is simplest in categorical terms.” (I dispute the assertion that the last three words are redundant, but that’s irrelevant), in which case “presheaf on the category of smooth manifolds” seems to fit the bill. However, I feel that the designation of “category of generalised smooth objects” is something that needs to be justified a little more strenuously than “look, I can write down the definition in seven words!” (paraphrasing Urs here a little!). In other words, I am quite happy with the statement

Presheaves on the category of smooth manifold (or open subsets of Euclidean spaces, or convex subsets of Euclidean spaces) behave a little like smooth manifolds so let’s see what we can generalise from the one to the other.

What I am uncomfortable with (and let me put it no stronger than that) is the statement

The category of presheaves on the category of manifolds (or whatever) is the category of generalised smooth objects.

Posted by: Andrew Stacey on January 25, 2008 10:26 AM | Permalink | Reply to this

### Re: Sheaves and Smootheology

Andrew,

thanks for this comment. I very much agree with what you say, including that there is an issue of pure taste here, on which we may differ right this moment (but tastes may change), but which shouldn’t prevent us from making progress on understanding some interesting cross-relationships between various ideas.

I very much appreciate your insistance on the Frölicher idea. I did mention the idea of having compatible pre-sheaf and co-presheaves for this reason, and am glad that you picked that idea up.

So what I am trying to understand here is: are those presheaves which made me pass from quasi-representables to general presheaves, are they

- possibly naturally equipped with a co-presheaf structure satisfying some Frölicher-like compatibility condition?

- possibly then, following your argument, actually “close” to quasi-representable presheaves in some useful sense.

That would be good!

So let me say more about these presheaves, since you are asking how I am thinking about them as smooth objects probed by ordinary smooth subsets.

These things should be smooth versions of classifying spaces of Lie $n$-groups.

Consider the simplest one which I mentioend: for $g$ an ordinary Lie algebra, the presheaf

$X_g: U \mapsto flat g-valued 1-forms on U \,.$

How is that the classifying space $B G$ of the simply connected Lie group $G$ which integrates $G$?

In this sense (what I am saying now I think is true, but still needs to be written out cleanly and properly, so handle with a bit of care):

we can form the fundamental path groupoid of $X_g$, i.e. we can “loop” it.

To do so, we look at all smooth maps of generalized smooth spaces:

$[0,1] \to X_g$

and divide out homotopies between them. In the obvious manner: two such maps are homotopic, if there is a smooth map

$[0,1]^2 \to X_g$

which interpolates between them, in the obvious manner.

The claim is, forming the fundamental groupoid of $X_g$ this way amounts to doing nothing but performing the “integration without integration” of the Lie algebra $g$ to the Lie group $G$ which we discussed for instance here.

You can see this as follows:

by looking at the definitions, you find that smooth maps

$[0,1] \to X_g$

are nothing but choices of a $g$-valued 1-form on the interval.

A homotopy between two such smooth maps is then nothing but a flat (here is where that condition comes in) smooth 1-form on $[0,1]^2$, which restricts to the two given ones on the boundary.

By the nonabelian Stokes theorem, we know that two $g$-valued 1-forms on $[0,1]$ have the same $G$-valued parallel transport (“holonomy”) over the interval if they are related by a flat $g$-valued 1-form on $[0,1]^2$ this way.

So we find: a homotopy class of smooth maps from the interval into $X_g$ is nothing but an element of the group $G$.

So $X_g$ is, apparently, a smooth model for the classifying space $B G$ of $G$.

And that, actually, suggests that you are right, and that there might be a quasi-represenabtle presheaf isomorphic or otherwise closely related to $X_g$: namely one which comes from a set-version of $B G$ which is equipped with a smooth structure somehow.

Let’s think about that! That would be interesting.

Posted by: Urs Schreiber on January 25, 2008 1:10 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

Let me say one more thing about the generalized smooth space given by the presheaf $X_g : U \mapsto Hom_{dg-alg}(CE(g),\Omega^\bullet(U)) \,,$ which I said is like a smooth version of the classifying space $B G$ of the simply connected Lie group $G$ integrating the Lie algebra $g$.

One striking aspect is, that is has just a single point $\{\mathrm{pt}\} \to X_g$ since there is just a single flat $g$-valued one form on the point: the vanishing one.

Still, there are many paths $[0,1] \to X_g$ in this space, one for each $g$-valued 1-form on the interval.

This makes it quite vivid how $X_g$ fails to be quasi-represenable: by the axiom that every constant map is a plot, a quasi-representable presheaf has one point for each element of its underlying set.

The fact that $X_g$ has a single point also says that there is an operation of composition defined globally on all paths in $X_g$: all paths have the same start- and endpoint.

That’s how we find that the fundamental groupoid of $X_g$ is indeed just a group! The fundamental groupoid has a single object, and is still a nontrivial groupoid. (That’s not true for fundamental groupoids of ordinary manifolds, of course.)

The fundamental groupoid of $X_g$, which has a single object, is indeed the one-object groupoid version of the group $G$. Which I write $\mathbf{B} G$.

Posted by: Urs Schreiber on January 28, 2008 7:09 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

I’d like to simplify this example a little. I think that the Lie algebra bit is extraneous for the purposes of this discussion. So are the general forms. I propose looking at the presheaf:

(1)$X \mapsto \Omega^1(X)$

We could define this on any reasonable category of genuine smooth objects (i.e. manifolds, open subsets of Euclidean spaces, Euclidean spaces themselves, or (locally) convex subsets of Euclidean spaces), I don’t think that that is important at this stage. It’s a bit simpler than Urs’ example above but, I think, contains the kernel of the matter: the elements of $\Omega^1(X)$ are maps from $X$, but the target varies with $X$ and so we may not be able to paste it together to make it quasi-representable.

So, two things that I would like to explore for this example.

1. Can we make this into something a little more Frolicher-like? Namely, can we add in a copresheaf and suitable compatibility relations? Although we can use the fact that we know that this presheaf is 1-forms to get a form of an answer, the answer should in the end be independent of this fact. Also, can we reduce the domain somewhat?

2. Can we find, essentially, a representing space? If so, how does it related to the original presheaf?

How does this sound, Urs? Is this a reasonable simplification or har jeg slått barnet ut med badevannet? Any other questions that you can think of?

Posted by: Andrew Stacey on January 30, 2008 9:23 AM | Permalink | Reply to this

### Re: Sheaves and Smootheology

I’d like to simplify this example a little.

Good idea. I was thinking of mentioning this, too, and in fact just did over in the other thread, over there. But let’s discuss it here.

I propose looking at the presheaf $U \mapsto \Omega^1(U) \,.$

May I change this to $U \mapsto \Omega^1_{closed}(U) \,.$ ?

That’s the presheaf I am calling $X_{CE(u(1))} \,.$ It comes from the Lie algebra $u(1)$.

We could define this on any reasonable category of genuine smooth objects (i.e. manifolds, open subsets of Euclidean spaces, Euclidean spaces themselves, or (locally) convex subsets of Euclidean spaces), I don’t think that that is important at this stage.

Yes, exactly. I kept changing my mind a bit lately about which kind of test domains would be most convenient. Currently I am thinking that open subsets of Euclidean spaces is a convenient choice. It has a couple of computational advantages over admitting all manifolds. But it shouldn’t really matter much.

are maps from $X$, but the target varies with $X$

I think I see where you are coming from here, but personally I wouldn’t put it like this. I gave a heuristic interpretation of this presheaf here.

But in any case, it is not quasi-representable, yes. Another question is if maybe it is isomorphic, in some suitable sense (for instance possibly after passing to cohomologies) to a quasi-representable one.

Can we make this into something a little more Frolicher-like? Namely, can we add in a copresheaf and suitable compatibility relations? Although we can use the fact that we know that this presheaf is 1-forms to get a form of an answer, the answer should in the end be independent of this fact. Also, can we reduce the domain somewhat?

I was thinking about that, too. This is what I could say:

One way to think of this presheaf is that it assigns to each test domain $U$ the set of algebroid morphisms from the tangent algebroid of $U$ to the Lie algebra of $U(1)$ $T X \to u(1) \,.$ Which is the same as the set of dg-algebra morphisms between the corresponding Chevalley-Eilenberg algebras $\Omega^\bullet(X) \leftarrow CE(u(1)) \,.$

So there is an obvious co-presheaf waiting to be identified here, namely not

$X_{CE(u(1))} : U \mapsto Hom_{algebroids}(T U , u(1)) = Hom_{DGCAs}(CE(u(1)), \Omega^\bullet(U)) = \Omega^1_{closed}(U)$

but

$coX_{CE(u(1))} : U \mapsto Hom_{algebroids}(u(1), T U) = Hom_{DGCAs}(\Omega^\bullet(U), CE(u(1))) \,.$ On general grounds, this seems to be like the thing to do.

So what are the dg-morphisms from $\Omega^1_{closed}(U)$ to $CE(u(1))$? The latter contains just a single degree 1 generator which is closed. So such a morphism is just an algebra homomorphism from $\Omega^1(U)$ to the dg-algebra on a single degree 1-generator. But that’s just a point in $U$.

I am not entirely sure what to make of that at the moment…

Can we find, essentially, a representing space? If so, how does it related to the original presheaf?

Yes, that’s something we should try to figure out. I am thinking that $X_{CE(u(1))}$ should be a model for $K(U(1),1)$. If so, there should be a quasi-representable smooth model of $K(U(1),1)$, along the lines John talks about here to which it is closely related.

Posted by: Urs Schreiber on January 30, 2008 11:16 AM | Permalink | Reply to this

### Re: Sheaves and Smootheology

A small comment on the above:

Currently I am thinking that open subsets of Euclidean spaces is a convenient choice. It has a couple of computational advantages over admitting all manifolds. But it shouldn’t really matter much.

There’s a sense in which it doesn’t matter at all.

Lawvere once remarked that the category of (finite-dimensional) smooth manifolds is the idempotent-splitting completion of the full subcategory of open subsets of Euclidean space. This means that every smooth manifold $M$ is the image of some smooth idempotent map $p: U \to U$ operating on an open set $U$, so that $p$ splits as

$U \stackrel{r}{\to} M \stackrel{i}{\to} U$

where $(r, i)$ is a retraction-inclusion pair. (Conversely, every idempotent $p$ factors in this way in the category of smooth manifolds.)

The thrust of his remark is that whenever you want to define some functor on the category of smooth manifolds, say for example De Rham cohomology

$H^n(-): Man^{op} \to Vect,$

it suffices to do it just on the open sets $U$, provided that in the receiver category (in this case $Vect$), all idempotents split. The extension of the functor (defined on the category of open sets) to all manifolds will be unique up to isomorphism.

For the same reason, every functor

$Open^{op} \to Set$

extends uniquely (up to isomorphism) to a functor

$Man^{op} \to Set$

so that there is an equivalence of toposes

$Set^{i^{op}}: Set^{Man^{op}} \stackrel{\sim}{\to} Set^{Open^{op}}$

induced by the inclusion $i: Open \hookrightarrow Man$.

This remark fits generally in the context of Morita theory, or the theory of Cauchy completions, as we have talked about elsewhere on this blog.

Posted by: Todd Trimble on January 30, 2008 2:31 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

Todd,

thanks a lot. That’s helpful.

What can be said in this regard with respect to open contractible and/or open convex subsets of Euclidean spaces, the other two popular kinds of test domains in this busines?

Posted by: Urs Schreiber on January 30, 2008 2:54 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

I think for the category of open contractibles, the Cauchy completion is just the category of contractible manifolds (without boundary). At any rate, a retract of a contractible space is contractible, so Cauchy completion doesn’t take you outside the world of contractible spaces.

From the standpoint of Cauchy (or idempotent-splitting) completion, I’m having a hard time telling apart open contractibles and open convex sets. Is every open contractible diffeomorphic to an open convex set? Are they even all diffeomorphic to standard open balls?

Posted by: Todd Trimble on January 30, 2008 4:05 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

But can we say anything about the relation between presheaves on open subsets versus presheaves on contractible open subsets?

We don’t have your above argument to show that they are equivalent. But does it mean they will be inequivalent?

I am hoping they are in fact equivalent. Maybe only if we restrict attention to actual sheaves (as opposed to mere presheaves)?

Posted by: Urs Schreiber on January 30, 2008 5:29 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

Sorry, Urs, I should have been clearer. Those two toposes are inequivalent.

The Morita theorem which lies in the background is that there is an equivalence

$Set^{C^{op}} \simeq Set^{D^{op}}$

iff $C$ and $D$ are Morita equivalent, meaning that their idempotent-splitting completions are equivalent categories. Speaking somewhat loosely (although it can be made precise), that would mean that retracts of objects of $C$ are isomorphic to retracts of objects of $D$.

But retracts of contractible opens are still contractible, whereas retracts of general opens obviously need not be contractible. So those two presheaf categories are not equivalent.

Posted by: Todd Trimble on January 30, 2008 6:01 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

You also asked a companion question about whether we get an equivalence when we restrict to sheaves on opens and sheaves on contractible opens. This I don’t know.

First I’m not sure if you have specific topologies (on the sites) in mind. But second, I don’t know, off the bat, nice general methods for detecting ‘Morita equivalence of sites’ (i.e., when sheaf toposes are equivalent). I’d really have to think hard about that one (and no guarantee I’d come up with anything useful).

Posted by: Todd Trimble on January 30, 2008 7:31 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

First I’m not sure if you have specific topologies (on the sites) in mind.

Hm, apparently I am missing something then: aren’t there “obvious” “canonical” Grothendieck topologies on these categories of subsets of Euclidean spaces?

I was tacitly assuming there are, without having really thought much about it.

Posted by: Urs Schreiber on January 30, 2008 8:00 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

Oh sure! There are ‘canonical’ topologies (with a technical meaning of ‘canonical’), and there are often ‘obvious’ topologies (as in the cases we’re looking at). There are typically many topologies; to play it safe, one should specify which one. In fact, it probably wouldn’t hurt to talk a little about topologies.

‘Canonical topology’ in the technical sense has a somewhat dry definition: a topology on a small category $C$ is subcanonical if every representable $hom(-, c)$ is a sheaf for that topology. The canonical topology is then by definition the largest subcanonical topology. It takes some practice to get a sense of what it may look like in actual examples.

There is also, as you say, an ‘obvious’ topology for each of the categories we’re looking at here, namely the ‘open covering’ topology. And, as it happens, I think the (informally) ‘obvious’ topology coincides with the canonical one.

Maybe it’s a good idea if I recall some definitions (for anyone who might need them). Let $C$ be a small category.

• A sieve on an object $c$ of $C$ is a subfunctor of a representable $hom_C(-, c): C^{op} \to Set$.
• A Grothendieck topology on a small category $C$ assigns to each object $c$ a collection of sieves on $c$ which are called covering sieves for that topology. These must satisfy certain axioms which I won’t go into right now, but which can be found in any book on topos theory. A site is a small category equipped with a Grothendieck topology.
• A sheaf with respect to a Grothendieck topology is a presheaf which ‘thinks’ that covering sieves really are coverings. ;-)

This last definition obviously requires some amplification, which I’ll explain with an example. Let’s take the open covering topology (on the category of open sets in Euclidean space, and smooth mappings between them). Here, a covering sieve on an open $U$ is by definition a subfunctor (an embedding $j$) of the form

$\bigcup_i hom(-, U_i) \stackrel{j}{\hookrightarrow} hom(-, U)$

where the $U_i$ are open subsets which cover $U$ in the usual sense. The union here is a coequalizer of the form

$\sum_{i, j} hom(-, U_i) \times_{hom(-, U)} hom(-, U_j) \stackrel{\to}{\to} \sum_i hom(-, U_i) \to \bigcup_i hom(-, U_i)$

which in our case simplifies to a coequalizer of the form

$\sum_{i, j} hom(-, U_i \cap U_j) \stackrel{\to}{\to} \sum_i hom(-, U_i) \to \bigcup_i hom(-, U_i).$

A sheaf, then, is a presheaf $X$ which can’t tell the difference between the left and right sides of the inclusion $j$ which marks a covering sieve, or in other words ‘thinks’ $j$ is an isomorphism, in the sense that ‘probing’ maps

$hom(-, U) \to X$

are in natural bijection with maps

$\bigcup_i hom(-, U_i) \to X$

by restriction along $j$. Now our good friend Yoneda tells us that the set of maps $hom(-, U) \to X$ is in natural bijection with the set $X(U)$. And since the union is a coequalizer, the set of maps from the union into $X$ is given by an equalizer; again according to Yoneda, this is the equalizer of a pair of maps

$\prod_i X(U_i) \stackrel{\to}{\to} \prod_{i, j} X(U_i \cap U_j).$

Thus the sheaf condition, that $X$ thinks covering sieve inclusions $j$ are isomorphisms, translates to the condition that $X(U)$ is the equalizer of that pair of maps (in the evident way).

It is easy to check that the open covering topology is subcanonical, i.e., that representables $hom(-, V)$ are sheaves for this topology. By the above, we have to check that $hom(U, V)$ is the equalizer of the evident pair of maps

$\prod_i hom(U_i, V) \stackrel{\to}{\to} \prod_{i, j} hom(U_i \cap U_j, V)$

whenever the collection $\{U_i\}$ covers $U$ in the usual sense. But this is obvious because the usual sense of ‘cover’ says exactly that $U$ is the coequalizer in

$\sum_{i, j} U_i \cap U_j \stackrel{\to}{\to} \sum_i U_i \to U$

and homming into $V$ takes coequalizers to equalizers!

It’s a little twisted, and it may take some thought to see that I’m not making up a complete tautology. :-)

Now: I think it’s true that the open cover topology is in fact the canonical topology (in the technical sense). We’ve just seen that it’s at least subcanonical, and I’ll assume I’m right that it’s the canonical one.

