## May 9, 2009

### Smooth Structures in Ottawa II

#### Posted by John Baez

guest post by Alex Hoffnung

Hi everyone,

I am going to even further neglect my duties to the journal club and take a moment to report on the Fields Workshop on Smooth Structures in Logic, Category Theory and Physics which took place this past weekend at the University of Ottawa. The organizers put together a great series of talks giving an overview of the past and current trends and applications in smooth structures. I should right away try to put the idea of smooth structures in some context. Further, I should warn you that I may do this with some amount of bias.

For my purposes the study of smooth structures should be the study of generalizations of smooth manifolds in the context of (quasi)topos theory and the attempt to transport structures from differential geometry to this setting. I will start by describing the parts I know best and then see where we go from there. It may be useful to state a slogan for the conference. Of course, I do not claim to speak for everyone, but Anders Kock quoted the following dictum of Grothendieck which has served as a useful motivation for me:

“It’s better to have a good category with bad objects than a bad category with good objects.”

John Baez gave a nice talk which gave some global perspective to this workshop and which I will try to use as a template for recalling and discussing the various other talks. I will also try to discuss some other topics that fell slightly outside of the range of John’s talk, including Rick Blute’s tutorial on differential linear logic. I will not be able to discuss everyone’s talks because I was not taking very diligent notes (actually, I went with the no note taking at all approach, gulp!). I would like to discuss the more homotopy theoretic talks by Dorette Pronk and Kristine Bauer especially since Kristine’s talk was somewhat reminiscent of a conversation with David Spivak on derived smooth manifolds who was at UCR last week visiting Julie Bergner. Maybe someone else wants to jump in here and mention something about these talks.

Since the room was mostly filled with category theorists and not differential geometers, it was safe to start off with the idea that the category of finite dimensional manifolds is in some ways a “bad” category. Of course, one takes this with a fairly large grain of salt.

With our salt in hand we can now ask why this is a bad category and what would constitute a “good” category.

#### Cartesian Closed

John began by noting that this category is not cartesian closed. In particular, given a pair of finite dimensional manifolds $M$, $N$, the category of smooth maps between these $C^\infty(M,N)$ is rarely again a finite dimensional manifold. This, along with related issues, has been discussed at some length in the “Comparative Smootheology” posts here at the café. $C^\infty(M,\mathbb{R}^n)$ is a vector space and if $M$ is compact then it’s a Fréchet space. In general, it’s a locally convex topological vector space. And more generally, for $M$ compact, $C^\infty(M,N)$ is a Fréchet manifold.

It turns out that much of the hard analytic work one needs to do to understand smooth structures can be found in a free online book by Kriegl and Michor called The Convenient Setting of Global Analysis; In particular, one should begin here to find a kind of topological vector space such that manifolds modelled on these form a cartesian closed category — though this doesn’t seem to have been done yet, and John posed that as a challenge.

While on the subject of convenient settings, Rick Blute also referred to theorems of Kriegl and Michor while describing differential linear logic. Since the talk was a tutorial he led us through many basic notions from linear logic including a categorical axiomatization of differential linear logic. Rick was mainly describing work of himself, Robin Cockett and Robert Seely based on work of Ehrhard and Regnier. Main features he desired were a *-autonomous category with a differential combinator. I cannot really say much more about this without getting even deeper into things I do not understand, so I am going to need to bother Rick when I am back up in Ottawa for the workshop next month to try and straighten some ideas out in my head.

#### Subspaces

Everyone knows from their first class on smooth manifolds that there is some work to be done in constructing a manifold. So, it is obvious that an arbitrary subset of a manifold will not again be a manifold. A closely related fact is that this category does not have equalizers. This leads us naturally to a main theme in one of the other tutorials given by Anders Kock. Many of the categories of smooth spaces presented at the conference (in fact, all if I am not mistaken) have equalizers. In the synthetic differential geometry of Lawvere and Kock, there are several special features not seen in other approaches.

Maybe I can first say something about synthetic differential geometry. Synthetic geometry begins by choosing a topos $\mathcal{E}$ with a commutative ring object $R$. The idea here will be exactly opposite to the sets with extra structure approach. Here one does not want to describe a smooth object as a set of elements, but instead treat smoothness as a property of morphisms. This, of course, favors morphisms over objects as the important structure to study. This is how you can tell the approach developed by category theorists from those developed by other mathematicians who define these categories to solve certain problems. So an object is determined by all the maps into it. The special commutative ring object acts as an environment in which we can “add and multiply” not elements, but morphisms from a common probing domain into $R$. This ring object should also have chosen $0$ and $1$ elements given by maps from the terminal object.

Now there are plenty of very strange diffeological spaces, for instance the Cantor set, but there are a number of non-concrete sheaves that one might want as objects in the category. One is the sheaf of differential $p$-forms. Another, which plays a big role in the synthetic approach is what James Dolan first introduced to me as the “walking tangent vector”. Let’s call this object $D$. It will live in our topos. Arbitrary maps from $D$ to a smooth space $M$ will be defined to be the tangent vectors to $M$. This $D$ is the equalizer of these two arrows: $x^2:\mathbb{R} \rightarrow \mathbb{R}$ $0:\mathbb{R} \rightarrow \mathbb{R}$ If we took this equalizer in the category of smooth manifolds, we would get the one-point set. So, it is important that the category of smooth manifolds embeds fully and faithfully into the topos for synthetic geometry but this embedding does not preserve equalizers. Our one-point equalizer acquires extra structure and becomes the basis of the notion of infinitesimals.

Gonzalo Reyes, in his talk, gave us bijections between vector fields $x: M \rightarrow M^D$ on $M$, autonomous differential equations $x: D\times M \rightarrow M$ and “infinitesimal transformations of $M$”, $D\rightarrow M^M$. Both Kock and Reyes talked about much more that I have skipped over including the main part of Kock’s lectures, which was on Kähler differentials for Fermat theories.

#### Quotient Spaces

Similar to the problem with subspaces, manifolds do not generally have quotients. In this case, there are popular ways of dealing with such things, such as orbifolds. Eugene Lerman has given a nice writeup on orbifolds that has been mentioned here before. John explained that one can take an approach where orbifolds are though of not as sets with extra structure, but groupoids with extra structure. The category of sets do have quotients (or coequalizers). Since it is becoming clear that the category (topos) of sets has these nice properties that we desire for smooth spaces, such as the cartesian closed property, subsets, and quotients, it is reasonable to try to take a set and endow it with some sort of generalized smooth structure. In fact, this is what many people worked on in the seventies and eighties. Each had his own application in mind, but here we are mostly concerned with the general strategies of defining these spaces.

John led us nicely into a talk by Konrad Waldorf with an example of a Lie groupoid which arises as a action groupoid of $\mathbb{R}^2$ weakly mod the group $\mathbb{Z}_5$. More generally, taking the weak quotient of a manifold by a Lie group gives a Lie groupoid with source and target surjective submersions. Once on the subject of Lie groupoids, we should describe there morphisms, and a possibly surprising fact is that the morphisms are not just “smooth” functors. Instead, the correct notion of a morphism between Lie groupoids is something like a principal $G$-bundle $f: M \rightarrow G-Tor$. It is in fact a “Morita equivalence” from $M$ to $G[1]$ through $\tilde{M}$, where $G[1]$ is just the group as a category and $\tilde{M}$ is the Lie groupoid formed by choosing a cover of $M$.

I am particularly fond of bicategories and most that I know of are of the bimodule/span flavor. Lie groupoids with morphisms as described as above form a bicategory called the bicategory of differentiable stacks.

#### Connections

Let me try to say a little more with a short overview of Konrad’s talk. First I should say something about my talk since it was somewhat of a precursor to his. The second part of my talk (I will get to the first part later) described how to use categories of smooth spaces to define smooth categories and smooth functors by internalization. I then talked about the smooth path groupoid $P_1(M)$ of a smooth space $M$ and a theorem relating connections on trivial bundles to smooth functors. The smooth path groupoid is the groupoid with a smooth space of objects $M$ and thin homotopy classes of paths in $M$ as morphisms. Thin homotopy is an equivalence relation between paths with homotopies that “do not sweep out any area”.

Konrad basically told us how to think of a connection on a principal $G$-bundle over $M$ as a smooth anafunctor from the path groupoid of $M$ to $G$. He never said “anafunctor”, but I think this is the most concise way to convery the general goal of his talk. This is part of a larger story of categorification, bundles and gerbes which you can read a lot about here from Bartels, here from Baez and Schreiber and here from Waldorf and Schreiber.

#### Comparative Smootheology

So, the first part of John’s talk led us through some of the properties we desired for our categories of smooth spaces and nicely set the stage for various other speakers. Then John gave us the short tour of categories of smooth spaces, leaving the grand tour for a later talk by Andrew Stacey. John outlined three approaches to smooth spaces; I should make it clear that there seem to be two main approaches. The synthetic differential geometry works inside a topos and can have spaces that are weird in the sense that they are not sets with extra structure. This, of course, is one of the charms of the subject, and I will try to say more about this later. The other approaches all consist of spaces which are sets with extra structure. This extra structure is the smooth structure obtained by modelling your space on spaces whose smooth structure we already know and love. The so-called “sets with extra structure” approach then has three flavors of its own. They are the “maps out”, “maps in”, and “maps in and out” approaches.