Now that we’ve gotten this out of the way, we can have a look at your question. You know, I think maybe you’re right: you’d get the same sheaf categories whether you use the contractible opens (with the open cover topology) or just plain old opens (with the open cover topology).

We probably have to be a wee bit careful about what we mean by the ‘open cover’ topology in the case of contractible opens: I think we’d want the coverings to be ‘nice’ in the sense that all finite intersections $U_{i_1} \cap \ldots \cap U_{i_n}$ that are nonempty are in fact contractible. For example, if all the (open) inclusions $U_i \hookrightarrow U$ are between open convex sets, this is automatic (and I’m guessing this is what you had in mind?).

Anyway, I’ll bet it comes down to some simple topological fact along the following lines: every open covering of an open $U$ can be refined to a ‘nice’ covering by contractible opens. Let’s see: there are more covering sieves if we use the site of all opens, which would put more stringent conditions on what it means to be a sheaf. So I think a sheaf for the site of all opens gives, tautologically, a sheaf for the site of contractible opens. But by the refinement property above, it should be true that the sheaf condition using the site of all opens is no more stringent than the condition using the site of contractible opens. So, I’ll bet you get the same sheaf toposes up to equivalence.

Posted by: Todd Trimble on January 31, 2008 12:02 AM | Permalink | Reply to this

### Re: Sheaves and Smootheology

(with a technical meaning of ‘canonical’)

By the way: is there, generally, at all anything like a general technical interpretation of “canonical” in terms of some universal property?

Given structures of kind $X$ which can be equipped with structures of kind $Y$, is there a technical way to say that the choice of $Y$ is canonical for $X$?

Posted by: Urs Schreiber on January 31, 2008 1:42 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

So, I’ll bet you get the same sheaf toposes up to equivalence.

That sounds really good. I need to think about if I may want/need to move my entire application from presheaves to sheaves, actually.

Chen-smooth structure for instance are, after all, not just presheaves but actually sheaves on their test domains.

And I’d think the same is true for all presheaves of the form $U \mapsto Hom_{DGCA}(A, \Omega^\bullet(U)) \,,$ which is the other class I want to get under better control.

Posted by: Urs Schreiber on January 31, 2008 4:44 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

Yes, this is a point I’d like cleared up. When I asked, “Why presheaves?” Todd answered (paraphrasing wildly), “The category of presheaves is very nice and, what is more, some presheaves are actually sheaves!”.

To me, that sounded a little in disharmony with other things I read here. Unpacking it a bit, the argument goes:

1. We’ve got these really nice objects (smooth manifolds) but in a nasty category.

2. Grothendieck says, better to work with a nice category so we do; say, presheaves on some site.

3. And you know what? The really great thing about this category is that some of its objects are nice!

It seems to me that if we work with presheaves then the source category is extremely important. If I enlarge the category it is quite likely that I will enlarge the category of presheaves (unless I am extremely careful). On the other hand, it may be possible to enlarge the category quite considerably without changing the sheaves, as Urs and Todd have been discussing.

If we work with something more restrictive than presheaves, then there may be lots of choices for the source category. If so, it is obviously of personal opinion as to which is chosen. As I don’t have a particular problem to which I am trying to apply this theory (despite what Urs said at the start, all my spaces are honest manifolds – just some are infinite dimensional), I’d quite like to know what is the source category with least redundancies.

At this point it becomes important what ones restrictions are. If sheaf-like then we need to ensure that we have something in each dimension. This suggests the category of Euclidean spaces as the simplest source category.

However, why are we focussing on sheaf-like conditions? Two reasons spring to mind. Firstly, because a lot is known about sheaves. Secondly, because we traditionally view smoothness as a local property which seems to fit in very nicely with the sheaf-like conditions.

But, due to Boman’s result, it turns out that smoothness is something finer than a local property (but still weaker than pointwise). So although a sheaf-like condition will work, it is weaker than is needed. Boman’s result suggests that a one-object category with object the real line is the simplest source category (the Fundamental Frolicher Category). Of course, given (a) suitable functor(s) on this category, one can define a sheaf on whichever source category one likes in a very obvious way. One can then work with this sheaf as one wants to.

The point is, though, that this sheaf is not the fundamental thing but is derived from it.

So .. do you want presheaves or something more restrictive? If more restrictive, you/we need to justify the restrictions.

Posted by: Andrew Stacey on February 1, 2008 8:42 AM | Permalink | Reply to this

### Re: Sheaves and Smootheology

Hi Andrew!

all my spaces are honest manifolds – just some are infinite dimensional

Just to make sure I am completely following: what exactly do you mean when you say that? Do you just mean

-that all your spaces are actually sets with extra bells and whistles (i.e. quasi-representable generalized things)?

- or do you mean that all your spaces are actually Fréchet-manifolds and/or Hilbert-manifolds or the like?

Just so that I know what you have in mind.

Boman’s result suggests that a one-object category with object the real line is the simplest source category

Let me just paraphrase, in order to check that I exactly understand what you are suggesting:

you are saying we might want to restrict attention to sheaves on the site of open subsets of the real line. Is that right?

I understand (or hope I do) that there is this cool fact that says that quasi-representable sheaves on open subsets of Euclidean spaces are already entirely fixed by their restriction to open subsets of the real line. And that’s what you are advertizing.

And I agree, that’s a remarkable fact which we should be aware of.

It just so happens that in the applications I need, I need sheaves which are not fixed by their restriction to 1-dimensional test domains.

(Like, for instance, the sheaf which sends each test domain to the set of closed 5-forms on that test domain.)

But that shouldn’t distract us. Whether or not we can characterize a given sheaf by its restriction to a sub-site matters in concrete examples, I’d think, but not so much for the general setup.

What we should try to understand is

a) if, when and how non-quasi-representble sheaves can be “encoded” by quasi-representable ones

b) if, when and how we can makes sense of a structure consisting of pairs of sheaves and cosheaves with a certain compatibility condition on them.

You made a suggestion that using b), a) may be accesible.

I have the following vague idea how to approach a) directly:

I am all motivated here by trying to understand the ??? in

rational homotopy theory : simplicial spaces :: ??? : sheaves .

So one strategy might be this:

given any non-quasi-representable sheaf $X$, there is canonically associated a simplicial space $S X_\bullet$ to it: namely its space of simplices:

$S X_n := Hom_{sheaves}( standard n-simplex, X ) \,.$

Chances are that from this simplicial space we obtain a space which is the set underlying a quasi-representable sheaf $\hat X$ which is “as close as possible” to the original $X$ as a quasi-representable sheaf can get.

Or something like that.

Posted by: Urs Schreiber on February 1, 2008 12:30 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

I wrote:

all my spaces are honest manifolds – just some are infinite dimensional

What I mean is that the spaces that I use (loop spaces, path spaces) are locally diffeomorphic to convenient vector spaces. That is, locally complete locally convex topological vector spaces. These are the ones used in Kriegl and Michor’s tome.

Urs wrote:

you are saying we might want to restrict attention to sheaves on the site of open subsets of the real line. Is that right?

Not quite. The bit I’m not sure about is the “sheaves”. I think that the compatibility will be a little more intricate.

Urs wrote again:

It just so happens that in the applications I need, I need sheaves which are not fixed by their restriction to 1-dimensional test domains.

Yes, I know. And I need to start thinking seriously about this example. What I suspect is that by throwing in a saturation argument then one would find that the restriction morphisms that you have in your functor are not the only ones present. By enlarging the restriction morphisms one would again find that everything is determined by its restriction to lines. This is what I need to think about and keep getting distracted from.

Urs enscribed:

But that shouldn’t distract us. Whether or not we can characterize a given sheaf by its restriction to a sub-site matters in concrete examples, I’d think, but not so much for the general setup.

On the contrary. I think it matters for the general setup but not for concrete examples. For concrete examples, one should probably take the setting in which it seems most natural (so sheaves on open subsets of Euclidean spaces for your forms example) but just have at the back of ones mind the knowledge that this is a particular angle on a more general object. Just like we can view ordinary manifolds as submanifolds of Euclidean spaces but we don’t define them that way as it is not canonical. However, if we are given a particular manifold as a submanifold of some Euclidean space then we’d be foolish to throw away that extra information purely on moral grounds.

Urs carved:

What we should try to understand is [something]

Yup, I agree. I feel as though we’re done on converting the unknown unknowns to known unknowns and now need to convert these known unknowns to known knowns. Hmm, I think a functor would be appropriate here.

Posted by: Andrew Stacey on February 4, 2008 9:40 AM | Permalink | Reply to this

### Re: Sheaves and Smootheology

Andrew, I don’t have time now to go into details (I have to catch a plane soon), but let me just say quickly that the ‘wild paraphrasing’ is a distortion of what I was saying. Usually (not always, but usually) I put a lot of care into my mathematical comments. Perhaps you could quote something I said that suggests your reading?

I’ll say again that I’m not particularly plumping for one approach over another. For some time I’ve been meaning to try to say something about the Frölicher space approach from the standpoint of something called the Chu construction, but that will have to wait.

Posted by: Todd Trimble on February 1, 2008 1:52 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

Perhaps you could quote something I said that suggests your reading?

Let us recall that this thread was originally based on Andrew’s review of various notions of smooth spaces which did originally not include presheaves, and that then I went ahead and hijacked the discussion by keeping going on about presheaves, receiving lots of helpful comments from you, Todd.

So I think Andrew’s “wild paraphrasing” was not so much a paraphrasing of something you said concretely, but rather an attempt to summarize the attitude he felt was the reason for that hijacking of the original discussion, which is really entirely my fault.

So, in case I haven’t succeeded in making myself clear so far, let me say very explicitly again why I, personally, are so interested in looking at generalized smooth spaces as (pre-)sheaves over test domains:

There is a beautiful connection between (finite dimensional) Lie $\infty$-algebras and differential graded commutative algebras (DGCAs), where these DGCAs are best thought of as the algebras of “left invariant forms on an $n$-group”, and a bunch of ideas in a) Sullivan models in rational homotopy theory, b) integration of $L_\infty$-algebras and c) integration of Lie algebroids all boild down to the fact that it is useful to consider the sheaves which assign to each test domain the dg-morphisms from the given DGCA to the forms on the test domain.

So whatever notion of generalized smooth spaces we settle for, I want these things to be an example of them. That’s why I am interested in presheaves here.

And also, because for these examples the in principle available induced structures of co-presheaves seem to be of little interest (at least I don’t see what these would be good for at the moment) I tend to be not primarily interested in conception s of generalized smooth spaces in terms of co-presheaves (hence in terms of “out-maps”).

Posted by: Urs Schreiber on February 1, 2008 2:59 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

Todd, you are absolutely right. I apologise. It was Urs who said it here:

But the point of presheaves is, as the name suggests, that some of them are in fact sheaves.

The closest that you said was this:

More sophisticated sheaf models would repair such defects by considering not all presheaves on S, but sheaves with respect to a Grothendieck topology on S which takes into account such coverings.

At that point in the discussion I was looking for a reason why one would bother with presheaf models. Your and Urs’ answers (same comments) were of the flavour of:

there are a number of pleasant features of [the category of presheaves]

which didn’t really satisfy me as categories such as the category of Frolicher spaces have all the properties that you both listed for presheaves.

By now, however, there is an answer that does satisfy me: Urs has a specific example which is not obviously quasi-representable but is obviously a sheaf. Whether it is something else as well is one of these known unknowns.

Okay, I was being a bit facetious with my opening remarks. My point was that while you and Urs are discussing the relative merits of one site over another and sheaves over presheaves, I’m still not convinced that this is the heart of the matter.

The way I see it is that there should be something extremely simple at the heart which captures the smooth, the whole smooth, and nothing but the smooth. For set-based theories, my feeling (again, let me put it no stronger) is that this is Frolicher spaces. For non-set-based theories, I don’t really have much of a clue – except that I want it to restrict to Frolicher spaces when it happens to be set-based.

However, I’m not saying that one should only ever work with Frolicher spaces. If ones theory has a natural description on, say, Chen spaces then it seems a little harsh to have to rephrase it in terms of Frolicher spaces. Rather, just use the functor from Frolicher spaces to Chen spaces, et voila! (Ooops, wrong language; I meant “der ser du!”.)

So if we can work out the non-set-based version of Frolicher spaces then there should be a natural functor from these to presheaves on your given choice of site (probably by taking natural transformations between functors).

So I don’t really want to get bogged down in choosing a particular site. What I want to know is what the compatibility relationships are. Sheaf-models give one set of compatibility relations, but I have yet to see a reason why these are the right relationships in this case.

I guess that here we come to a difference between continuity and smoothness. Continuity is blatantly a local phenomenon so sheaves on sites is a natural generalisation. Smoothness is stronger than local so something stronger than sheaves might be called for.

Of course, working out the details for Urs’ example(s) would help in this and I’ll shut up now and get on with that.

So, once again, I’m sorry, Todd, for misrepresenting you. I was just trying to insert a spanner in the works (sorry, wrong language again; I meant “wrench”) to derail the discussion and getting heading in the direction that I want it to go in. As I’m outnumbered two-to-one perhaps I should shut up until I’ve thought more about Urs’ example!

Yeah right; like that’s going to happen.

Posted by: Andrew Stacey on February 4, 2008 8:59 AM | Permalink | Reply to this

### Re: Sheaves and Smootheology

I’ve thought more about Urs’ example!

We’ve already at least mentioned the important aspects that need to be considered, I think.

It seems that we want to consider pairs consisting of a sheaf $plots_X$ and co-sheaf $coplots_X$ on test domains in the category $S$, together with a pairing, which is a transformation

$pairing : plots_X \times coplots_X \to Hom_S$

between two functors $S^{op} \times S \to Set$.

I like that idea, that sounds good.

I am pretty sure such a pairing does exist canonically for those non-quasi-representable examples of the form $X_{CE}(g)$:

as I mentioned, in these examples the sets of plots in questions are the sets of dg-algebra morphisms $U \mapsto Hom_{dg}(CE(g),\Omega^\bullet(U))$ so that the sets of co-plots would be simply the opposed Hom-sets $U \mapsto Hom_{dg}(\Omega^\bullet(U), CE(g)) \,.$

The pairing then is simply induced by the composition of these dg-morphisms. Hence the pairing yields an assignment

$U \times V \mapsto plots(U) \times coplots(V) \hookrightarrow Hom_{dg}(\Omega^\bullet(V),\Omega^\bullet(U)) \simeq Hom_S(U,V) \,.$

The last isomorphism says that dg-algebra morphisms between algebras of (smooth) differential forms come from (smooth) maps on the underlying test domains, so its a consistent pairing.

What I haven’t thought about is the “saturation” condition in this context. It could turn out that there are more coplots in these examples than I am seeing currently. That might be worthwhile to think about.

Posted by: Urs Schreiber on February 4, 2008 2:45 PM | Permalink | Reply to this

### Re: Sheaves and Smootheology

Urs (or anyone else), this is just to let you know that I’ll be on vacation from Feb. 1 to Feb. 9, with somewhat less regular internet access than I’d have at home. So in case you don’t hear from me much during that time, you’ll know why.

Posted by: Todd Trimble on February 1, 2008 12:05 AM | Permalink | Reply to this

### Re: Sheaves and Smootheology

I should have added to the comment on idempotent-splitting completion that the typical way to exhibit a manifold $M$ as the image $p(U)$ of an idempotent $p$ on an open $U$ is to embed $M$ in Euclidean space, and take $U$ to be a tubular neighborhood of the embedding. Then equip $U$ with a structure of normal bundle over $M$, so that the bundle projection $U \to M$ gives a retraction $r$ which is left inverse to the inclusion $i: M \to U$.

Posted by: Todd Trimble on January 30, 2008 2:39 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Comments on Mostow’s paper.

I’ve just skipped through Mostow’s paper (linked to by Urs above). Here’s my initial comments.

As Mostow says, his definition is the same as that of Sikorski’s of a differentiable space. So including Mostow spaces in my paper really only involves adding a reference.

He comments on the difference between his definition and two others, due to Smith and Chen. His comparison is more of a “they do that but I do this and for this reason” type. Chen’s definition is in my paper.

Smith’s definition is intriguing. His “closure” condition is very close to that of Frölicher but ever so subtly different. He starts with a topology and insists that all functionals be continuous. Thus Smith’s set of functionals is the intersection of Frölicher’s with the set of continuous functionals.

This highlights a point that I try to make in my article. If one wants to do smoothness then one gets into a mess if one starts with a topological space and tries to impose a compatible smooth structure on top. Rather, one should start with the smooth structure and then underlie that with a compatible topology.

Andrew

Posted by: Andrew Stacey on January 4, 2008 2:54 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Ok, I’ve had a decent look at Andrew’s notes on different notions of “smooth”. There are many things to discuss and ask here, and sadly there are also many unfinished, historical threads on this business here at the n-category cafe. One just needs to consider our most recent discussion on these affairs (which began roughly here when Urs and John were wondering if every Lie algebroid integrates to a Chen-smooth groupoid).

So, like the presidential elections, the central issue of “comparative smootheology” is framing the debate of what we are discussing!

I don’t know how to do that, so I’m just going to throw out a few random comments on some of the threads I see.

1. Boman’s theorem . This is a crucial part of Andrew’s notes, and the whole debate (consider John’s question here , for instance). My understanding is that Boman’s theorem says, roughly,

A map $f : U \rightarrow V$ between Euclidean spaces is smooth if and only if it takes smooth curves to smooth curves.

Sadly Andrew doesn’t spell out Boman’s theorem for us in his notes (gulp?!), though he does spell out the generalization he will need. Is this the gist of it?

2. Confusion of definitions . I believe there is a confusion between Andrew’s use of the term “Chen space” and the way we’ve been using it here at the n-category cafe. My understanding is that we have been thinking in terms of Chen’s “last variant” as outlined in Definition 4 of John and Urs’ notes on Higher Gauge Theory. That is, a “Chen space” for us has been just a diffeological space, where the domains of the plots are arbitrary convex sets (and not just open subsets of Euclidean spaces). I think Andrew is using some earlier definition of Chen. In fact, on this point, I am confused why Andrew’s definition of a diffeological space has the words “continuous map” in it, although the spaces involved are just sets. Isn’t the definition of a diffeological space supposed to be topology-free?