John gave the example of Sikorski spaces as a “maps out” approach. Sikorski spaces start out as topological spaces and are then equipped with a subsheaf of the sheaf of continuous real-valued functions. Functions in this subsheaf are thought of as “smooth”, and they must be closed under smooth operations. The Sikorski spaces include abstract examples such as the space with just one point and the ring of first-order Taylor series, $\mathbb{R}[x]/\langle x^2\rangle$, as the smooth functions. This is another version of the “walking tangent vector” in synthetic differential geometry.

Now that I have mentioned these, I will forget about the “maps out” approach for the rest of this post. Andrew Stacey gave an extensive advertisement for Frölicher spaces telling us why he thinks they form such a nice category of smooth spaces. I will get to this shortly. The day before I gave a description of Souriau’s diffeological spaces and some of the nice properties of this category and some hints towards possible applications. I will say a bit more about this as well.

The final talk of the workshop was a very entertaining “party political broadcast for the Frölicher party” by Andrew Stacey. Just like Chen, Souriau, Sikorski, Smith and Frölicher, Andrew had a particular problem in mind when he began considering generalizations of manifolds. In particular, I believe he was looking for a category with nice properties which contained loop spaces, which are certain spaces of maps. The goal which I know next to nothing about (so Andrew if you are reading and want to jump in here and straighten me out or just tell me something cool, go right ahead) was to define Dirac operators on these loop spaces.

The properties Andrew was looking for are similar to those mentioned earlier. He wants a complete, cocomplete, cartesian closed category. Andrew nicely led us through the process of distilling the concept of being smooth. He called this “the smooth, the whole smooth, and nothing but the smooth”. I will try to stop transcribing his jokes here because I will ruin them all. The most useful part of this talk was a flow chart which you can see here outlining relationships between different categories of smooth spaces. Part of the reason Andrew likes Frölicher spaces so much is that they sit at the center of this chart and embed nicely into each of the other categories. There are three levels in his chart. The bottom level represents the approaches where topology is derived from or given along side the smooth structure. The second level approaches smooth spaces by endowing topological spaces with smooth structure. The passing from underlying sets to underlying topological spaces is responsible for the conspicuous absence of adjoint functors between the first and second level. If I recall correctly, the third level is one of K.T. Chen’s four attempts at defining a satisfactory notion of smooth space. This being one of his earliest attempts he had not yet realized what a small role the topology played and went as far as defining smooth structures on Hausdorff spaces. Of course, this is not an unreasonable place to start, but his approach became more refined with time.

The bottom line of this chart is my favorite. John Baez and I wrote a paper detailing the Chen spaces and diffeological (Souriau) spaces. These two categories are quasitopoi with all limits and colimits. This means they are locally cartesian closed and have a weak subobject classifier. These things are explained in detail in our paper. Chen spaces and diffeological spaces are very similar although not equivalent, but as Anders Kock pointed out, the Frölicher spaces have one significant difference from these. They are cartesian closed, but not locally cartesian closed. This brings me to a recurring point that I have mentioned in passing so far. Synthetic differential geometry works inside a topos whereas the categories of Chen spaces and diffeological spaces are quasitopoi of some nice sort. In particular, they are categories of concrete sheaves on a concrete site. The idea of this presentation is an attempt at some kind of balancing act. Sheaves on a site are a relatively abstract notion, but concrete sheaves are a very simple thing. So explaining diffeological spaces, for instance, to a broad audience, one could get away with saying they are concrete sheaves on the site of open subsets of $\R^n$ with smooth maps as morphisms. This would require a small bit of explanation but would make most mathematicians fairly happy. Of course, there is another side of the room (the topos theorists) who will stand up and say “Why should I restrict to concrete sheaves?”. There reason for asking is because they know very well that by dropping this concreteness condition on the sheaves they are immediately enhanced from a mere quasitopos to a full-blown topos. There are lots of answers to this question. The first one is “You shouldn’t!”. Concrete sheaves are for people who like their spaces to have points. My attitude is that concreteness comes with a little switch. It can be turned on or off at will. So we should understand both the topos of sheaves and the quasitopos of concrete sheaves. So hopefully this can keep everyone happy.

That was the workshop in a nutshell, although I did leave out some talks, mostly due to my own ignorance, so I apologize for that and hope someone else can fill in the blanks.

Posted at May 9, 2009 7:57 PM UTC

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### Re: Smooth Structures in Ottawa II

He never said “anafunctor”, but I think this is the most concise way to convery the general goal of his talk.

That’s right. I don’t know what Konrad said, but chances are he said [[descent object]] instead of anafunctor.

One should be aware that the term [[anafunctor] is understood only in a comparatively small circle, which happens to be centered roughly around the community found here at the Café.

And one should be aware that strictly speaking anafunctor is really a special case of a concept which was known and termed before in one way or other. It seems that a good and comprehensive discussion of anafunctors in this more general context is David Roberts’ thesis chapter 2 Internal categories and anafunctors.

But, right, in whichever way we phrase it, the upshot is that with $H(\mathbf{X},\mathbf{A})$ denoting the collection of suitably general morphisms from generalized space $\mathbf{X}$ to $\mathbf{A}$, we have

$H(X,\mathbf{B}G) \simeq G Bund(X)$

$H(P_1(X),\mathbf{B}G) \simeq G Bund_\nabla(X)$

$H(\Pi_1(X),\mathbf{B}G) \simeq G Bund_{flat}(X)$

$H(X,\mathbf{B}^2 U(1)) \simeq U(1) BundGerb(X)$

$H(P_2(X),\mathbf{B}^2 U(1)) \simeq U(1) BundGerb_\nabla(X)$

$H(\Pi_2(X),\mathbf{B}^2 U(1)) \simeq U(1) BundGerb_{flat}(X) \,.$

Here $P_n(X)$ denotes the [[path $n$-groupoid]] and $\Pi_n(X)$ the [[fundamental $n$-groupoid]].

This is also described at the end of [[motivation for sheaves, gerbes and higher stacks]].

Posted by: Urs Schreiber on May 9, 2009 10:45 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Thanks Urs.

Here is a link to Konrad’s talk so now we can see exactly what he said.

Posted by: Alex Hoffnung on May 10, 2009 4:52 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Maybe I should add that “parallel transport functor” as used there is a way to describe the stackification of $[P_1(-),\mathbf{B}G]$, so it is something that may be represented by an anafunctor, but is a more invariant concept in that it does not come with a specified choice of cover.

Actually, I think one good way to think of those parallel transport functors the way we talked about them is as providing the natural rectified $n$-stacks of $n$-bundles with connection, i.e. those which as $\infty$-groupoid valued pseudofunctors on $Diff^{op}$ are actually ordinary functors: $Trans(-,\mathbf{B}G) : Diff^{op} \to \infty Grpd$.

One advantage is that that makes them fit into the [[model structure on simplicial presheaves]].

So,

- an ordinary functor $P_1(X) \to \mathbf{B}G$ is a trivial $G$-bundle with connection;

- a transport functor $P_1(X) \to \mathbf{B}G$ is a possibly non-trivial $G$-bundle with connection;

- and an anafunctor $P_1(X) \leftarrow Codesc(Y,P_1) \to \mathbf{B}G$ is a presentation of the latter by the former on a cover.

Posted by: Urs Schreiber on May 11, 2009 10:25 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Thanks for keeping the rest of us up-to-date here Alex.

Posted by: Bruce Bartlett on May 10, 2009 5:23 PM | Permalink | Reply to this

### To solve a problem

I couldn’t say it better than Alex:

This is how you can tell the approach developed by category theorists from those developed by other mathematicians who define these categories to solve certain problems.

This highlights the PR problem for category theory ‘for its own sake’. Compare Adams’ work on Hopf invariant 1. That full blown development of secondary operations would not have been done ‘for its own sake’.

Posted by: jim stasheff on May 11, 2009 3:49 PM | Permalink | Reply to this

### Re: To solve a problem

I should say that my remark in regards to synthetic differential geometry, that

The idea here will be exactly opposite to the sets with extra structure approach.

was probably a bit of an overstatement. Probing objects for smooth structure is about deciding which maps out of that object are smooth, but there is still a strong distinction along these lines between the approaches.

“Concreteness” is important for sets with extra structure. One aspect of this is that all maps from the one-point set to a smooth space $X$ must be considered to be plots. This is the underlying set. The synthetic approach requires one to understand smooth maps from any object into $X$ as a “generalized element” of $X$.

Posted by: Alex Hoffnung on May 11, 2009 6:03 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Jeff Morton also has a report on this conference.

Posted by: David Corfield on May 15, 2009 9:31 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

A few corrections and clarifications:

The space $C^\infty(M, \mathbb{R}^n)$ is a Fréchet space for any finite dimensional manifold. The topology is that of uniform convergence on compact sets so to get a Fréchet structure you just need a countable exhaustion by compact sets (Fréchet means that the topology is specified by a countable number of semi-norms; it’s equivalent to metrisable for locally convex topological spaces). The problem with $C^\infty(M,N)$ for $M$ non-compact is that maps that are “close” in the topology can be “far apart” in $N$. The topology only specifies what happens on compact sets and so, if $M$ is not compact, has nothing to say about what happens elsewhere. This destroys any hope of local linearity; to have a local addition one needs to be able to add maps that are close in the topology and the standard way of doing this is to say that close in the topology means that the points are close in $N$ so one can use local additions on $N$ to add the maps. This fails if $M$ is not compact. The way around this is to change the topology on $C^\infty(M,N)$. Kriegl and Michor cover this in chapter IX of their book.