3. What I am interested in . In fact, of the four notions of “smooth” that Andrew delimits, I am only interested in the comparison between Frolicher spaces and diffeological spaces. I am philosophically prejudiced against (Andrew’s variant of) Chen spaces, and differentiable spaces, because they include the word “topology”, which I think should not be relevant. In fact, I obtained this prejudice from Andrew, so I’m sure he doesn’t mind :-)

4. Sheaves. My understanding is that a diffeological space is exactly the same thing as a concrete sheaf $\mathcal{F}$ on the site $Euc$ of open subsets of Euclidean spaces. “Concrete” means of course that the plots can actually be interpreted as maps from the Euclidean subsets into some fixed set $X$; in other words a diffeological space is a good-ol’ “set with structure”.

It would be nice if Andrew would make more explicit this “sheafy” interpretation of diffeological spaces, and Frolicher spaces.

5. Wire diffeology . My understanding is that when you convert a diffeological space to a Frolicher space, you lose information. The example Andrew gives is that $\mathbb{R}^2$ with its standard diffeology and with its “wire” diffeology are not isomorphic as diffeological spaces, but they become isomorphic if you convert them to Frolicher spaces.

What is the wire diffeology? Can you explain it Andrew? If I understood this example better, then I could make more progress.

6. Asymmetry . When you turn a diffeological space $(X, \mathcal{D})$ into a Frolicher space, something disturbing happens. One defines the set of test functions simply as the set of difeological maps from $X$ into $\mathbb{R}$,

(1)$\mathcal{F} := Dlog ( (X, \mathcal{D}), (\mathbb{R}, \mathcal{D}_{\mathbb{R}}))$

You might think that the flip-side of things, namely the set $\mathcal{C}$ of test curves, would just be the other hom-set,

(2)$\mathcal{C} = Dlog ( (\mathbb{R}, \mathcal{D}_{\mathbb{R}}), (X, \mathcal{D})).$

But that’s not the case! Rather you have to construct $\mathcal{C}$ abstractly. There’s an asymmetry here which is disturbing. What’s going on?

7. John and Urs’ tough-to-find paper . I remember reading a draft paper by John and Urs, something like “Higher Gauge Theory II”, which had big grey letters “draft” written on it. This paper is important to the debate because it contains explicit calculations of “smooth spaces in action”. Urs, you’ve told me this before, but can you tell me again where I can find this paper (sorry!)

Posted by: Bruce Bartlett on January 5, 2008 8:24 PM | Permalink | Reply to this

### Re: Comparative Smootheology

I remember reading a draft paper by John and Urs, something like “Higher Gauge Theory II”, which had big grey letters “draft” written on it. This paper is important to the debate because it contains explicit calculations of “smooth spaces in action”. Urs, you’ve told me this before, but can you tell me again where I can find this paper (sorry!)

That paper is essentially the article on hep-th by both of us, which you can easily find, which is also reproduced in my thesis.

Since the discussion in there was partially incomplete concerning some technicalities, John kept (and I guess still keeps) an updated version of the file on his website, with more details added.

But one big chunk of details that was missing in the original version has been worked out in Konrad and mine Parallel transport and functors. The other chunk has been worked out in an article called the The geometry of smooth 2-functors which is sitting essentially finished here on our desks, waiting only for the section on transgression of 2-functors to 1-functors to be finished.

The stuff from the theory of smooth spaces that is needed again and again in these considerations is the nature of the internal hom in Chen-smooth spaces. This is just the restriction of the internal hom in general presheaves to Chen-smooth spaces.

So the important fact to which all proofs concerning smooth functors make recourse to is that the collection of plots on test domain $U$ in the loop space of $Y$ is $Hom_{Manifolds}(U \times S^1, Y)$ when $X$ is a manifold.

More generally, it is the internal hom that describes mapping spaces $hom(X,Y) : U \mapsto Hom_{Set^{S^{op}}}(U \times X , Y)$

which is used all the time.

For any model of smooth spaces for which the plots on mapping spaces are like this, all our results about smooth $n$-functors work.

Posted by: Urs Schreiber on January 6, 2008 6:42 PM | Permalink | Reply to this

### Re: Comparative Smootheology

That paper is essentially the article on hep-th by both of us, which you can easily find, which is also reproduced in my thesis.

Thanks Urs. It’s very confusing though, and I still haven’t been able to find the magic one I am looking for.

The first confusion is that if one clicks on “John Baez” or “Urs Schreiber” at the archive, you get a list like this. The only higher gauge theory paper which appears here is this one… but that’s not the one I’m looking for.

I remember one which has appendices where you guys were working out explicity the categorical properties of smooth spaces (like the internal homs, etc).

The second confusion is that there is another paper by Baez and Schreiber,

Higher Gauge Theory: 2-Connections on 2-Bundles

and this doesn’t show up in that arxiv search at all! It is referenced in the 0511710 paper, although under the name “Higher gauge theory II” (the “II” doesn’t show up in the archive title, which is also confusing). It’s still not the one I’m looking for though :-)

So you see, in my mind, there are at least three higher gauge theory papers by Baez and Schreiber. One of them is perhaps a figment of my imagination and one doesn’t show up on archive searches (unless you point directly to it). What’s going on?

Posted by: Bruce Bartlett on January 6, 2008 7:07 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Bruce Bartlett wrote:

The second confusion is that there is another paper by Baez and Schreiber…

Let me explain. In 2004, Urs and I wrote a paper called Higher Gauge Theory: 2-Connections on 2-Bundles and put it on the arXiv. This paper has lots of imperfections, so we decided to keep polishing it before publishing it.

At about the same time, Toby Bartels came out with a paper based on the PhD thesis he did with me. This straightens out the theory of 2-bundles.

So, we decided to call his paper Higher Gauge Theory I: 2-Bundles and rename the paper by Urs and me Higher Gauge Theory II: 2-Connections on 2-Bundles. The current draft of HGT2 is on my website — click on the link I just gave.

As it seemed to be taking forever to complete all this work, near the end of 2005 I used the proceedings of Ross Street’s 60th birthday conference as an excuse to summarize what we’d done so far. This led to a paper by Urs and me called Higher Gauge Theory. Unfortunately, it doesn’t correctly explain Toby’s work on 2-bundles! I may fix that someday.

Since then, Urs and Konrad Waldorf have written a paper on Parallel Transport and Functors. My student Alex Hoffnung and I are about to come out with a paper called something like Chen Spaces: A Convenient Category for Differential Geometry. Alex is also writing a paper about connections on bundles and smooth anafunctors, which covers material similar to the paper by Urs and Konrad, but from a somewhat different viewpoint — the viewpoint explained in my seminar. And, Danny Stevenson and I are about to release a paper called A Classifying Space for 2-Bundles.

The existence of all these papers means that when HGT2 is finally done, it won’t need to be packed with proofs of basic facts about the tools we use! We can focus on the actual subject matter: higher gauge theory.

If you think this is a complicated story, just wait: my first paper on this stuff, Higher Yang–Mills Theory, dates back to 2002! I never published it because I wasn’t happy with it.

It’s just taken a long time to get all the details straightened out.

Posted by: John Baez on January 11, 2008 9:56 PM | Permalink | Reply to this

### Re: Comparative Smootheology

In fact, on this point, I am confused why Andrew’s definition of a diffeological space has the words “continuous map” in it, although the spaces involved are just sets.

Bruce is right, there are various definitions floating around which differ only ever so slightly.

I think (but don’t have the book “collected works of K.T.Chen”) with me right now, in some version of his definition he required the set $X$ underlying a Chen-smooth space $X$ to be a topological space, and all plots to be continuous.

The details of the definition of general Chen-smooth spaces don’t actually matter much in applications, because there we always need and use the canonical smooth structure on mapping spaces, which says that plots on $U$ of $hom(X,Y)$ are those in the set $Hom(U \times X , Y)$.

That satisfies the definition of a Chen-smooth space with our without the requirement that the underlying set $Hom_{Set}(X,Y)$ be a topological space, because we can always think of it as being equipped with the compact-open topology of mapping spaces.

In fact, all of this makes me think that we should simply be thinking in terms of presheaves on manifolds in general. That’s the slickest definition, which just takes half a line and does everything for you you ever need in the kind of applications that we care about here (namely mapping spaces).

Posted by: Urs Schreiber on January 6, 2008 7:08 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Bruce wrote:

a lot of stuff

which I would like to reply to.

First, let me note a bit of history. The original intention for the paper under discussion was that it would be a 3-page note. It then, like the enormous turnip, grew and grew. But as it started out small I used the definitions and resources immediately available to me and didn’t go hunting in the library for further material (hence not referencing Mostow’s paper). Some of Bruce’s points highlight this deficiency.

1. Boman’s theorem.

This was published in Math. Scand. in 1967. MathSciNet does not link to the article (it’s a bit odd that here in Trondheim I don’t have access to an online version of an article in Math. Scand., but who said the system had to be logical?). However, the key result is stated and proved in Kriegl and Michor’s book which can be found online. The result is theorem 3.4 on page 26 (chapter I if you don’t want to download the whole lot). In my paper I miscited this as theorem 2.4 of Kriegl and Michor.

Bruce is right that he has the gist of it.

2. Confusion of definitions.

This is where my ‘historical remarks’ above is relevant. I have a copy of Chen’s paper on iterated integrals and I took the definition of a ‘Chen space’ from that. I had picked up the fact that he changed his definition a little, but I do not have the later definition in front of me. Obviously, it would be worth commenting on how the definition changed as time progressed.

If the later definition is, essentially, to include the sheaf condition then you can get an idea of how that would fit in with the other definitions by looking at section 5 (The Differences). There is a short discussion on what an appropriate locality axiom for Chen spaces would be. What might surprise you is that the one most similar to that for diffeological spaces is not “something that is locally a plot is a plot” but rather “something that is locally extendible to a plot is a plot”.

Given the interest, I am more than happy to add in a section on Chen’s later definition(s). Could someone give me a definitive reference for the preferred definition?

As for the inclusion of the word “continuous” in the definition of a diffeological space, that’s a cut-and-paste error. Mea culpa. I think I’m right in saying that only the stated definition is wrong, in the rest of the document I use (implicitly) the correct definition.

3. What I am interested in.

Your prejudice is absolutely correct!

I’m only interested in Frolicher spaces, so I’m even more prejudiced than you, Bruce.

4. Sheaves.

See the comment above about sheaves. Basically, I didn’t put them in because I didn’t know about them. Now I know that people are interested in them I’ll add something about these interpretations. I think that with the right definition of Frolicher spaces in terms of sheaves then the overall comparisons will be the same.

5. Wire diffeology.

This is an example I picked out from Patrick’s book on diffeology. The Wire diffeology of a Euclidean space is the diffeology containing all the smooth curves but no (non-degenerate) smooth maps with higher dimensional domain.

Yes, when going from, say, diffeological spaces to Frolicher spaces you do lose information. But that is only relevant if that information was important in the first place. When I consider a problem about a ball rolling on a surface then it generally is not important what colour the ball is, so I forget it. Or a closer analogy would be with pairs $(X, \mathcal{B})$ where $\mathcal{B}$ is a basis for a topology versus topological spaces.

6. Asymmetry.

That the functionals appear as a hom-set is the coincidence here (well, that’s a slight oversimplification). For some definitions this is not true. One can read this as saying that, unlike some definitions, a diffeological space contains no spurious extra information. When defining a diffeological space one says “I want all these maps to be smooth.”. There is a corresponding Frolicher space for which all those maps are smooth, the problem is that there may well be more smooth maps than those you first thought of.

7. John and Urs’ tough-to-find paper.

I agree with Urs’ comment that often in practice it doesn’t matter which definition is used since all form “nice” categories.

On the other hand, it would be nice to have a little consistency here and have a good reason for using one definition rather than another. I make the case for Frolicher spaces. I have yet to see a rejoinder for any other - even the presheaf versions above are not really a rejoinder since the reasons stated for liking presheaves is that the resulting categories are nice, as is the category of Frolicher spaces! Moreover, it may well be possible to recast Frolicher spaces as Frolicher sheaves in which case all we have done is replace the word “space” by “sheaf” in the discussion without fundamentally changing the point.

That point, as I see it, is the following:

Is smoothness just something that I declare it to be, or is there something more fundamental going on?

Chen, Mostow, Sikorski, Souriau, all say: We declare these things and no other to be smooth.

If it looks like a duck, quacks like a duck, but isn’t in my list of ducks, it ain’t a duck.

Frolicher says: hang on, there may be some other things that ought to be smooth but aren’t in your list. I’m going to add them in.

If it looks like a duck, quacks like a duck, looks good in orange sauce, then let’s just call it a duck.

(If the “duck” line is starting to wear thin, I’ll start quoting a sketch from long ago about Thatcher, Reagan, and Gorbachov discussing democracy.)

To end, I’d like to highlight an example that I consider in the paper: the pinched plane. Take $\mathbb{R}^2$ and squash the $x$-axis to a point. In the paper I construct a curve in this space which is Frolicher-smooth but not diffeologically-smooth. This curve approaches the squashed point very slowly (so that it is infinitely slow when it reaches it) and then goes out again. Thus it is continuous, is smooth on both half lines, and all derivatives (such as they are) agree at the point where it goes through the squash. However, it does not lift to a smooth curve in $\mathbb{R}^2$ and so is not diffeologically smooth.

Andrew

Posted by: Andrew Stacey on January 7, 2008 10:55 AM | Permalink | Reply to this

### Re: Comparative Smootheology

MathSciNet is out of date. Boman’s article can be found online. Mathematica Scandinavica has back issues available online, free for issues up to five years ago. Boman’s article is in issue 20, here is a direct link to it as a pdf.

Andrew

Posted by: Andrew Stacey on January 7, 2008 11:06 AM | Permalink | Reply to this

### Re: Comparative Smootheology

I’m only interested in Frolicher spaces, so I’m even more prejudiced than you, Bruce.

I like the idea of Frölicher spaces, and, while I haven’t to date really used it to do anything, I’ll keep it in mind and see if the application will appear where I find myself wanting to use it.

But let me add one more point concerning presheaves:

for quite a while I was very happy with the fact that Chen-smooth spaces (the version we used, at least) are quasi-representable or concrete (is there a good standard terminology?) sheaves:

while there isn’t an ordinary smooth set such that all plots of my Chen-smooth space are maps into this ordinary smooth set, there is, at least, a mere set, such that all plots are at meast maps of sets into that fixed set.

The presence of this underlying set is a common feature of most of the definitions we are talking about, Chen, diffological, also Frölicher – but is not required for general presheaves.

For quite a while I was very happy with this requirement of quasi-representability. It certaily looks like a good first step away from ordinary manifolds to spaces with a more general smooth structure.

But at some point, I became dissatisfied with it. That point was reached when I finally noticed – here (last subsection) – that an immensely useful presheaf on manifolds, which would work wonders if I could regard it as a generalized smooth space, is not manifestly of this form (and I doubt that it is secretly of that form, but should I be wrong about this, I’ll take everything back).

That presheaf is the following: For $g$ any Lie $\infty$-algebra (for instance an ordinary Lie algebra), it is the presheaf I am calling $X_g$ in the discussion here, which is defined by the fact that its plots are “all maps that look like maps into a space on which the Chevalley-Eilenberg algebra $CE(g)$ of g is the algebra of differential forms”:

$X_g : U \mapsto Hom_{dg-algebra}(CE(g), \Omega^\bullet(U)) \,.$

(Can anyone prove or disprove that presheaves of this kind can or cannot be equivalent to quasi-representable presheaves? I haven’t even seriously tried to think about this.)

These kinds of presheaves seem to be very important for everything gauge theoretic. In fact, this presheaf plays the role of something like a smooth version of the classifying space $B G$, though I cannot at the moment make that statement fully precise. But this is something I want to think about more.

Actually, it seems to me that more is true here: it seems to be that all of what is called rational homotopy theory can be regarded as the study of simplices in smooth spaces of the above form.

The technical details which I am still slightly unsure about here is how to exactly see that the above prescription gives just the right amount of maps, not too many. But this means that at worst some tuning of the technicalities here is required to make this true.

And if its true, that would be mighty reason to study smooth spaces which do not necessarily have an underlying set as Chen spaces, diffeological spaces and Frölicher spaces have.

Posted by: Urs Schreiber on January 7, 2008 12:57 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Actually, it seems to me that more is true here: it seems to be that all of what is called rational homotopy theory can be regarded as the study of simplices in smooth spaces of the above form.

JIM: why restrict to smooth spaces? or rather why APPLY rational homotopy theory
only to smooth spaces? or even just to spaces at all? It’s a perfectly good theory in its own rite (not misspelled).

Posted by: jim stasheff on January 7, 2008 1:47 PM | Permalink | Reply to this

### Re: Comparative Smootheology

why restrict to smooth spaces? or rather why APPLY rational homotopy theory only to smooth spaces? or even just to spaces at all? It’s a perfectly good theory in its own rite (not misspelled).

Okay, right. What I mean is:

Sullivan models in rational homotopy theory seem to be about simplices in generalized spaces which come to us as certain presheaves.

Posted by: Urs Schreiber on January 7, 2008 1:52 PM | Permalink | Reply to this

### Re: Comparative Smootheology

I’ll start quoting a sketch from long ago about Thatcher, Reagan, and Gorbachov discussing democracy.

Posted by: jim stasheff on January 7, 2008 1:49 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Andrew Stacey said:

Take $\mathbf{R}^2$ and squash the x-axis to a point. In the paper I construct a curve in this space which is Frolicher-smooth but not diffeologically-smooth. This curve approaches the squashed point very slowly (so that it is infinitely slow when it reaches it) and then goes out again. Thus it is continuous, is smooth on both half lines, and all derivatives (such as they are) agree at the point where it goes through the squash. However, it does not lift to a smooth curve in $\mathbf{R}^2$ and so is not diffeologically smooth.

Is it true that this space is not locally diffeomorphic to a closed subspace of $\mathbf{R}^n$? More generally, is there a theory of local models for Froelicher spaces? If not, does this make them hard to work with? My general feeling is that geometries are built out of gluing certain kinds of local models together using certain kinds of maps, but maybe I’m too set in my ways.

Posted by: James on January 8, 2008 1:13 AM | Permalink | Reply to this

### Re: Comparative Smootheology

There appears to be a general consensus that “smoothness” is a weak phenomenon; or at least that strong smoothness is too fragile a concept for many purposes. That is to say, “smoothness” is something that should, as Urs put it, be probed by test functions into or out of known test spaces. I think that this leads to three mutually independent questions.

1. What are the test spaces?

2. Should the test functions be functions from the test spaces, to the test spaces, or both? If both, what compatibility relations should there be? The obvious one is that the composition of a function from a test space followed by a function to a test space should be smooth, are there any more relations?

3. Is the theory set-based?

Posted by: Andrew Stacey on January 9, 2008 10:23 AM | Permalink | Reply to this

### Re: Comparative Smootheology

This may be a little odd, posting a comment and then immediately replying to it, but I wanted the above comment to stand on its own without any obvious political bias. This comment is highly biased. I want to concentrate on the first question that I asked, though I’ll mention the others in passing.

1. What are the test spaces?

This is where I see the real asymmetry in the various definitions.

When taking maps into the test spaces there seems to be only one candidate: $\mathbb{R}$. I believe that the reason for this is that everyone knows that a map into a suitable subset of $\mathbb{R}^n$ is smooth if and only if each of the projections to $\mathbb{R}$ is smooth so for set-based smootheologies no-one ever considered anything bigger as it was clearly just “bloat”.

For maps from the test spaces then the analogous result holds but is much less well-known (and harder to prove). For open subsets it is Boman’s theorem; for convex subsets one can find it in Kriegl and Michor (as referenced by my paper).

For set-based smootheologies, this distinction is minor. For more general smootheologies then it becomes important because the larger one’s test category then the more information that one’s theory can hold. But that is only useful if that information is pertinent. If it is not, one may as well do without it.

So here’s a question: what do I gain by taking plots defined on spaces other than $\mathbb{R}$?

Note that this is not a Frolicher vs Chen issue. One can define a Chen-like theory using plots from $\mathbb{R}$.

2. Test functions in, out, or both? If both, what relations?

This is the real Frolicher vs the rest question. To avoid repeating myself, I’ll refer you to the introduction of my paper.

3. To set or not to set?

I have no real opinion on this, though I’ve been pondering it (and no one has commented on my first ponderings above so I’m not sure how much of an issue it is for the rest of you either). I think that there will be a Frolicher-type definition of a presheafy smootheology but what the exact definition is, I’m not sure yet. I’ll keep you all posted (unless you get so bored of me harping on about this that I get thrown out of this blog).

So in summary. The fact that I’m even asking qn 1 is really an historical issue. Qn 2 is the heart of the Frolicher vs The Rest. Qn 3 is an interesting development that ought to be able to be applied to any smootheology.

Andrew

PS By the way, Urs - do you mind if I steal the title of this entry? “Comparative Smootheology” is a fantastic phrase!

Posted by: Andrew Stacey on January 9, 2008 10:43 AM | Permalink | Reply to this

### Re: Comparative Smootheology

By accident I discover that, also by accident, I missed that remark by Andrew:

By the way, Urs - do you mind if I steal the title of this entry? “Comparative Smootheology” is a fantastic phrase!

I feel honored. Please feel free to use it!

Actually, I’d feel like talking much more with you about comparative smootheology than we are doing here. I think that idea of Frölicher presheaves might have some potential.

For instance, I agree very much with you that it looks a little disconcerting that there are presheaves on smooth spaces, like those coming from just continuous plots, which look like they do not actually deserve to be addressed as generalized smooth spaces.

I would also like to better understand if and how every “good” presheaf might be related to a quasi-representable one.

Lots of reasons to further think about comparative smootheology…

Posted by: Urs Schreiber on January 26, 2008 11:59 AM | Permalink | Reply to this

### Re: Comparative Smootheology

naively I ask: why bother about smoothness?

Posted by: jim stasheff on January 9, 2008 1:37 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Jim Stasheff writes:

naively I ask: why bother about smoothness?

There’s a branch of math known as calculus which has proved useful here and there.

Posted by: John Baez on January 9, 2008 6:53 PM | Permalink | Reply to this

### Re: Comparative Smootheology

But where (other an applications to things we already consider smooth) is formal = algebraic calculus not enough? cf. Gel’fand or H Cartan

Posted by: jim stasheff on January 10, 2008 12:45 AM | Permalink | Reply to this

### Re: Comparative Smootheology

But where (other an applications to things we already consider smooth) is formal = algebraic calculus not enough? cf. Gel’fand or H Cartan

You might have to expand on that for me to be able to give a decent reply, but for the moment let me say this:

the main application in the background here, both for Andrew with his concern with Dirac operators on loop space, which can be thought of as arising as the supercharges of 2-dimensional surpsymmetric sigma-models with the target base manifold in question, as well as for John, Burce and myself in the contexts that we keep discussing here, that main application is the understanding of mapping spaces $hom(X,Y)$ for $X$ and $Y$ ordinary manifolds, since it’s these kind of spaces which appear as configuration space for physical systems of sigma-model type (most every fundamental system you have ever seen).

I am not entirely sure what you are referring to by formal algebraic calculus here. What I know is that there are a couple of ways to get finite algebraic approximations to mapping spaces $hom(X,Y)$. One sneaky and very useful one is the one used in what is called topological conformal field theory (TCFT), where $hom(X,Y)$ is essentially replaced by its cohomology complex.

That’s goiod and useful but not, as far as I can see, a full substitute for the real thing $hom(X,Y)$.

Ultimately we want to be able to handle, rigorously as they say (namely really: handle) action functional defined on spaces of the form $hom(X,Y)$, be able to integrate them over $hom(X,Y)$ in some good sense such as to compute partition functions and do other differential geometric operations here, or there generalizations to generalized smooth spaces.

But please let me know more precisely what you had in mind, if what I just said does not properly address your remark.

Posted by: Urs Schreiber on January 10, 2008 9:42 AM | Permalink | Reply to this

### Re: Comparative Smootheology

just for starters:
Frank Adams had a purely algebraic construction for calculating the homology
of the based loop space on a simply connected space
KTChen had a smooth construction for calculating the cohomology
of the based loop space on a smooth manifold

M need not be simply connected (though I suspect Adams could be upgraded at least for coefficients Q or R)

`analytic’ results could be obtained (see work of his student Hain) especially for complex manifolds

Posted by: jim stasheff on January 10, 2008 2:03 PM | Permalink | Reply to this

### Re: Comparative Smootheology

a purely algebraic construction for calculating the homology

That’s similar to the TCFT example which I mentioned: if we decide to be intersted only in the (co)homological aspects of a space, finite algebraic models suffice.

But in general we need the entire smooth space, not just its (co)homology, I think.

Of course the (co)homologica aspect is the comparatively most accessible one. That’s why topological quantum field theory is so well developed, while already 2-dimensional non-topological QFT is such a challenge.

But I must admit that I am actually not entirely sure how much of full 2-dimensional CFT is actually be reproduced from just 2-dimensional TCFT.

Probably quite a bit.

Posted by: Urs Schreiber on January 10, 2008 2:17 PM | Permalink | Reply to this

### Re: Comparative Smootheology

me thinks you want to head toward operators on Hilbert spaces?

Posted by: jim stasheff on January 10, 2008 8:40 PM | Permalink | Reply to this

### Re: Comparative Smootheology

*crawl out of hiding*

As a former numerical analysis guy, I found it fascinating when I learned you can do calculus without smoothness.

*crawl back into hiding*

Posted by: Eric on January 10, 2008 5:50 AM | Permalink | Reply to this

### Re: Comparative Smootheology

I’m late to this party, and I’ve arrived nearly empty-handed, but…

One thing I like in the theory of differentiable manifolds is the symmetric approach to defining tangent and cotangent spaces. (You’ll find this, for instance, in Mac Lane and Moerdijk’s topos theory book.) This goes as follows.

Take a differentiable manifold $X$ and a point $x$ of $X$.

• Let $V_x$ be the vector space of germs at $0$ of differentiable maps $\mathbb{R} \to X$ sending $0$ to $x$. (I don’t know if I’m using the germ terminology quite right; I mean an element of $V_x$ to be an equivalence class of pairs $(U, \gamma)$ where $U$ is an open neighbourhood of $0$ in $\mathbb{R}$ and $\gamma: U \to X$ is a differentiable map such that $\gamma(0) = x$.)
• Let $V^x$ be the vector space of germs at $x$ of differentiable maps $X \to \mathbb{R}$ sending $x$ to $0$. (An element is an equivalence class of pairs $(W, f)$ where $W$ is an open neighbourhood of $x$ in $X$ and $f: W \to \mathbb{R}$ is a differentiable map such that $f(x) = 0$.)
Then there is a bilinear pairing \begin{aligned} V^x \times V_x & \to & \mathbb{R} \\ ([f], [\gamma]) & \mapsto & \langle [f], [\gamma] \rangle = (f \circ \gamma)'(0). \end{aligned} Of course, this pairing is very degenerate, but like all pairings you can force it to be nondegenerate: quotient $V^x$ out by the subspace consisting of elements $[f]$ such that $\langle [f], - \rangle = 0$, and similarly $V_x$. And that’s the definition of the tangent and cotangent spaces: it gives the usual nondegenerate pairing $T^*_x \times T_x \to \mathbb{R}.$

So, because this is something I like, I wonder whether anything remotely similar happens in any of the definitions of “smooth space”? If so, it will make me like that definition!

Posted by: Tom Leinster on January 9, 2008 6:32 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Wow, those are cool definitions. Didn’t know you could do that.

Posted by: Bruce Bartlett on January 10, 2008 1:46 AM | Permalink | Reply to this

### Re: Comparative Smootheology

I’m just off to teach so no time to post a long reply … have you looked at the final section of my paper that started all this debate? What you say above seems very similar to what I put there. I’ll have to check the reference you gave to see what the overlap is.

Andrew

Posted by: Andrew Stacey on January 10, 2008 7:56 AM | Permalink | Reply to this
Read the post Differential Forms and Smooth Spaces
Weblog: The n-Category Café
Excerpt: On turning differential graded-commutative algebras into smooth spaces, and interpreting these as classifying spaces.
Tracked: January 30, 2008 10:16 AM

### Re: Comparative Smootheology

I’m climbing down from the comment tree, first because the branches get pretty thin the higher up you go, but also because I want to start a new thread by asking Andrew some questions.

First, let me clear the slate by saying (to Andrew) that perhaps I misrepresented myself earlier. At some point Urs had said something about presheaves and sheaves and I interjected a comment in an attempt to clarify, and then Urs and I got involved in a little side discussion – this may have given a misleading impression that I was “taking sides” (favoring sheaf-theoretic approaches) in a general discussion between you two, comparing different approaches to smootheology. Actually, I’m interested in all these approaches! And just now I’m having some fun contemplating Frölicher spaces, which is what I wanted to talk about now from a general point of view.

By the way, Andrew – have you talked with other categorists about this stuff? I know that distinguished categorists like Lawvere and Schanuel have spoken positively of Frölicher spaces and the Kriegl-Michor book, and I expect many others have thought hard about smootheology. On the Continent not too far from you is Anders Kock (I believe in Aarhus, Denmark) – he may very well be interested, and knowledgeable, about this stuff.

It seems to me that the mechanism underlying Frölicher spaces must be very general, and that’s what I wanted to talk about. So, let $M$ be any concrete monoid, that is, a pair consisting of a set $R$ and a submonoid of the monoid of all endofunctions, $M \subseteq hom(R, R)$. For classical Frölicher spaces, $R$ will be the set of reals and $M$ the monoid of smooth endofunctions on the reals.

Then, there is a notion of saturation given by what categorists call a Galois connection, as follows. Let $X$ be any set; given subsets $F \subseteq hom(X, R)$ and $C \subseteq hom(R, X)$, there is a product $F \cdot C \subseteq hom(R, R)$ consisting of all composites $f c$ where $f \in F$ and $c \in C$. This gives a map

$(-) \cdot (-): P(hom(X, R)) \times P(hom(R, X)) \to P(hom(R, R))$

where $P$ denotes the covariant power set functor; this product $F \cdot C$ preserves colimits (unions) in each of its separate variables $F$, $C$. It follows on very general grounds that there are accompanying “divisions” or internal homs which are adjoint to products, in the sense of there being equivalences

$(F \subseteq E/C) \Leftrightarrow (F \cdot C \subseteq E)$

$(C \subseteq F\backslash E) \Leftrightarrow (F \cdot C \subseteq E)$

whenever $E \in P(hom(R, R))$. Explicitly,

$E/C = \{f: X \to R | \forall_{c: R \to X} c \in C \Rightarrow f c \in E\},$

$F\backslash E = \{c: R \to X | \forall_{f: X \to R} f \in F \Rightarrow f c \in E\}.$

From this, it follows that for any $E \subseteq hom(R, R)$, say $E =$ the given monoid $M$, we have a Galois connection between the posets $P(hom(X, R))$ and $P(hom(R, X))$, according to the equivalence

$(F \subseteq M/C) \Leftrightarrow (C \subseteq F\backslash M).$

This can be reinterpreted as an adjoint pair

$(M/(-): P(hom(R, X)) \to P(hom(X, R))^{op}) \dashv ((-)\backslash M: P(hom(X, R))^{op} \to P(hom(R, X)))$

from which we get a closure or saturation operator on $P(hom(R, X))$, which is just the monad taking $B$ to $(M/B)\backslash M$. There is a similar saturation operator on $P(hom(X, R))$.

Then, we can define an $M$-Frölicher space as a triple $(X, F, C)$ where $F \subseteq hom(X, R)$ and $C \subseteq hom(R, X)$ are saturated with respect to division or homming into $M$, in the sense given above.

It is extremely tempting to speculate that the category of $M$-Frölicher spaces (with the obvious notion of morphism) has all the formal properties one has for ordinary Frölicher spaces, e.g., completeness, cocompleteness, cartesian closure, … But note that this construction is very general!

Any thoughts on this? Or is there something special about the monoid of smooth endofunctions on $R$ that one needs to invoke to make everything come out right?

There are other natural questions to ask (especially if one is comparing/contrasting with sheaf-theoretic approaches), but I’ll stop here for now.

Posted by: Todd Trimble on February 5, 2008 5:17 AM | Permalink | Reply to this

### Re: Comparative Smootheology

Arrgh… how dumb of me. Obviously in the definition of $M$-Frölicher space $(X, F, C)$, the saturation conditions should have read: $F = M/C,$ $C = F\backslash M$

(whence it follows that $F$ and $C$ are closed in the sense I gave before). I think the reason I made such a dumb mistake is that $C$ and $F$ determine one another by these formulas if they are closed, and then I simply forgot to give these formulas! Duh.

Posted by: Todd Trimble on February 5, 2008 6:45 AM | Permalink | Reply to this

### Re: Comparative Smootheology

Hi Todd,

First let me say that there’s absolutely nothing wrong with taking sides; and there’s also nothing wrong with being an interested bystander (though perhaps you should wear a flak-jacket for when the integral signs start flying around). Your input is extremely useful.

You are absolutely right about more general spaces than Frolicher spaces. On my trip to the library the other day, I found SLN 1174 in which there is an article by Chen promoting his differentiable spaces immediately followed by an article by Frolicher promoting his view! As the volume is the proceedings from a conference, it is clear that each knew of the other’s point of view so it’s an interesting read.

Anyway, in Frolicher’s paper is the general description that you give. In fact, it’s even more general: the set $M$ need not be a monoid, and it can be a set of morphisms $A \to B$ with $A \ne B$. The general discussion proceeds pretty much as you outline. Frolicher refers to the resulting category as the Petermann category of $M$. He also name-drops Lawvere, Schanuel, and Zame.

You are right that this is too general to guarantee cartesian closedness. There is a condition on $M$ that needs to be satisfied. Let me see if I can distill it from Frolicher’s paper.

We start with a set $M \subseteq B^A$ with $B$ and $A$ two sets. We assume that $M$ contains the constant maps. The sets $A$ and $B$ have natural $M$-structures, with plots in $A$ being those maps $\alpha : A \to A$ such that $\phi \alpha \in M$ for all $\phi \in M$.

Playing the game “what if we had cartesian closedness” leads one to the conclusion that $M$ would also have a natural $M$-structure. Its plots are the maps $\gamma : A \to M$ for which $\gamma \circ (\alpha, \beta) \in M$ for all $\alpha, \beta$ plots of $A$. Here, $\gamma \circ (\alpha, \beta)$ uses a bit of adjointness: as $\gamma$ is a map from $A$ to $M$, which is itself a subset of maps from $A$ to $B$, $\gamma$ defines in the obvious way a map $A \times A \to B$.
The coplots are then those maps $\phi : M \to B$ for which $\phi \circ \gamma \in M$ for all plots $\gamma$ as above.

For this to be a genuine $M$-structure it must satisfy the saturation condition: that a map $\gamma : A \to M$ is a plot if and only if $\phi \circ \gamma \in M$ for all coplots $\phi$. This turns out to be equivalent to cartesian closedness.