On the subject of Kriegl and Michor’s book, I think it is worth quoting from the introduction of chapter IX so that it’s clear what they have and haven’t done.

If we insist that the exponential law should hold for manifolds of mappings between all (even only finite dimensional) manifolds, then one is quickly lead to a more general notion of a manifold, where an atlas of charts is replaced by the system of all smooth curves. One is lead to further requirements: tangent spaces should be convenient vector spaces, the tangent bundle should be trivial along smooth curves via a kind of parallel transport, and a local addition as in (42.4) should exist. In this way one obtains a cartesian closed category of smooth manifolds and smooth mappings between them, where those manifolds with Banach tangent spaces are exactly the classical smooth manifolds with charts. Theories along these lines can be found in [Kriegl, 1980], [Michor, 1984a], and [Kriegl, 1984]. Unfortunately they found no applications, and even the authors were not courageous enough to pursue them further and to include them in this book. But we still think that it is a valuable theory, since for instance the diffeomorphism group $Diff(M)$ of a non-compact finite dimensional smooth manifold $M$ with the compact-open $C^\infty$-topology is a Lie group in this sense with the space of all vector fields on $M$ as Lie algebra. Also, in section (45) results will appear which indicate that ultimately this is a more natural setting.

I haven’t read the cited papers, but having been reminded of their existence by wanting to clear up this confusion, I’m minded now to do so.

On to Sikorski spaces. Do they really contain the space that you describe? The definition that I have seen (from Mostow’s paper) says that the functions must be a subset of the continuous functions on a topological space. That rules out your “walking tangent vector” Sikorski space. Is there another definition that includes such more bizarre Sikorski spaces?

This is probably as good a place as any to say that my Comparative Smootheology paper is now up to version 0.4 and can be found via my webpage. I haven’t updated the arXiv version (probably shan’t bother for 0.4) due to their “five strikes and you’re out” policy [1].

In particular in this version there’s a unified way to describe all the different approaches to smooth spaces so that all of them become “maps in and maps out”. This makes it clearer what the actual differences are and what is merely down to presentation.

I’m tempted, for version 0.6, to remove the “party political broadcast for the Frölicher party” from Comparative Smootheology and make it purely objective. That would also shorten it a little which might not be a bad thing!

[1] Yeah, I know. This is just my warped sense of humour getting in the way again.

Posted by: Andrew Stacey on May 19, 2009 1:08 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Sorry for the extremely slow reply. Thanks for the post!

You wrote:

On to Sikorski spaces. Do they really contain the space that you describe? The definition that I have seen (from Mostow’s paper) says that the functions must be a subset of the continuous functions on a topological space. That rules out your “walking tangent vector” Sikorski space. Is there another definition that includes such more bizarre Sikorski spaces?

Thanks for pointing this out. It was just a mistake.

The answers are “no” and something like “not that I know of”. I guess for the existence of a more general definition a little more can be said. For diffeological spaces you already know the more bizarre definition. It is non-concrete sheaves on the Euclidean site. I think it would be not hard to generalize the definition of Sikorski spaces in a somewhat similar way, which I will not dare to suggest. You just need a way of convincing yourself and others that the generalization is still just a bizarre example of a smooth space.

Posted by: Alex Hoffnung on May 25, 2009 5:46 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

The generalisation of Sikorski spaces (and other “maps out” variants) that immediately springs to mind is non-commutative geometry, but then I don’t know a lot about that so I don’t know if my intuition is correct on that score.

Posted by: Andrew Stacey on May 26, 2009 8:35 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

The relative of Sikorski spaces that immediately springs to my mind is ‘schemes’.

Like a Sikorski space, a scheme is a topological space equipped with a sheaf of commutative rings. Unlike a Sikorski space, we don’t demand that these commutative rings are $C^\infty$-rings (i.e., closed under smooth operations). But more importantly, we don’t demand that these sheaves are subsheaves of the ring of continuous functions. So, the “walking tangent vector” is a perfectly fine scheme. And in this case we actually have a sheaf of $C^\infty$ rings.

So, I think the only reason the walking tangent vector fails to be a Sikorski space is that it lacks is the ‘concreteness’ property, by which I now mean the property that its sheaf of $C^\infty$-rings is a subsheaf of the ring of continuous functions.

If so, this might be an interesting property to drop.

Posted by: John Baez on May 26, 2009 5:56 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

The relative of Sikorski spaces that immediately springs to my mind is […]

Posted by: Urs Schreiber on May 26, 2009 6:33 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Andrew wrote:

I’m tempted, for version 0.6, to remove the “party political broadcast for the Frölicher party” from Comparative Smootheology and make it purely objective. That would also shorten it a little which might not be a bad thing!