The proof of this for $C^\infty(\mathbb{R},\mathbb{R})$ is non-trivial but Frolicher does give a couple of examples where it is straightforward: $M = C(\mathbb{R},\mathbb{R})$ and $M = l^\infty$ (viewed as a subset of $\mathbb{R}^\mathbb{N}$.

I am not surprised that the heavy-weights know of and have opinions on the various approaches to smooth objects. The difficulty lies in finding out those opinions! I haven’t yet made it down to Aarhus, but I’m sure I will and when I do I’ll nobble Anders Kock (incidentally, in the same volume of SLN there’s an article by him on Synthetic Geometry so it’s an extremely useful find! I’m actually glad that the original article that I wanted to find is not available online.) Also, when I was going around talking about Dirac operators on loop spaces I found that people didn’t really know how to make the loop space into a smooth manifold; they tended to think of Milnor’s paper on infinite dimensional Lie groups as the definitive statement. I only came across this viewpoint by accident: I was looking in the AMS bookstore and saw that ‘The Convenient Setting of Global Analysis’ was quite cheap so bought a copy. I only got through it because I know a bit more functional analysis that the average topologist. For someone who doesn’t know much FA it’s quite a hard read, I think (which is why I wrote up some notes on the differential topology of loop spaces - which one can find on my homepage).

I freely admit that I’m not doing anything particularly new here in promoting Frolicher spaces; though I don’t think a mathematical comparison has been done before in the lines of my paper which started all this discussion. Even the tangent-cotangent stuff at the end has its germ (weak$^*$ pun intended) in Frolicher’s paper in SLN (though I based it on ideas the Kriegl and Michor book as I hadn’t seen this article then).

On the other hand, there does seem to be an increased interest in loop/path spaces as manifolds/smooth spaces so I think that there’s genuine value in what we’re doing, even if the experts knew it all a long time ago!

But maybe the experts can save us a bit of time. Is there a non-set-based version of these $M$-structures? If so, we can just adapt it to our purposes.

Posted by: Andrew Stacey on February 5, 2008 8:52 AM | Permalink | Reply to this

### Re: Comparative Smootheology

I should also say, as it may help people to understand my methods, that I’m not much of a “big picture” mathematician. I tend to work by finding problems to solve and then looking for ways to solve them. Whenever I try to look at the “big picture” or try to say “I’m going to learn all about XYZ” without a motivating problem, then I tend to feel as though I’ve stepped into the total perspective vortex (sadly the wiki entry for that is better than the h2g2 one). After a glimpse at the totality of mathematics, I quickly find myself running for my cave with a towel wrapped firmly around my head (though I’m no fool, I do grab the fairy cake to eat later).

So it’s extremely useful to have someone like Todd saying, “This is very similar to the ideas of ABC.” or “Have you looked at CDE?”. I just hope no one says “This is all done in FEG and done much better.”!

Posted by: Andrew Stacey on February 5, 2008 9:40 AM | Permalink | Reply to this

### Re: Comparative Smootheology

I completely agree with Todd about the comment tree. The thin air at high altitute is getting a little rare so I’m climbing down as well. This is really a reply to Urs’ comment.

There’s a danger involved in starting out with both a set of plots and a set of coplots, compatible but not necessarily saturated. The danger is that there are two ways to saturate such a pre-structure and they may not be equivalent.

The simplest example is to take a set $X$ and take plots and coplots to be simply constant functions. Then the two saturations are $(X, X, U^X)$ and $(X, X^U, U)$ (where I wrote the codomain in place of the set of constant functions into that codomain). These are at opposite ends of the scale.

So when you give a not-necessarily-saturated pair then you also need to declare yourself: are you a robber or a cop? That is, are you involved in plotting or coplotting. Whichever you view yourself as, you may as well just throw away the other family as we’ll simply ignore it. For a good example of this, see Smith’s paper on de Rham theory. His definition of a differentiable space involves both functionals and plots but it is clear from his treatment that he is a coplotter (a good way to see if someone is a plotter or coplotter is to see how they define morphisms).

I suspect that you are a robber, Urs. So chuck your coplots in the bin. Your statement

It could turn out that there are more coplots in these examples than I am seeing currently.

is back-to-front. It is true, but since you view plots as primary, it is irrelevant. The statement you should have made is

It could turn out that there are more plots in these examples than I am seeing currently.

Here’s the recipe for turning a pre-structure into a structure, even though we aren’t yet sure what that is. I’ll explain it for robbers, but the adaptation for cops is obvious.

Start with a presheaf, say $X$, on your favourite category, say $\mathcal{S}$. Each object in $\mathcal{S}$ defines a canonical presheaf by taking homs into it, $\mathcal{S}(-,S)$. Define a copresheaf on $\mathcal{S}$ by taking natural transformations from $X$ to $\mathcal{S}(-,S)$. That is,

$F_X(S) = nat(X,\mathcal{S}(-,S)).$

Each object in $\mathcal{S}$ also defines a copresheaf by taking homs out of it, $\mathcal{S}(S,-)$. Define a presheaf on $\mathcal{S}$ by taking natural transformations from this to $F_X$. That is,

$C_X(S) = nat(\mathcal{S}(S,-), F_X(S))$

I think, but I haven’t checked, that if we continued this process then it would stabilise here. That is, if we took the obvious copresheaf again then it would be $F_X$. This is what we should mean by saturated.

There is an obvious natural transformation $X \to C_X$ and an obvious pairing $C_X \times F_X \to \mathcal{S}(-,-)$.

The questions that this raises in my mind are:

1. Do all the details check out?

2. What does this look like for Urs’ example (or any of the simplifications)?

3. Is $C_X$ actually a sheaf, $F_X$ a cosheaf (what does that mean)? How do they behave if we change the original category $\mathcal{S}$?

4. Is the natural transformation $X \to C_X$ injective?

5. How close to quasi-representability is the resulting thing?

Any thoughts anyone? Or am I barking up the wrong tree (always a danger that having climbed down the comment-tree, I’ve gotten lost in the forest).

Todd, does this have any resonance with anything that you know about in the wider context?

Posted by: Andrew Stacey on February 5, 2008 9:19 AM | Permalink | Reply to this

### Re: Comparative Smootheology

This is all rather cool. Thanks Andrew and Todd, for all this input.

I have so far just quickly glanced at your latest comments, will read them later in more detail.

Just one quick comment:

I find very interesting Andrews latest remark about how to saturate any given presheaf.

If our discussion boils down to the statement that we want to address as a generalized smooth space not all presheaves on test domains, not even all sheaves, but all saturated (pre?)sheaves, I’d be rather happy. That sounds like a good statement.

By the way: there must be some general topos-theoretic nonsense dealing with this notion of saturation, which is somewhing one can consider for presheaves on arbitrary domain categories. How do topos theorists think of that? What does it mean fully generally?

Indeed, if it turns out that we want to settle for saturated presheaves, I will want to understand what the saturation of the sheaf of flat $g$-valued forms. Right now I have no real clue how to approach that question, though.

Posted by: Urs Schreiber on February 5, 2008 10:34 AM | Permalink | Reply to this

### Re: Comparative Smootheology

Andrew, thanks for your explanation of necessary and sufficient conditions for cartesian closedness. They made a lot of sense.

Way back when, I asked a question about local cartesian closedness; I’d like to amplify on that. Quite often in mathematics, one encounters situations where one wants to consider structures varying over a structure. Examples spring readily to mind, for example in algebraic geometry where one considers the category $Sch/S$ of schemes varying over a scheme $S$, or in fiberwise topology as developed say by Ioan James (who spells it ‘fibrewise’ ;-)), where spaces vary over a space. I think it’s arguable that a truly convenient setting for global smooth analysis would be a category $Smooth$ such that each slice category $Smooth/S$, of smooth spaces varying over a smooth space $S$, is also cartesian closed. In categorical parlance, one would say in that case that $Smooth$ is ‘locally cartesian closed’.

So my question is: has this condition been studied by Frölicher, Chen, et al.? I can tell you now that certainly the sheaf models have this property: it’s a basic theorem that slices of toposes are toposes.

On another front: speaking to Urs, you wrote

There’s a danger involved in starting out with both a set of plots and a set of coplots, compatible but not necessarily saturated. The danger is that there are two ways to saturate such a pre-structure and they may not be equivalent. The simplest example is to take a set $X$ and take plots and coplots to be simply constant functions. Then the two saturations are $(X, X, U^X)$ and $(X, X^U, U)$ (where I wrote the codomain in place of the set of constant functions into that codomain). These are at opposite ends of the scale.

I’m a little confused here. I thought that to saturate a non-saturated pre-structure, you had to saturate back and forth. For instance, in your example where you start with constant functions as plot and coplots, let’s say you start with the plots and then saturate the coplots; then you get the pre-structure $(X, X, U^X)$ as you say. But then, in general the plots are no longer saturated with respect to the coplots (as one can see taking $X = U = \mathbb{R}$), so you have to saturate again the other way to get an honest structure (and mercifully, things stabilize at that point). [In the lingo I used earlier, the closure or saturation operator was $C \mapsto (M/C)\backslash M$, where $M/-$ saturates the coplots with respect to plots, and $-\backslash M$ saturates plots with respect to coplots.] Or did I misunderstand something?

Lots of interesting things to discuss… I need to have a think about them!

Posted by: Todd Trimble on February 5, 2008 3:52 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Apologies to Andrew: in my previous comment, I think I had misread his notation. If one saturates functions with respect to constant curves, one gets all functions (not necessarily smooth), which I guess he (Andrew) would denote by $R^X$. But then, if one goes back and saturates curves with respect to all functions, one gets just constant curves. So he gave the back-and-forth saturations after all, and his example works as he said. Sorry about that.

I hope to come back to some other points in Andrew’s comment later.

Posted by: Todd Trimble on February 6, 2008 3:44 PM | Permalink | Reply to this

### Re: Comparative Smootheology

To partly answer your question on local cartesian closedness: I don’t know if it has been studied before. I’ve been trying to see if it has a simple answer in a similar vein to that of cartesian closedness but no inspiration has yet struck.

in fiberwise topology as developed say by Ioan James (who spells it ‘fibrewise’ ;-))

The word you are missing there is correctly, as in “who spells it correctly ‘fibrewise’”.

Posted by: Andrew Stacey on February 12, 2008 8:42 AM | Permalink | Reply to this

### Re: Comparative Smootheology

Todd, does this have any resonance with anything that you know about in the wider context?

Yes! In fact I wrote something about this on another thread that David Corfield started, which is headed I think for the categorified (categlorified?) view of modal logic.

The process you describe for obtaining a copresheaf $F$ from a sheaf $X$ is a particular case of something general, called conjugation by Lawvere. For any small category $S$, there is a pair of contravariant functors

$Set^{S^{op}} \to Set^S: X \mapsto [s \mapsto hom_S(X, S(-, s))]$

$Set^S \to Set^{S^{op}}: F \mapsto [s \mapsto hom_{S^{op}}(F, S(s, -))]$

and these are part of an adjunction (now using covariant functors)

$(Set^{S^{op}} \to (Set^S)^{op}) \dashv ((Set^S)^{op} \to Set^{S^{op}})$

which is analogous to the saturation process, as you intimate.

However, I believe there is an important difference: in the situation where we deal with Frölicher spaces, the adjunction is between posets (in this case power sets, e.g., $P(hom(X, R))$ and $P(hom(R, X))$). Here, monads $M$ are always idempotent: we have $M M = M$. This is why saturation stabilizes. In the more general case of conjugation, one cannot expect this phenomenon: the monad/comonad is not idempotent, and stabilization does not occur.

I haven’t had a chance to think yet about your other questions.

Posted by: Todd Trimble on February 8, 2008 6:22 PM | Permalink | Reply to this

### Re: Comparative Smootheology

This looks like something very basic. And it’s definitely something I haven’t seen before.

Does conjugation (in this sense) come up anywhere in geometry? I think cosheaves can be used to give a general point of view on homology, much like sheaves do for cohomology (conservation of co). But duality, when you have it, says homology and cohomology are just different ends of the same dog. Which seems to suggest that the same could be said of sheaves and cosheaves.

Does this have anything to do with conjugation?

Posted by: James on February 9, 2008 1:58 AM | Permalink | Reply to this

### Re: Comparative Smootheology

It must. I have to admit that I have not thought deeply about (Isbell) conjugation, but clearly Lawvere has – he has talked about it probably hundreds, maybe even thousands of times over the decades in lectures, seminar talks, conferences, private conversations… He clearly considers it absolutely fundamental (in algebraic geometry, in general topology, in functional analysis…). Perhaps now is a good time for me and others here to start taking it seriously.

In Taking Categories Seriously, Lawvere writes (p. 17):

Now an extremely fundamental construction in enriched category theory is the adjoint pair known as Isbell conjugation $(-)^*: V^{A^{op}} \stackrel{\leftarrow}{\to} (V^A)^{op}: (-)^#$ … The general significance of this construction is somewhat as follows: if $V$ is the category of sets, or simplicial sets, or (properly construed) topological spaces, or bornological linear spaces, and if the $V$-category $A$ is construed as a category of basic geometrical figures, then $V^{A^{op}}$ is a large category which includes general geometrical objects that can be probed with help of $A$, but that would inevitably come up in a thorough study of $A$ itself. On the other hand, $V^A$ includes very general algebras of quantities whose operations (à la Descartes) mirror the geometric constructions and incidence relations in $A$ itself. Then the conjugacies are the first step toward expressing the duality between space and quantity fundamental to mathematics: $(-)^*$ assigns to each general space the algebra of functions on it, whereas $(-)^#$ assigns to each algebra its “spectrum” which is a general space. Of course neither of the conjugacies is usually surjective; the second step in expressing the fundamental duality is to find subcategories with reasonable properties that still include the images of the conjugacies [here Lawvere has a diagram consisting of a geometric subcategory $\mathcal{X} \hookrightarrow V^{A^{op}}$ containing $A$, an algebraic subcategory $\mathcal{A} \hookrightarrow V^A$ containing $A^{op}$, and an adjoint pair $\mathcal{X} \stackrel{\leftarrow}{\to} \mathcal{A}^{op}$ which is a restriction of the conjugation adjoint pair $(-)^* \dashv (-)^#$, such that $(-)^*$ factors through $\mathcal{A}^{op}$ and $(-)^#$ factors through $\mathcal{X}$]. For example, if $V$ is the category of sets and $A$ is a small category with finite products, we could take $\mathcal{X}$ to be the topos of “canonical sheaves” on $A$, and $\mathcal{A}$ to be the algebraic category of all finite-product-preserving functors.

As usual, Lawvere expresses himself in highly concentrated form, but it seems the last two sentences in particular has resonance for the particular conversation taking place between Andrew and Urs (and me a little bit). It’s probably worth pursuing.

PS: I’m not at home, so I’m not sure whether the math calligraphy came out as I intended – Urs, would you mind having a look?

Posted by: Todd Trimble on February 9, 2008 3:09 PM | Permalink | Reply to this

### Re: Comparative Smootheology

OK, I think I catch the drift of this if I pretend $V$ is the category of sets and $A$ is the category of affine schemes, that is, the opposite of the category of commutative rings. But I don’t really see the point. Maybe the point is to do something in general which already exists in algebraic geometry?

It would interest me if there was some way to get measures or distributions into the picture. They’re dual to functions and probably form a cosheaf.

Posted by: James on February 10, 2008 12:19 AM | Permalink | Reply to this

### Re: Comparative Smootheology

the conjugacies are the first step toward expressing the duality between space and quantity fundamental to mathematics: $(-)^*$ assigns to each general space the algebra of functions on it, whereas $(-)^#$ assigns to each algebra its “spectrum” which is a general space.

This sounds great. And it, indeed, relates nicely with what we were talking about.

I assume the question which concerned us most lately, concerning the need/desire to pass from presheaves to saturated (pre?)sheaves is addressed by Lawvere’s remark that one should

find subcategories with reasonable properties that still include the images of the conjugacies

Just so that I am sure I am following the current state of the discussion:

do we think that the saturation property that Andrew brought is the right way to go about “Frölicher presheaves”. I.e., is it reasonable to trust that the “good” definition of generalized smooth space is “saturated (pre?)sheaf on smooth test domains”?

I’d want some concrete examples to practice on here, but need to think more about it. In particular, I’ll want to know if sheaves like “closed $n$-forms on test domains” are saturated and, if not, what their saturation is.

Unfortunately, I won’t find much time to think about this in detail in the next two weeks, probably.

Posted by: Urs Schreiber on February 11, 2008 7:53 AM | Permalink | Reply to this

### Re: Comparative Smootheology

A-ha! That’s extremely interesting; thanks, Todd.

By analogy with normed vector spaces, we could call a pair $(C,F)$ where $C = F^\sharp$ and $F = C^\flat$ reflexive. The analogy seems to be quite strong: every pair $(C,F)$ embeds in its “bidual” and once we are one step along the process, every pair $(C,F)$ is actually a retract of the next pair.

Before Todd posted the stuff from Lawvere above, I’d tried to figure out when a pair is reflexive and I think I can get it down to a condition on points. From that, I would hazard the following conjectures.

Conjecture

1. If a pair $(C,F)$ stabilises eventually then it stabilises after one full iteration.

(analogous statement: a normed vector space $X$ is reflexive if and only if it is complete and its dual is reflexive; we might need to do one iteration to complete it.)

2. If a pair $(C,F)$ stabilises then it is quasi-representable.

(this doesn’t have an analogous statement but is suggested by the point condition.)

Whilst this analogy suggests that we look at the category of reflexive whatevers, it also suggests that that category might not be a “nice” subcategory of all whatevers. I don’t think that reflexive Banach spaces is a “nice” subcategory of all Banach spaces.

On the other hand, the category of Banach spaces isn’t all that nice either. If we enlarge it to the nearest nice category then we get the category of locally convex topological vector spaces.

So let us try to extend the analogy to locally convex topological spaces. Now we find that we might be in better shape. If we allow ourselves the possibility that our initial LCTVS structure was not the “right” one then we can always modify it a bit to make it reflexive: namely take the weak topology induced by the dual (and the weak topology on the dual).