That might be good — and I’m not just saying this as a fan of diffeological spaces. Right now your paper has two goals: saying what’s good about Frölicher spaces, and comparing different approaches to smooth spaces. These goals conflict to some extent. So, you might want to start with a more even-handed version of ‘Comparative Smootheology’, and then write a paper lauding the virtues of Sikorski spaces.

The talk you gave was a wonderful pedagogical introduction to Frölicher spaces, showing how the definition could be reached by a series of small improvements starting from something quite obvious. This would make a very nice paper of the expository or ‘conference proceedings’ sort. (There’s a nice journal at Göttingen that might like this paper.)

More hard-nosed journals demand ‘results’. If that’s the way you want to go, you could supplement your exposition with a bunch of results that say what you can do with Frölicher spaces — and especially, what you can do with them better than with other notions of smooth space!

As far as abstract categorical properties, it’s hard to see how Frölicher spaces can beat diffeological spaces. Since diffeological spaces are concrete sheaves on a concrete site, a vast arsenal of weaponry springs into play whenever you use them. If you want even better abstract categorical properties, the only route I see is to drop the concreteness condition, which lets you get a topos of sheaves, instead of a mere quasitopos of concrete sheaves. But if you want a topos of smooth spaces, you might as well join Kock and Lawvere and Reyes and work on synthetic differential geometry.

So, if Frölicher spaces really have advantages over diffeological spaces — and your propaganda on that score is convincing — I suspect that these advantages are not category-theoretic in nature, but rather, something very much to do with differential geometry. There should be some differential-geometric theorems that hold for all Frölicher spaces — but not, say, diffeological spaces.

But I’m not sure you want to spend your time working in this direction…

Posted by: John Baez on May 26, 2009 6:08 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

I think you are confusing Frölicher and Sikorski spaces a bit here. Andrew’s talk was a “party political broadcast for the Frölicher party”. Although, I think the points you make about the lack of categorical properties should remain valid when talking about either.

Posted by: Alex Hoffnung on May 27, 2009 7:07 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

cat John | perl -pe ‘s/Sikorski/Frölicher/g’ > cafe

Nice to know you were paying attention in my talk, John!

Posted by: Andrew Stacey on May 27, 2009 8:19 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

I wasn’t really confusing Frölicher spaces and Sikorski spaces in my mind; I just spelled ‘Frölicher’ in a strange way — ‘Sikorski’ — throughout this comment. To reduce the total amount of confusion in the world I’ll use my superpowers and fix it.

Posted by: John Baez on May 27, 2009 2:49 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

A word of warning: I’m going to exaggerate a few of my views here to make my points a bit clearer. Usually when I do this someone misunderstands me so this time I’m giving you fair warning. For those at the Ottawa conference, take it in the same spirit as the “party political broadcast”.

I’m beginning to think that I’ve been approaching this issue from the wrong side. I’ve been trying to convince all of you that Frölicher spaces are the correct set-based generalisation of smooth manifolds and I don’t really feel that I’ve been getting anywhere.

However, maybe it’s not my job to convince you but your job to convince me! After all, I’m the differential topologist here and you are the category enthusiasts. So you should be trying to convince me that I need to use your category theory in my work. Otherwise, what’s the point of it all?

Saying that diffeological spaces form a quasitopos is like saying that Australia has sheep. Good for them, but so what? How does that help me in my study of Norwegian moose? - I mean, manifolds.

The frustrating aspect of it all for me is that it feels as though a category with nice properties is not just a good thing but the only thing (please remember my warning at the start of this post). As a differential topologist I want to say that there is something special about smoothness and whatever my category is then I want to capture that specialness. This is what I mean by my slogan: “the essence of smoothness”. I accept that it is a little indistinct, but I mean that the category should smell right.

Let me try to explain what doesn’t smell right about sheaves on a site. Sheaves and sites, as I understand it, are a generalisation of the concept of topology. That is, they are based on the concept of continuity. Smoothness is separate from continuity, we accept that when it comes to individual spaces but not, it appears, when it comes to the category as a whole. Smoothness is much tighter than continuity. To test smoothness at a point, I don’t even need to know its germ at that point, I only need to know its jet. There’s a huge difference between jets and germs!

The evidence is that there is something special about smoothness. Frölicher’s work - and Boman’s, Kriegl’s, and Michor’s as well - points to curves as controlling smoothness. So why would I want to complicate matters by adding in more junk that I don’t need.

As an aside, if it truly is desirable to have a quasi-topos then why not take sheaves on open sets on $\mathbb{R}$? What’s to be gained from all the others? And if something is to be gained, why stop at finite dimensional spaces? Why not take all convenient vector spaces?

Kriegl wrote a paper (MR0785022) called A Cartesian closed extension of the category of smooth Banach manifolds. The reviewer for the AMS made the following comment (emphasis mine):

The main part of the paper presents the proof that this category, which is a sufficiently large extension of the category of Banach manifolds, is small enough not to include abstruse objects, too.