The analogy here is good. Such topologies are determined by families of test maps into or out of finite dimensional vector spaces ($\mathbb{R}$ will do, but let’s not spoil the analogy with a fight about test domains!) which is precisely the sort of structure we have for smooth objects!

So if the conjecture above is correct: that all reflexive pairs are quasi-representable, then there may well be a reflexify functor that essentially replaces any “strong” structure by the associated “weak” structure.

Which backs up my assertion earlier that smoothness is an essentially weak phenomenon.

Posted by: Andrew Stacey on February 12, 2008 8:29 AM | Permalink | Reply to this

### Re: Comparative Smootheology

On the other hand, the category of Banach spaces isn’t all that nice either. If we enlarge it to the nearest nice category then we get the category of locally convex topological vector spaces.

I’d like to hear a little more what you have in mind when you say the category of Banach spaces is not all that nice (which I’m perfectly prepared to believe). Over at the blog Ars Mathematica, there was a brief exchange on Banach spaces, where I opined that the category of Banach spaces and bounded linear maps of norm less than or equal to 1 is in some respects ‘nicer’ (being complete, cocomplete, symmetric monoidal closed, and monadic over $Set$), but maybe not yet nice enough according to your lights.

But I’m even more intrigued by ‘nearest nice category [being] the category of locally convex TVS’. Can you say a few words about that that (particularly about ‘nearest’ and ‘nice’)? I guess you address this a bit in your next paragraph:

So let us try to extend the analogy to locally convex topological spaces. Now we find that we might be in better shape. If we allow ourselves the possibility that our initial LCTVS structure was not the “right” one then we can always modify it a bit to make it reflexive: namely take the weak topology induced by the dual (and the weak topology on the dual).

where you imply that locally convex TVS have the requisite flexibility to accommodate reflexification. And then this:

The analogy here is good. Such topologies are determined by families of test maps into or out of finite dimensional vector spaces ($\mathbb{R}$ will do, but let’s not spoil the analogy with a fight about test domains!) which is precisely the sort of structure we have for smooth objects!

Aha! Rather an interesting observation.

By the way (and possibly somewhat off-topic): Lawvere often talks about bornological linear spaces as being a nice category. Any thoughts on that?

Posted by: Todd Trimble on February 12, 2008 12:59 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Well, no. The category of Banach spaces with contractions may be ‘nicer’ than that of Banach spaces with linear maps but it seems to me to be a hack to get the category right without any real feeling for the original objects and morphisms! I’m reasonably happy to enlarge a given not-nice category to a nicer one (at least whilst perusing this blog) but to viciously truncate the not-nice category just because it didn’t match one’s current definition of ‘nice’ seems a little … well … brutal.

I suppose that’s a bit subjective; I admit that I haven’t thought much about “Banach spaces with contractions”.

So why are LCTVS better? Well, it is complete and cocomplete without having to modify anything; indeed, every LCTVS is the projective limit of normed vector spaces which is why I called it the ‘nearest nice category’. I suppose the nearest nice category to Banach spaces is complete LCTVS but let’s not quibble on that.

There are also a couple of symmetric monoidal structures on LCTVS’s. In fact, there are quite a few ways of defining a tensor product of LCTVS’s. I think that the inductive tensor product is the ‘correct’ one for the adjoint to the hom-functor (though, again, there are various ways to put an LCTVS structure on the hom-sets so a choice here will determine a choice of tensor product).

Bornological LCTVS’s are, essentially, inductive limits of normed vector spaces. There is a bornological tensor product as well, and a “bornologification” functor. So, yes, unsurprising Lawvere is right when he says that bornological LCTVS’s form a nice category. Bornological LCTVS’s are also characterised by the assertion that “bounded $\implies$ continuous”.

Kriegl and Michor in their book devote a fair amount of time and effort to bornological spaces because they turn out to be quite closely linked to smooth (convenient) vector spaces.

As you admit somewhere over there to not being an analyst, I’m not sure how much detail to give so I’ll stop there for now.

Posted by: Andrew Stacey on February 12, 2008 1:54 PM | Permalink | Reply to this

### Re: Comparative Smootheology

So why are LCTVS better? Well, it is complete and cocomplete without having to modify anything; indeed, every LCTVS is the projective limit of normed vector spaces…

Are you saying that in the category of all TVS, the locally convex ones are characterized by the property that they are projective limits of normed vector spaces? (I seem to recall that local convexity is equivalent to being definable by a family of seminorms; is that memory correct?) Can we change “normed” to “normed finite-dimensional”?

Sorry if these are naive questions. But I’ve never heard it put this way, and it’s really nice!

There are also a couple of symmetric monoidal structures on LCTVS’s. In fact, there are quite a few ways of defining a tensor product of LCTVS’s. I think that the inductive tensor product is the ‘correct’ one for the adjoint to the hom-functor (though, again, there are various ways to put an LCTVS structure on the hom-sets so a choice here will determine a choice of tensor product).

I was quite unaware of that. Where can I read about that? Is this material in Kriegl-Michor?

Bornological LCTVS’s are, essentially, inductive limits of normed vector spaces. There is a bornological tensor product as well, and a “bornologification” functor. So, yes, unsurprising Lawvere is right when he says that bornological LCTVS’s form a nice category.

That, again, sounds just marvelous. But again, inductive limits (by which I guess you mean colimits, right?) are taken relative to which ambient category – all TVS?

Thanks for the link to the free copy of Kriegl-Michor!

This was a very helpful comment, Andrew. Thanks very much.

As you admit somewhere over there to not being an analyst, I’m not sure how much detail to give so I’ll stop there for now.

I also said over there that I don’t mind analysis (and I’m not afraid of analysis) if I think I can get a grip on the conceptual point, and so far you’re doing a great job of conveying that. So please feel free to say as much as you’d like, as long as you don’t mind stupid questions later. :-)

Posted by: Todd Trimble on February 12, 2008 3:45 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Todd thrashed out on his keyboard:

Are you saying that in the category of all TVS, the locally convex ones are characterized by the property that they are projective limits of normed vector spaces? (I seem to recall that local convexity is equivalent to being definable by a family of seminorms; is that memory correct?) Can we change “normed” to “normed finite-dimensional”?

Yes and no (in that order). In a TVS the topology is completely determined by a neighbourhood base at the origin. In a LCTVS that base can be taken to be convex, radial, and absorbing. That is to say,

1. for $x, y \in U$, $t \in (0,1)$ then $t x + (1 - t)y \in U$;

2. for $\vert t \vert \le 1$, $t U \subseteq U$;

3. for $\bigcup_t t U = X$.

From these properties one can deduce that the associated gauge function (aka Minkowski functional) is a continuous semi-norm. Conversely, given a continuous semi-norm, the open ball satisfies the above conditions and is open. Thus there is a family of semi-norms on $X$ such that the topology on $X$ is the projective topology for the identity map viewed as maps $X \to (X, \rho)$.

As $\rho$ is a semi-norm, its kernel is a subspace of $X$. Quotienting out by this kernel we obtain a vector space and the descent of $\rho$ to this quotient is a norm. Moreover, the topology on $(X, \rho)$ is the pull-back topology via the quotient map.

Hence the original topology on $X$ is the projective topology for the maps $X \to X/ ker \rho$. The latter are now normed vector spaces which we can complete to Banach spaces if we so desire without changing the fundamental result.

Sadly, we cannot take the normed vector spaces to be finite dimensional. The simple case of an infinite dimensional normed vector space is a counterexample (the projective family in the above is simply the space $X$ again).

I was quite unaware of [tensor products]. Where can I read about that? Is this material in Kriegl-Michor?

Kriegl and Michor have an appendix on functional analysis which contains all the results that they need but not a lot else, and it’s fairly dense (understandably). They go into a bit of detail on bornological spaces early on in the book because it is important for the main topic. I don’t recommend that book as an initial source on any of this; rather, if one already knows a bit of FA outside Banach space theory then one can follow the Kriegl-Michor stuff on bornological spaces without too much difficulty.

They cite Schaefer and Jarchow, two books that I use as my basic references for this material. If there are any cafe regulars left reading this thread then they might be pleased to know that the tensor products of LCTVS’s dates back to Grothendieck.

That, again, sounds just marvelous. But again, inductive limits (by which I guess you mean colimits, right?) are taken relative to which ambient category – all TVS?

Hmm, don’t have any of my sources in front of me right now. I think that the ambient category is LCTVS (though if you take filtered colimits then it doesn’t matter). Thus a bornological LCTVS is one that is the inductive limit, as a LCTVS, of normed vector spaces. Basically, for any LCTVS one can define a natural associated family of normed vector spaces which map into the given LCTVS in a similar manner to the maps out given above (one starts with suitable bounded subsets rather than open subsets). Unlike the outward bound maps, this may not generate the topology. If it does, the space is bornological. This isn’t usually taken as the definition of bornological, by the way, but is an easy consequence of it.

As will probably come as no surprise, in the category of LCTVS, projective limits are straightforward but inductive limits are more complicated. The naive inductive limit may not be a LCTVS so we have to cast around to find a nearest LCTVS to pass to. This can produce some surprises.

I also said over there that I don’t mind analysis (and I’m not afraid of analysis) if I think I can get a grip on the conceptual point, and so far you’re doing a great job of conveying that. So please feel free to say as much as you’d like, as long as you don’t mind stupid questions later. :-)

Great! And as for stupid questions, I could repeat the “no stupid questions, only stupid answers” line, but I prefer this quote about Pauli:

“One could ask Pauli anything without worrying whether or not he would think it was a stupid question because he thought that all questions were stupid.”

But more seriously, here’s my take on FA.

1. We have these huge vector spaces that we’d like to know something about. We know a fair bit about finite dimensional vector spaces so would like to extend that knowledge to the infinite cases.

2. Secretly, all finite dimensional vector spaces are topological spaces so we start with topological vector spaces.

3. One of the best ways to extend knowledge is via test functions into and out of known spaces. To know that there are “enough” test functions we need the Hahn-Banach theorem to hold; for this we need locally convex topological spaces.

4. At this point, we have to decide what part of finite dimensional theory we feel is most important. A common choice here is the norm as it is so versatile and gives us so many tools. This leads us to Banach spaces, and to Hilbert spaces.

5. Sadly, Banach and Hilbert spaces are not all that common “in the wild” – one often has to construct them out of a more natural space. Also, although they allow one to transfer a lot of the structure of finite dimensional vector space theory to infinite dimensions, not all of it transfers all of the time. There are “exotic” Banach spaces. There are other types of LCTVS which generalise other aspects of finite dimensional theory; Kriegl and Michor find that bornological spaces are particularly suited to their work. I currently favour nuclear spaces which have a lot of nice properties, particularly in regard to tensor products and mapping spaces. Indeed, my construction of the Dirac operator on a loop space depends on the fact that $C^\infty (S^1, \mathbb{R})$ is a complete, reflexive, nuclear space and absolutely definitely does not work if we try to pass to a nearby Banach or Hilbert space.

Posted by: Andrew Stacey on February 13, 2008 8:54 AM | Permalink | Reply to this

### Re: Comparative Smootheology

Andrew, thanks for your expert reply. My knowledge of general TVS theory doesn’t extend much beyond what I remember from a year-long functional analysis course with François Trèves (not a run-of-the-mill course on Banach and Hilbert alone – it was more on the theory of distributions aspects with an eye toward PDE; there was quite a bit on general LCTVS as well, with a lot of buzzwords like barrelled, bornological, Montel, … my memory of which has faded over the years). Still, I could follow what you explained just fine, and I really appreciate it.

Thanks much for the references to Schaefer’s and Jarchow’s texts. I’ll have to get hold of them.

I have a bunch of questions.

I’ve never looked at Grothendieck’s work in this area; for some reason I thought his tensor products were restricted to nuclear spaces. Just to be absolutely clear (since neither of us quite came out and said it): the claim is that LCTVS is symmetric monoidal closed, using the inductive tensor product? That would be sort of amazing to me, although if it’s the case then I imagine some of the Café regulars would have known it already (e.g., John Baez). Is that result in Schaefer and/or Jarchow?

While we’re on the topic, I have some naive questions about nuclear spaces. I have a vague memory that the category of nuclear spaces is supposed to be what we category theorists call “compact closed”, meaning that the canonical map

$V^* \hat{\otimes} W \to hom(V, W)$

where $\hat{\otimes}$ is some tensor product (inductive? projective?), is an isomorphism. (As a purely formal statement, that would appear to be borne out by Theorem 5.4, page 9, here, which I have not examined in detail.) But I’m worried that I must be missing something. For one thing: can the tensor and/or internal hom be taken to be the ones for LCTVS? For another: in the case of a symmetric monoidal category, compact closure implies that the double dual embedding is an isomorphism. I would interpret that as implying here that nuclear spaces are reflexive, whereas you seem to intimate (in your point 5) that such is not generally the case; a quick glance at the Wikipedia article on nuclear spaces doesn’t seem to support that either.

Any responses (from Andrew or any other expert) to this barrage of questions and naiveté are appreciated. Are the Schaefer and Jarchow books good places to learn about nuclear spaces?

Posted by: Todd Trimble on February 13, 2008 4:27 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Let’s see if we can make some headway on this now. When you originally posted, Todd, I didn’t have a copy of either Schaefer or Jarchow. I’ve now snaffled the library copy of Schaefer so can say something sensible.

Nuclear spaces are not all reflexive. According to Schaefer, quasi-complete nuclear spaces are semi-reflexive and so barreled quasi-complete nuclear spaces are reflexive (so nuclear and Fréchet would do). Most of the ones “in the wild” are such, but not all. That’s in ch IV, 5.5 and to the end of section 5 in Schaefer.

A bit later on we find that if $E$ is a complete nuclear space and $F$ any complete lctvs then the completion of $E \otimes F$ with respect to the projective topology on the tensor product can be canonically identified with $\mathcal{L}_e(E'_{\tau}, F)$. This needs a little explanation. The space $E'$ is the continuous linear dual of $E$. The subscript $\tau$ denotes its topology. This is the Mackay topology. It is the finest locally convex topology on $E'$ such that $E$ is its continuous dual. Next, $\mathcal{L}(E'_{\tau}, F)$ is the space of continuous linear maps from $E'_{\tau}$ to $F$. The subscript $e$ also denotes a topology. In this case it is the topology of uniform convergence on the equicontinuous subsets of $E'$.

So I think that the answer to your question about tensor products of nucelar spaces is an extremely qualified yes. The qualifications are probably sufficient to show that your worry about this implying too much are baseless.

By the way, the dual of a nuclear space need not be nuclear. Any product of lines is nuclear but its dual is the sum of lines which is not nuclear if the indexing set is not countable.

Right, I’m posting this half now. I’ll return to the monoidal structure later. As a teaser, look at section 5 of chapter I of Kriegl and Michor’s book. For bornological lctvs then the bornological tensor product is the adjoint to the bornological hom functor (Ch I, 5.7).

Andrew

Posted by: Andrew Stacey on March 7, 2008 3:50 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Right, now for the monoidal structure. Let’s see if we can make it work.

There are two things to select. A topology on the tensor product of two lctvs and a topology on the space of linear maps from one lctvs to another. These must be somehow compatible. We want

$\mathcal{L}(E, \mathcal{L}_{\rho}(F,G)) \cong \mathcal{L}(E \otimes_{\tau} F, G)$

where $\rho$ and $\tau$ denote topologies.

Dropping the topology, we have an adjuntion

$L(E,L(F,G)) \cong L(E \otimes F, G)$

with map in one direction given by

$f \mapsto \bigg(x \mapsto \Big(y \mapsto \big(f(x \otimes y)\big)\Big)\bigg)$

This implicitly uses the natural set map

$E \times F \to E \otimes F$

together with the set adjunction

$Map(A, Map(B,C)) \cong Map(A \times B, C)$

Right, so, first we want a topology on $E \otimes F$ such that the resulting map $y \mapsto f(x \otimes y)$ is continuous for all $x \in E$. Let’s aim at the outset for symmetry in $E$ and $F$. This means that the bilinear map

$E \times F \to G, \qquad (x,y) \mapsto f(x \otimes y)$

must be separately continuous. By universality, we therefore demand that the universal bilinear map $E \times F \to E \otimes F$ be separately continuous. Again, by universality, we want this topology to be the finest topology with this property. This exists and is called the inductive topology on $\otimes$ and written $\otimes_i$ (don’t confuse this with the corresponding completion. We are not completing the tensor product.).

Now we have made sense of

$\mathcal{L}(E \otimes_i F, G)$

and ensured that the natural map takes this into

$L(E, \mathcal{L}(F,G))$

Next we need to topologise $\mathcal{L}(F,G)$ such that the image of $\mathcal{L}(E \otimes_i F, G)$ is precisely the continuous linear maps $E \to \mathcal{L}(F,G)$. As our topology on $\otimes_i$ was symmetric, we know that for $y \in G$, the map $x \mapsto f(x \otimes y)$ is continuous as a map $E \to G$. Appealing again to universality, we therefore want to specify a topology on $\mathcal{L}(F,G)$ such that if $T : E \to \mathcal{L}(F,G)$ is a linear map such that $x \mapsto T(x)(y)$ is continuous as a map $E \to G$ for all $y \in F$ then $T$ is continuous. Again, there is such a topology and it is the weak topology on $\mathcal{L}(F,G)$; also known as the simple or pointwise topologies, and in some circumstances the weak* topology. A subbasis of $0$-neighbourhoods for this topology is given by the sets $\{T : T(y) \in U\}$ for $U$ a $0$-neighbourhood in $G$ and $y \in F$. This topology is sometimes denoted by $\mathcal{L}_s(F,G)$.

We therefore have our adjunction

$\mathcal{L}(E \otimes_i F, G) \cong \mathcal{L}(E, \mathcal{L}_s(F,G))$

Of course, the pointwise topology is also pretty pointless so this isn’t necessarily a useful adjunction. One would often prefer a stronger topology on the space of linear maps.

Can one show that such adjunctions are unique? I mean, as we have found an adjunction using the inductive topology on the tensor product and the simple topology on the mapping spaces then there won’t be one if we chose different topologies (but don’t change the underlying unit/counit natural transformations).

Andrew

Posted by: Andrew Stacey on March 10, 2008 10:18 AM | Permalink | Reply to this

### Re: Comparative Smootheology

Andrew,

thanks for all the input. I have to admit I need to catch up a little here.

In our desired application to smooth spaces. What would “reflexive” amount to? When would we call a sheaf on our test domains “reflexive”?

Posted by: Urs Schreiber on March 11, 2008 2:16 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Hi Urs,

In the analogy (and I stress that it is only such), reflexive locally convex topological vector spaces were supposed to correspond to saturated (generalised) spaces; that is, ones which if you do the Isbell conjugation twice then you get back where you started. I guess that these are “Very Nice Spaces” in your current document.

The analogy is getting quite tenuous; I no longer believe the “if it stabilises then it does so (almost) immediately” line. Also, you only get reflexivity of lctvs because there is a topology involved. If you throw out the topology then taking duals just keeps on getting bigger, and bigger, and bigger, and …

So I’m not sure whether this particular part of the discussion is very relevant any more; it had meandered a bit into considering the properties of the category of lctvs – which is something I find very interesting, although Todd seems to have gone strangely quiet.

I’m currently trying to grok your document. As I think I said a long time ago (in a galaxy far far away), I’m a little slow sometimes. I’ll try to think of something useful to say soon, but I’ll do that over there.

Andrew

Posted by: Andrew Stacey on March 11, 2008 3:42 PM | Permalink | Reply to this

### Re: Comparative Smootheology

although Todd seems to have gone strangely quiet.

Well, while I think I know what you mean, let’s acknowledge that Todd has been fantastically helpful!

One reason why I started writing that document is in the desire to have a place where all the help he provided can be found summarized comprehensively (and, that too, in the context which I am trying to fit all these things into).

He has been so very helpful that one tends to feel he is quiet if he doesn’t react every day to our latest messages…

Posted by: Urs Schreiber on March 11, 2008 3:57 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Let me second that, Andrew – thanks for taking the time to explain.

I haven’t checked all the details, but this symmetric monoidal closed structure on LCTVS is a lot more straightforward than I was expecting it to be. (One thing I haven’t checked is that these constructions yield locally convex TVS; perhaps it will be clear on reflection.)

It’s a little curious to me that the tensor is the ordinary tensor on the underlying vector spaces, suitably topologized; I guess I wasn’t expecting that.

Can one show that such adjunctions are unique?

Of course, once you settle on the tensor product, the internal hom will be uniquely determined up to isomorphism. And the selection of the tensor product topology (the maximal one rendering the universal bilinear map separately cocontinuous) does seem, from one point of view, like an almost inevitable choice. But then, the topology on the resulting internal hom $\mathcal{L}_s(F, G)$ is pretty weak, as you say. In particular, the definition doesn’t itself take into account the topology on $F$.

Maybe there’s a more sensible or useful topology on the hom $\mathcal{L}(F, G)$? (That would of course change the tensor product.) Know of any natural candidates? Would the topology of uniform convergence on compact sets be a ridiculous choice?

Posted by: Todd Trimble on March 11, 2008 3:58 PM | Permalink | Reply to this

### Re: Comparative Smootheology

The inductive topology on the tensor product is a locally convex topology by construction. When I said “take the finest topology such that $\chi : E \times F \to E \otimes F$ is separately continuous” I meant “take the finest locally convex topological vector space topology”. Of course, one needs to prove that such exists but this is standard (back to Grothendieck, I guess).

[note added in proof: actually, as the above process expresses the desired topology as the projective limit of a family of lctvs topologies then it is automatically a lctvs topology]

That the underlying vector space of the adjoint is the algebraic tensor product follows from the existence of some reasonable topology on the tensor product and the fact that the adjunction exists on the non-topological level. I suppose it is possible that there is another closed structure which is not compatible with that arising from the non-topological adjunction, but it would probably be even more obscure than the one given above. What can one say about multiple closed structures on a category?

So if we want an adjunction that is the restriction of the non-topological one then all we can play around with are the topologies. Above, I went for “find something that works”. Now let’s try the more ambitions “find something that’s vaguely useful”.

There is a standard way of topologising the space of continuous linear maps, $\mathcal{L}(F,G)$. To do this, find a family $\mathfrak{S}$ of bounded subsets of $F$ that is total (linear hull is dense in $F$). Then the topology of uniform convergence on the sets in $\mathfrak{S}$ is a locally convex topology on $\mathcal{L}(F,G)$. This is denoted by $\mathcal{L}_{\mathfrak{S}}(F,G)$.

Popular choices are:

1. The family of all finite subsets
2. The family of all convex, circled, compact subsets
3. The family of all precompact subsets (i.e. subsets whose completion is compact)
4. The family of all bounded subsets

The first and last are the extremes and are called the weak and strong topologies respectively.

Of course, we want to choose a topology functorially so need to specify the family functorially. The above are okay, I’m not sure if there are any others.

So we pick some functorial family, write it as $\mathfrak{S}$, and consider $\mathcal{L}_{\mathfrak{S}}(F,G)$. We want to find a topology on $E \otimes F$ such that a map $E \otimes F \to G$ is continuous if and only if the map $E \to \mathcal{L}_{\mathfrak{S}}(F,G)$ is continuous. Again, we want symmetry in $E$ and $F$.

So let $f : E \times F \to G$ be a bilinear map. We must have that $f$ is separately continuous to ensure that the maps $f_x$ for $x \in E$ are in $\mathcal{L}(F,G)$. Now let us see what we need to make the assignment $x \mapsto f_x$ continuous. For $S \in \mathfrak{S}$ and $V$ a 0-neighbourhood in $G$ we want the set

$\{x : f_x(S) \subseteq V\}$

to be a 0-neighbourhood in $E$. That is, there must be some open $U$ in $E$ such that $f(U \times S) \subseteq V$. Such a map is called $\mathfrak{S}$-hypocontinuous (this includes separate continuity).

As we want symmetry, we must have $(\mathfrak{S}, \mathfrak{S})$-hypocontinuity; i.e. in both variables (the notation allows for having a different family on $E$ as $F$). Note that this is stronger than uniform continuity on sets of the form $S_1 \times S_2$.

Appealing to universality we want the topology on $E \otimes F$ to be the finest lctvs topology such that the canonical bilinear map $\chi : E \times F \to E \otimes F$ is $(\mathfrak{S}, \mathfrak{S})$-hypocontinuous.

Does such exist? Schaefer doesn’t go into any detail on this, referring the reader to Grothendieck. As I don’t have Grothendieck’s book in front of me, let’s see if we can argue it.

Consider the family of all lctvs topologies on $E \otimes F$ with the property that the canonical bilinear map $\chi : E \times F \to E \otimes F$ is $(\mathfrak{S}, \mathfrak{S})$-hypocontinuous. Certainly this family is not empty as it contains the indiscrete topology.

Hmm, at this point I notice a subtlety of notation in Schaefer’s book that I had not previously considered. He reserves “locally convex topological vector space” for a Hausdorff topological vector space which is locally convex. The larger class of spaces are referred to as vector spaces with a locally convex topology.

No matter, it is easy to show that this family has a Hausdorff topology: clearly the projective topology is in it as for that topology the bilinear map $\chi : E \times F \to E \otimes F$ is continuous, whence $(\mathfrak{S}, \mathfrak{S})$-hypocontinuous (this follows from the fact that elements of $\mathfrak{S}$ are bounded; that is, absorbed by every open set). Thus if it has a limit topology, this limit is Hausdorff.

The family under consideration has an upper bound in the family of all topologies on $E \otimes F$. It is also easy to show (by general arguments) that this topology is locally convex and Hausdorff. It remains to show that it is an element of our family, namely that the bilinear map $\chi : E \times F \to E \otimes F$ is $(\mathfrak{S}, \mathfrak{S})$-hypocontinuous.

This upper bound topology, call it $\mathcal{T}_0$, is characterised by the fact that a map into $(E \otimes F, \mathcal{T}_0)$ is continuous if and only if the corresponding maps into $(E \otimes F, \mathcal{T})$ are continuous for every $\mathcal{T}$ in our family (sorry if I’m being a bit too detailed here!). So for $x \in E$, consider the map $\chi_x : F \to (E \otimes F, \mathcal{T})$ with $\mathcal{T}$ in our family. As $\chi$ is $(\mathfrak{S}, \mathfrak{S})$-hypocontinuous with respect to $\mathcal{T}$ it is separately continuous, hence $\chi_x$ is continuous into $(E \otimes F, \mathcal{T})$. Thus $\chi_x : F \to (E \otimes F, \mathcal{T}_0)$ is continuous and so (by symmetry) $\chi$ is separately continuous.

Now we need the hypocontinuity bit. Let $S \subseteq F$ be a $\mathfrak{S}$-set and $V$ a 0-neighbourhood in $(E \otimes F, \mathcal{T}_0)$. As $\mathcal{T}_0$ is a projective topology, it is formed by taking finite intersections of sets open in the topologies in its defining family. Thus $V$ contains a subset of the form $V_1 \cap V_2 \cap \cdots \cap V_n$ with each $V_j$ a 0-neighbourhood in some topology $\mathcal{T}_j$ in our family of topologies.

As $\chi$ is $(\mathfrak{S}, \mathfrak{S})$-hypocontinuous for each of these topologies, there are 0-neighbourhoods $U_j$ in $E$ for which $\chi(U_j \times S) \subseteq V_j$. As the index is finite, $U = U_1 \cap \cdots \cap U_n$ is a 0-neighbourhood in $E$ and satisfies $\chi(U \times S) \subseteq V$. Hence $\chi$ is $\mathfrak{S}$-hypocontinuous and thus, by symmetry, $(\mathfrak{S}, \mathfrak{S})$-hypocontinuous.

Phew!

We therefore have the desired topology on $E \otimes F$; let us denote it by $E \otimes_{\mathfrak{S}} F$.

In conclusion, for any functorial family of bounded sets, $\mathfrak{S}$, there is an adjunction

$\mathcal{L}(E \otimes_{\mathfrak{S}} F, G) \cong \mathcal{L}(E, \mathcal{E}_{\mathfrak{S}}(F, G))$

In particular, as Todd suggests, the family of convex, circled, compact subsets would do.

One thing to note, in partial response to one of Todd’ remarks, one can often vary the topology on $F$ a fair amount without changing the topology on $\mathcal{L}_{\mathfrak{S}}(F,G)$. This is the sort of thing that occurs with uniform boundedness results. Essentially, the family of bounded subsets of a lctvs is not unique to the topology. The process of bornologification is one way of rectifying that.

The more direct use of the topology on $F$ is that it is telling you which linear maps $F \to G$ should be considered continuous.

Posted by: Andrew Stacey on March 12, 2008 10:57 AM | Permalink | Reply to this

### Re: Comparative Smootheology

Thanks again! I’d like to think about this some more, but this is very clearly written, and I appreciate it.

What can one say about multiple closed structures on a category?

I don’t know of any general theory about that. This problem has certainly been studied for some specific categories; for example, the only smc structure on $Set$ is the usual cartesian one, and (only slightly harder) the only smc structure on $Vect$ is the usual one. There is more leeway if you pass to toposes; for example, for presheaves on a symmetric monoidal category, there is something called the Day convolution structure in addition to the usual cartesian structure. Then, in a different direction, for (strict) $n$-categories in the range $n \geq 2$, there are variations based on the theme of so-called Gray tensor products. And now here you’ve clearly outlined a multiplicity of closed structures for a certain topological category; it might be interesting to see whether/how your analysis generalizes to other topological categories. Hmm…

Posted by: Todd Trimble on March 13, 2008 2:17 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Regarding what one can say about multiple closed structures on a category, there are some nice techniques that apply to algebraic categories (i.e. categories of models for some finite limit sketch).

Foltz, F.; Lair, C.; Kelly, G.M. Algebraic categories with few monoidal biclosed structures or none. J. Pure Appl. Algebra 17 (1980) 171–177.

Incidentally, this paper appears to be the origin of the
result that there are precisely two monoidal closed structures on Cat: the usual cartesian one, and the one that is sometimes known, after Ross Street, as the “funny tensor product” (where the internal hom gives a category whose objects are functors and whose arrows are “transformations”: natural transformations without the naturality condition!).

Posted by: Robin on March 13, 2008 4:06 PM | Permalink | Reply to this

### Re: Comparative Smootheology

And how could I have neglected to mention that Steve Lack refers to the two tensors on Cat as the “black” and “white” products – paving the way for the insertion of a “gray” tensor product of 2-categories!

(Apologies for drifting from the topic of this thread.)

Posted by: Robin on March 13, 2008 4:14 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Welcome to the discussion, Robin. No apology needed for the drift! It may have escaped your notice but we’re a little off-topic anyway so it would be a bit hypocritical to object to moving further away.

Actually, though, you haven’t drifted off the current topic. The reference to the Foltz-Lair-Kelley paper is extremely useful in answering my question. In there, the authors observe that in a monoidal category, the endomorphism monoid of the identity of the monoidal structure injects into the endomorphism monoid of the identity functor. In the category of all locally convex topological vector spaces then we can easily show that the latter is $\mathbb{R}$ (that is, the ground field): given a natural transformation $\nu : 1 \to 1$ on the identity functor, let $\lambda = \nu_{\mathbb{R}}(1)$. Then for any $x \in X$, we have a continuous linear map $x : \mathbb{R} \to X$ such that $x(1) = x$. By naturality, we have

\begin{aligned} \mathbb{R} & \overset{\quad \lambda\quad}{\to} && \mathbb{R} \\ x\downarrow & &&\downarrow x \\ X & \underset{\quad \nu_X \quad}{\to} && X \end{aligned}

from which we deduce that $\nu_X(x) = \lambda x$ and so $\nu$ is completely determined by $\nu_{\mathbb{R}}(1)$. Conversely, given $\lambda \in \mathbb{R}$ it is easy to construct the corresponding natural transformation.

As we are working with Hausdorff locally convex topological spaces (see my comment earlier), the Hahn-Banach Theorem tells us that if $X$ is a locally convex topological vector space with $dim X \ge 2$ then $\dim \mathcal{L}(X) \ge 2$ also. Hence the only objects in our category that can be units for a monoidal structure are $\{0\}$ and $\mathbb{R}$.

It is easy to show that we cannot use $\{0\}$ for the unit of a closed monoidal structure. If it were the unit, writing $\mathfrak{L}$ for the internal hom and $\boxtimes$ for the monoidal product, we would have

$\mathcal{L}(X,Y) \cong \mathcal{L}(\{0\} \boxtimes X, Y) \cong \mathcal{L}(\{0\}, \mathfrak{L}(X,Y)) = \{0\}$

which is patently absurd.

Thus our unit must be $\mathbb{R}$. Let us write $\vert X\vert$ for the underlying vector space of an lctvs $X$. Then we see that

$\vert \mathfrak{L}(X,Y) \vert \cong \mathcal{L}(\mathbb{R}, \mathfrak{L}(X,Y)) \cong \mathcal{L}(\mathbb{R} \boxtimes X, Y) \cong \mathcal{L}(X, Y)$

and so $\mathfrak{L}(X,Y)$ is some topologised version of $\mathcal{L}(X,Y)$. This is pretty much what I was asking above.

If we could state categorically (pun intended) that the only way to functorially topologise $\mathcal{L}(X,Y)$ was via the $\mathfrak{S}$-topologies referred to earlier then we would be done as we have constructed the corresponding product for these ways of topologising the hom-sets and we have a complete characterisation.

Note that we have not assumed that the monoidal structure was symmetric, nor that it was bi-closed.

Okay, so how about the product? Let us call it $\boxtimes$ for the time being. It would be nice to show that the underlying vector space of $X \boxtimes Y$ is $\vert X \vert \otimes \vert Y \vert$. Can we prove this? Yes we can! Er … I think so.

Let us summarize our structure. We have a monoidal product, $\boxtimes$, with unit $\mathbb{R}$. This is closed with internal hom $\mathfrak{L}(-,-)$. Moreover, the underlying vector space of $\mathfrak{L}(-,-)$ is $\mathcal{L}(-,-)$.

The forgetful functor from lctvs to vector spaces has a left adjoint which assigns to any vector space the finest lctvs topology for which all inclusions of finite dimensional subspaces are continuous. For a vector space $V$ let us denote this lctvs by $V_s$ (as it is closely allied to the simple topology).

This functor, $V \mapsto V_s$, is strong monoidal. To see this we observe that

$\begin{split} \mathcal{L}\big((U \otimes V)_s, Z\big) &\cong L(U \otimes V, \vert Z \vert) \\ &\cong L(U, L(V, \vert Z \vert)) \\ &\cong L(U, \mathcal{L}(V_s, Z)) \\ &\cong \mathcal{L}(U_s, \mathfrak{L}(V_s, Z)) \\ &\cong \mathcal{L}(U_s \boxtimes V_s, Z) \end{split}$

In the reverse direction, we certainly have a suitable inclusion:

$\begin{split} \mathcal{L}(X \boxtimes Y, Z) &\cong \mathcal{L}(X, \mathfrak{L}(Y,Z)) \\ &\subseteq L(\vert X \vert, \mathcal{L}(Y,Z)) \\ &\subseteq L(\vert X \vert, L(\vert Y \vert, \vert Z \vert)) \\ &\cong L(\vert X \vert \otimes \vert Y \vert, \vert Z \vert) \\ &\cong \mathcal{L}\big((\vert X \vert \otimes \vert Y \vert)_s, Z) \end{split}$

from which we deduce the existence of an epimorphism

$(\vert X \vert \otimes \vert Y \vert)_s \to X \boxtimes Y.$

Category-purists may wish to look away for the next bit, but I’m working this out as I go along so am going with strategies that I’m more familiar with.