Here are some aspects of diffeological spaces that I consider as abstruse:

• There is a distinction between a diffeological space and the same one with a dimensionally-truncated diffeology (I see Patrick IZ is now calling these spaghetti and lasagne diffeologies). This means that I have, for each $n$, a complete copy of the category of ordinary manifolds wherein I essentially discard the diffeology above dimension $n$ (obviously I can’t discard it completely, but insist that everything higher locally factors through something lower). By Boman’s result, all of these are exact copies of the category of manifolds - even manifolds with corners, thanks to Kriegl and Michor. Why should I accept a category with so much junk in it?

• Take $\mathbb{R}^2$ with either its standard or the spaghetti diffeology and remove all plots that intersect the $x$-axis in other than a set of measure zero. Turns out I still have enough information to separate smooth functions from non-smooth functions from this space. I can still find partial derivatives along the $x$-axis and do everything that I might want to do with smooth functions. So why should I allow this object in my category? What is it for?

How I would like the debate to go is to start by discussing: “What is the essence of a manifold? What is it that makes a manifold so special?” This is, to me, the most interesting aspect of the whole debate. Let me give a closely related analogy. Given a manifold, possibly infinite dimensional, what is a $k$-form on it? In chapter 33 of their book, Kriegl and Michor discuss this and give 12 different possible answers (all of which agree in finite dimensions). However, only one has all the properties one would expect from a differential form and so, they argue, that is the true definition. However, such a conclusion can only arise once there has been a discussion of what one might want to do with a differential form and what it is trying to represent.

Once that is cleared up, then the category should be obvious. Saying that we happen to have a nice category already is a bit like saying that we’re going to use a hammer to make a hole in the wall because the hammer is right there and the drill is in the tool shed.

So what is the essence of a manifold? I tried to give a more categorical answer in my Ottawa talk and argue that if one focusses on the morphisms then one naturally ends up at Frölicher spaces. That doesn’t seem to have convinced anyone. Let me try another approach that may find some resonances here. From reading Michor’s papers: A convenient setting for differential geometry and global analysis (I,II) (available from his webpage), it is clear that Kriegl and Michor view(ed) parallel transport as being fundamental to what a manifold is. Their extension of manifolds (which is cartesian closed but probably not complete or co-complete) replaces charts by smooth curves plus a little extra (Michor admits that “The smooth curves alone are a thin structure, so we need a lot of other data as well”). That little extra amounts to parallel transport of tangent vectors.

John said above:

Since diffeological spaces are concrete sheaves on a concrete site, a vast arsenal of weaponry springs into play whenever you use them.

So what? Just because it’s readily available doesn’t mean that I should use it. If what I want is not readily available then I go out and invent some more. That’s what mathematics is all about. Otherwise I just end up solving the problem for a spherical horse in a vacuum.

So Frölicher spaces are not locally cartesian closed (which, if I have my definitions right, is the only objection to it being a quasi-topos). Why not? Is there a way to fix that? Are Hausdorff Frölicher spaces any better behaved? These are the questions that I would like to answer. Also, it’s not obvious how to generalise Frölicher spaces to the non-set-based situation but again, this is an interesting question. Much more interesting than going from concrete sheaves to all sheaves.

John also said:

If you want even better abstract categorical properties, the only route I see is to drop the concreteness condition, which lets you get a topos of sheaves, instead of a mere quasitopos of concrete sheaves. But if you want a topos of smooth spaces, you might as well join Kock and Lawvere and Reyes and work on synthetic differential geometry.

I clearly don’t know enough about topoi to understand this paragraph. If I want a topos, why wouldn’t I choose the route of synthetic differential geometry? It seems much more conceptually close to the “essence” that I’m after: just add the “walking tangent vector” to the real line and we’re done.

Finally, let me turn John’s penultimate paragraph back on you lot.

So, if Frölicher spaces really have advantages over diffeological spaces — and your propaganda on that score is convincing — I suspect that these advantages are not category-theoretic in nature, but rather, something very much to do with differential geometry. There should be some differential-geometric theorems that hold for all Frölicher spaces — but not, say, diffeological spaces.

If diffeological spaces really have advantanges over Frölicher spaces then I suspect that they are merely in the realms of extremely abstract and abstruse category theory that, in day-to-day differential topology, is extremely unlike to bother me. So there should be some differential-geometric theorems that show that diffeological spaces, and not Frölicher spaces, are necessary to understand ordinary smooth objects.

And these should be actual theorems, not just “it’s easy to do because it’s sheaves”.

Posted by: Andrew Stacey on May 27, 2009 9:49 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Hi Andrew,

you write:

I’ve been trying to convince all of you that Frölicher spaces are the correct set-based generalisation of smooth manifolds and I don’t really feel that I’ve been getting anywhere.