Consider an element of $X \otimes Y$. This is of the form

$\sum_{i=1}^k x_i \otimes y_i$

where we can assume that the $x_i$ and $y_i$ form linearly independent families. We can therefore find $f_i$ and $g_i$ in $X'$ and $Y'$ respectively such that $f_i(x_j) = \delta_{i j}$ and similarly $g_i(y_j) = \delta_{i j}$. Define a linear map $\vert X \vert \to \mathcal{L}(Y, \mathbb{R})$ by

$x \mapsto \sum_{i=1}^k f_i(x) g_i$

As this factors through $\mathbb{R}^k$, it lifts to a continuous linear map $X \to \mathfrak{L}(Y,\mathbb{R})$. Via the adjunction, it defines a map $X \boxtimes Y \to \mathbb{R}$. Pulled back to $(\vert X \vert \otimes \vert Y \vert)_s$ this map is clearly given by

$x \otimes y \mapsto \sum_{i=1}^k f_i(x)g_i(y)$

By construction, its value on our original element $\sum x_i \otimes y_i$ is $k$. Hence the map $(\vert X \vert \otimes \vert Y \vert)_s \to X \boxtimes Y$ is injective.

It is therefore a bimorphism. Unfortunately this is not enough to deduce that it is bijective on underlying sets. What we can deduce is that $\vert X \vert \otimes \vert Y \vert$ sits inside $X \boxtimes Y$ as a dense subspace.

Hmm. At this point I run into difficulties. I seem to need $\mathfrak{L}(X, -)$ to preserve extremal monomorphisms.

If it does, then we can show that $\vert X \vert \otimes \vert Y \vert = \vert X \boxtimes Y \vert$. This works as follows. Topologise $\vert X \vert \otimes \vert Y\vert$ with the subspace topology from $X \boxtimes Y$; for short-hand let us denote this by $X \otimes Y$ and let us write the inclusion as $i : X \otimes Y \to X \boxtimes Y$. Then this map $i : X \otimes Y \to X \boxtimes Y$ is an extremal monomorphism. The identity on $X \boxtimes Y$ adjoints to a continuous linear map $X \to \mathfrak{L}(Y, X \boxtimes Y)$. It is easy to see that the underlying linear map is simply the map $\vert X \vert \to L( \vert Y \vert, \vert X \vert \otimes \vert Y\vert)$, $x \mapsto (y \mapsto x \otimes y)$, followed by the inclusion $i : X \otimes Y \to X \boxtimes Y$.

The map $Y \to X \boxtimes Y$, $y \mapsto i(x \otimes y)$ is continuous. Hence, as we have given $X \otimes Y$ the induced topology, the map $Y \to X \otimes Y$, $y \mapsto x \otimes y$, is continuous. The map $X \to \mathfrak{L}(Y, X \boxtimes Y)$, $x \mapsto (y \mapsto i(x \otimes y))$ is also continuous. The assumption that $\mathfrak{L}(Y,-)$ preserve extremal monomorphisms now implies that $\mathfrak{L}(Y,i) : \mathfrak{L}(Y, X \otimes Y) \to \mathfrak{L}(Y, X \boxtimes Y)$ is an extremal monomorphism. Hence the map $X \to \mathfrak{L}(Y, X \otimes Y)$, $x \mapsto (y \mapsto x \otimes y)$, is continuous.

From this we see that the map $X \to \mathfrak{L}(Y, X \boxtimes Y)$ factors through $X \otimes Y$, whence the identity on $X \boxtimes Y$ factors through $X \otimes Y$. As the latter is a dense subspace of the former, we deduce that $X \boxtimes Y = X \otimes Y$.

I think that this is equivalent to $\mathfrak{L}(X, -)$ preserving extremal monomorphisms. For if $Y \subseteq Z$ is a topological subspace, we want to prove that a map $f : \vert W \vert \to \mathcal{L}(X,Y)$ is continuous (i.e. lifts to $W \to \mathfrak{L}(X,Y)$) if the composition $f : \vert W \vert \to \mathcal{L}(X, Z)$ is continuous. So suppose that this composition is continuous, then we obtain a continuous map $g : W \boxtimes X \to Z$. As, by assumption, $W \boxtimes X = W \otimes X$ and the original function $f$ factored through $Y$ we see that $g$ factors through $Y$. Moreover, this factorisation is continuous. The adjunction of $g$ is thus a continuous linear map $W \to \mathfrak{L}(X,Y)$ and it is easy to see that this recovers $f$.

So, my question to you all (whoever is still listening). Must an internal hom functor preserve extremal monomorphisms? If not, we have the possibility that there is a monoidal structure on lctvs in which the monoidal product is an enlargment of the tensor product. It must lie between the tensor product and its completion (with respect to the induced topology), but could be somewhere in between (say, sequential completion or quasi-completion).

Time for me to stop, I deem. Certainly this goes a long way to answering my question about monoidal structures on lctvs but not yet all the way.

Andrew

Posted by: Andrew Stacey on March 28, 2008 2:38 PM | Permalink | Reply to this

### Re: Comparative Smootheology

It’s worth saying that the weak topology does have it’s uses. I’ve just used it over here. The point being that if one can prove that something doesn’t work for the weak topology then it isn’t going to work for any other reasonably topology that one uses.

For positive results, I tend to favour the strong topology. For nice spaces, such as smooth loops in Euclidean spaces, a lot of the topologies that one would consider to be “reasonable” coincide so it doesn’t much matter which one chooses.

(And I’d just like to say: “Way-hey, 100 comments!”)

Posted by: Andrew Stacey on March 12, 2008 11:05 AM | Permalink | Reply to this

### Re: Comparative Smootheology

I am in the process of writing up some notes incorporating the results of all the discussion we had here, and the discussion of smooth algebras there.

Unless I am mixed up, in terms of that, the conjugation operation that Todd describes

$Spaces \to C^\infty Algebras : X \mapsto C^\infty(X) := Hom_{Spaces}(X,-)$

and

$C^\infty Algebras \to Spaces : A \mapsto Hom_{C^\infty Algebras}(A,C^\infty(-))$

and the Frölicher saturation condition which Andrew emphasizes would say that we regard a space $X$ as a smooth space, or Frölicher smooth space if

$X \simeq Hom_{C^\infty Algebras}( C^\infty(X), C^\infty(-) ) \,.$

Which seems to make very good sense indeed.

In particular, it seems this would imply that for such Frölicher smooth spaces $X$ and $Y$ we have

$Hom_{Spaces}(X,Y) \simeq Hom_{C^\infty Algebras}(C^\infty(Y), C^\infty(X)) \,.$

Is that right?

Posted by: Urs Schreiber on March 6, 2008 1:37 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Let me make some general comments on this.

I agree with the way you’ve rewritten the conjugation operation in this case; the only thing I’d consider adding is some notation to indicate that we are thinking of an object $U$ of the site $S$ as a space (a functor $S^{op} \to Set$) when we write

$C^{\infty}(X) = Hom_{Spaces}(X, -): U \mapsto Hom_{Spaces}(X, U).$

That is, we are implicitly using the Yoneda embedding $y: S \to Set^{S^{op}}$, so the functor above is really

$Hom_{Spaces}(X, y U): U \mapsto Hom_{Spaces}(X, hom(-, U)),$

just to be absolutely clear.

Then, I’d say that $X$ is a Frölicher smooth space if the unit of the conjugation monad (at the component) $X$,

$u_X: X \to Hom_{C^{\infty}Algebras}(C^{\infty}(X), C^{\infty}(-))$

is an isomorphism.

If we abbreviate the left adjoint part of the conjugation adjunction to $F: X \mapsto C^{\infty}(X) \in C^{\infty}Algebras^{op}$ and the right adjoint part to $G: A \mapsto Hom_{C^{\infty}Algebras}(A, C^{\infty}(-))$, then we have

$Hom_{Spaces}(X, Y) \stackrel{Hom(X, u_Y)}{\cong} Hom_{Spaces}(X, G F Y) \cong Hom_{C^{\infty}Alg^{op}}(F X, F Y) \cong Hom_{C^{\infty}Alg}(C^{\infty}(Y), C^{\infty}(X))$

as you claim. So yes, that’s right!

Posted by: Todd Trimble on March 6, 2008 4:52 PM | Permalink | Reply to this

### Re: Comparative Smootheology

I meant to add the following to the previous post. That for any space $X$, and $Y$ a Frölicher smooth space, the canonical map

$Hom_{Spaces}(X, Y) \to Hom_{C^{\infty}Algebras}(C^{\infty}(Y), C^{\infty}(X))$

is an isomorphism. That’s what my calculation at the end of the last post was supposed to show.

Posted by: Todd Trimble on March 6, 2008 5:00 PM | Permalink | Reply to this

### Re: Comparative Smootheology

So yes, that’s right!

Cool! So now we are in business.

That makes it clear that Andrew has been absolutely right (not that I doubted it anymore, but it seems we hadn’t nailed down the general abstract nonsense reason why so far): we want to say

$\;\;\;$ space = sheaf on $S$

and

$\;\;\;$ $C^\infty$-space = Frölicher sheaf on $S$.

Great. I’ll incorprate that in our file and then let’s see how to move on. This is becoming real fun…

Posted by: Urs Schreiber on March 6, 2008 5:39 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Todd, just remind me:

was there also a general abstract argument that

$X' := G (F (X))$

is Frölicher, for all spaces $X$, i.e. that the “saturation stabilizes” after one step, as Andrew said it should?

Posted by: Urs Schreiber on March 6, 2008 5:55 PM | Permalink | Reply to this

### Re: Comparative Smootheology

No, this part I denied back here. There is no reason that I can see to expect that.

The traditional saturation procedure for Frölicher spaces actually involved a Galois connection between posets, namely the power sets of “functions” and of “curves”:

$P(hom(X, \mathbb{R})), P(hom(\mathbb{R}, X)).$

Namely, to each $F \subseteq hom(X, \mathbb{R})$ we associated its set of curves $Curve(F)$, namely the largest subset $C$ of $hom(\mathbb{R}, X)$ such that the set of composites $F \cdot C$ is contained in $C^{\infty}(\mathbb{R}, \mathbb{R})$. And similarly, to each $C \subseteq hom(\mathbb{R}, X)$ we associated its set of functions $Func(C)$, namely the largest subset $F$ such that the set of composites $F \cdot C$ is contained in $C^{\infty}(\mathbb{R}, \mathbb{R})$. By some very general nonsense, the pair of poset maps $Curve(-)$, $Func(-)$ fits in a contravariant adjunction between these two power sets as posets, and the saturation procedure is obtained by following one of $Curve(-)$, $Func(-)$ by the other in either direction.

Its the fact that we are dealing with an adjunction between posets that implies that saturation stabilizes. That is to say: if you have a monad $M: P \to P$ on a poset $P$, then the unit of the monad gives

$u_{M x}: M x \leq M M x$

for all $x$, and the multiplication of the monad gives

$m_x: M M x \leq M x$

for all $x$, whence $M M x = M x$: we get stabilization after one step.

But in our situation, we are dealing with a monad on categories like the topos $Set^{S^{op}}$, where the above reasoning does not apply at all. I’m not sure how much more I can say about this situation.

Posted by: Todd Trimble on March 6, 2008 6:48 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Urs, if you look at this comment you’ll see that I modified by stabilisation conjecture to saying that if it stabilises then it does so after one iteration.

Moreover, I conjectured that every stable pairing is quasi-representable.

I’ve been a bit preoccupied with other things recently so haven’t thought much more about this; but that doesn’t mean that I’ve lost interest! Quite the contrary. I’m pleased to see that we’re back in business.

Here’s a thought to throw at you both (and anyone else who’s listening). If I take my conjugation pairing; Todd writes $F$ and $G$ above (I’d prefer $C$ or $P$ in place of $G$ for curves or plots; $F$ for functions is fine). If we start with an arbitrary $S$-space $X$ and keep applying $F$ and $G$ in the right order then we get nice maps $(GF)^k X \to (GF)^{k+1} X$ and so can take the colimit (we have cocomplete, right?). Is this space automatically Frölicher?

I guess its like taking the free $M$-monoid on an object $Y$ of some category, where $M$ is a monad on that category.

I would suspect that the resulting object might look a bit like my maximal chain construction as outlined here.

Andrew

Posted by: Andrew Stacey on March 7, 2008 12:14 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Urs, if you look at this comment you’ll see that I modified by stabilisation conjecture to saying that if it stabilises then it does so after one iteration.

Ah, I see. Thanks. This thread had gotten quite long…

I need to understand if the space $G(A)$ for $A$ any $qDGCA$, i.e. a DGCA whose underlying GCA is freely generated on a positively graded vector space is Frölicher or not.

First we should notice that any qDGCA is automatically a $C^\infty DGCA$, since the degree 0 part is just the ground field and hence functions on a point.

Then, the big theorem of rational homotopy theory suggests that it is not unreasonable to expect that the canonical inclusion

$A \hookrightarrow F(G(A))$

($F$ here now the “Functions” functor in the DGCA context, i.e. Forms)

is “close to being an isomorphism”. It ought to be an isomorphism on cohomology. I started discussing with Danny Stevenson how to adapt the analogous rational homotopy theory proof to the situation here, but we didn’t get very far before he left for LA…

But actually, I am wondering if it might not even be an isomorphism directly, here in this context. I don’t really have a clue about this except for the following observation: it seems to be pretty darn impossible to guess any element whatsoever in $F(G(A))$ which does not come from the inclusion $A \hookrightarrow F(G(A))$.

You know, to show that the inclusion is not an iso, one should simply give a counterexample, an element not in the image. But I don’t see any such.

(This doesn’t proof anything of course. I am just mentioning this in case it makes anyone see a little further than I can.)

Anyway, why is that relevant here? Clearly, because if this were an iso, then $G(A)$, the “classifiying space for $A$-valued forms”, would be Frölicher, because then

$G(F(G(A))) \simeq G(A) \,.$

Well, on the other hand, if point 2 of your conjecture is right, that Frölicher spaces are quasi-representable, then the above argument rather suggests that $A \hookrightarrow F(G(A))$ cannot be an iso, because $G(A)$ doesn’t seem to be quasi-representable at all.

(Mainly because it always has a single point but many higher cells. That can’t happen for quasi-representables.)

But it might still be an iso on cohomology (it ought to be, if there is any justice in this world). So I thought a bit about whether it might make sense to relax the Frölicher condition to just asking that

$X \simeq G(F(X))$

is a “weak (homotopy(?)) equivalence” of spaces. On the other hand, that doesn’t quite seem reasonable from the point of view of smooth structures.

So that’s a bunch of puzzlements. Hopefully we can eventually solve some of them.

I’m pleased to see that we’re back in business.

All this is high priority for me. But it may happen that you see me occupied with chewing on a different end of a big bone of which this here is one part. Be assured that this doesn’t imply that I lost interest in this discussion here.

Posted by: Urs Schreiber on March 7, 2008 12:44 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Here are two questions whose answer should follow from the information in Andrew’s article. But before I forget, let me ask them here:

given manifolds $X$ and $Y$, is $hom(X,Y)$ (inner hom in the category of sheaves over open subsets of Euclidean spaces) a Frölicher sheaf?

More generally:

given Frölicher sheaves $X$ and $Y$. Is $hom(X,Y)$ a Frölicher sheaf?

Suppose only $X$ is required to be Frölicher. Can we then say anything about the saturation of $hom(X,Y)$?

Posted by: Urs Schreiber on March 11, 2008 3:28 PM | Permalink | Reply to this

### Re: Comparative Smootheology

Minor correction, as should be evident from reading the subsequent comments by Todd I got the definition of the presheaf $C_X$ the wrong way around. It should be natural transformations out of $F_X$ not into it.

Posted by: Andrew Stacey on February 12, 2008 8:33 AM | Permalink | Reply to this
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### Re: Comparative Smootheology

I’ve updated the original article that sparked this thread. The new version is available from the arxiv as arXiv:0802.2225.

To continue the software theme, this is a beta release.

I’ve added Smith spaces and the later version of Chen spaces (I’ve retained the early definition as well as it makes for an interesting comparison). This has involved adding a few more examples to make precise the relationships between the various categories.

The precise nature of the relationship between Chen spaces and diffeological spaces may be of interest to some. In brief, the obvious functor from Chen spaces to diffeological spaces has both a left and right adjoint, both of which are embeddings of full subcategories. However, the two categories fail to be equivalent as one can exhibit two different Chen spaces which map to the same diffeological space. These are simple to describe so, at the risk of starting another branch in this comment-tree, here they are.

Take the closed unit interval, $[0,1]$. One of the two spaces is its standard Chen structure which contains the identity map. The other consists of all those $C^\infty$-maps $\phi : C \to [0,1]$ which are locally smoothly extendible. That is to say, each point $p \in C$ has a neighbourhood $C_p$ in $C$ such that the restriction of $\phi$ to $C_p$ extends smoothly to a neighbourhood of $C_p$ in its ambient affine space. One can easily check that this defines a Chen space. It is also easy to check that this has the same underlying diffeological space as the usual Chen structure on $[0,1]$. However, the identity on $[0,1]$ is not locally smoothly extendible at $0$ or $1$ so this is not the usual Chen structure on $[0,1]$.

Thanks to those who commented on the alpha version, your comments were extremely useful.

Andrew

PS For Todd and Urs, as you’ll see I didn’t put much in about non-set based theories beyond a ‘this is interesting, but forthcoming’. So let’s get it sorted out!

Posted by: Andrew Stacey on February 18, 2008 8:18 AM | Permalink | Reply to this
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