But you are getting somewhere. I pointed out how your Frölicher point of view nicely converges to that of “structured sheaves” in the sense of “structured generalized spaces”.

See the comment

Posted by: Urs Schreiber on May 27, 2009 10:44 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

I’m trying to catch up with the homework that you’ve set me on this stuff! I’m slowly absorbing it and will try to post something coherent soon. I’m afraid I’m a slow reader though.

Posted by: Andrew Stacey on May 29, 2009 8:36 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

If someone feels like reviewing for the spectators what the main goals are here, I for one would be grateful.

I know you’re trying to extend the concept of smooth space beyond manifolds, but what are the key properties you want? what are the less important things? and why do you want them?

Algebraic geometry deals with infinite-dimensional singular spaces all the time, in a very satisfactory way. Since an algebraic structure on a space is usually thought of as being stronger than a smooth structure, it might be a useful case study. But since I don’t know exactly what y’all are trying to accomplish, it’s hard for me to say.

Posted by: James on May 27, 2009 1:51 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

If someone feels like reviewing for the spectators what the main goals are here, I for one would be grateful.

I think Andrew originally (maybe still?) comes from the question: How can one formalize usefully the notion “Dirac operator on a smooth loop space”.

A different but related motivation that drives a good bit of the discussion, I think, is the question:

“What’s the right context for differential refinements of topological nonabelian cohomology?”

As in this standard motivating example: for $G$ a compact Lie group, a topological cohomology class in $H^4(B G, \mathbb{Z})$ classifies a topological 2-gerbe on the topological space $B G$. Now we want to in some sense regard $B G$ as a smooth space and talk about a refinement of that 2-gerbe to a smooth 2-gerbe with smooth connection, realizing the corresponding class in $H^4(X, \mathbb{R})$.

So far it is well known how to model this, using simplicial Deligne cohomology.

But we want more: the above picture so far glosses over lots of interesting structure. That class in the differential refinement of $H^4(B G,\mathbb{Z})$ is really an obstruction to lifting the “canonical nonabelian smooth $G$-cocycle on $B G$” (i.e. the universal smooth $G$-bundle on a smooth version of $B G$) to a smooth nonabelian 2-cocycle on $B G$ (the associated String 2-bundle). And we want to refine all that again to differential cohomology.

This, too, can still be modeled by mixing the tools of simplicial spaces, Deligne cohomology, etc. But the constructions this way become increasingly more unnatural and involved. They tend to miss the simple clear picture.

This becomes more manifest the further we continue generalizing. Next, say for $G = Spin$, there is an element in $H^8(B G, \mathbb{Z})$ which comes with its analogous story in nonabelian differential cohomology. Etc.

One wants one clean way to formulate all the expected phenomena here. A good notion of “generalized smooth space” should make that happen.

It’s really a standard example of the need for an $(\infty,1)$-topos: we want to refine the $(\infty,1)$-topos $Top$ with its cohomology given by $H(X,A) := Hom_{Top}(X,A)$ to an $(\infty,1)$-topos $T$ whose objects are such that

$\bar H(X, A) := Hom_{T}(X,A)$

is a differential cohomology.

Phrased this way it is clear that we want $T$ to be the topos of $\infty$-sheaves on something like $Diff$.

But it turns out there is one bit of extra structure we want: in differential cohomology we want to be able to distinguish between an $A$-cocycle on $X$ (smooth as it may be) with or without connection.

It turns out, I think, that this extra bit of structure is precisely the “structure sheaf” $\Pi(-)$ on $Diff$. Which may be thought of as a “Frölicher $\infty$-stack” in a sense, I think, in a sense.

This I was thinking nicely merged our sheafy approach to generalized spaces with Andrew’s favor for sheaf-cosheaf structures.

I think Andrew is entirely right with his intuition that concentrating on contravariant functors out of the site must mean some kind of prejudice. Why not bifunctors $C^{op} \times C \to ...$.

And indeed, that’s the picture “one” arrives at from the $\infty$-stack perspective:

- when doing cohomology, we want an $(\infty,1)$-topos of $\infty$-stacks $(Diff)^\op \to \infty Grpd$;

- when doing on top of that geometry, then we want to equip that with a “structure sheaf”, which is a bifunctor $Diff^{op} \times Diff \to \infty Grpd$.

So some kind of Frölicher $\infty$-space.

Posted by: Urs Schreiber on May 27, 2009 5:45 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

I think Andrew originally (maybe still?) comes from the question: How can one formalize usefully the notion “Dirac operator on a smooth loop space”.

I know how to make sense of the notion of a Dirac operator on a smooth loop space. I’ve done it. What’s a little more indistinct is how best to think of the space of sections of an infinite dimensional vector bundle over an infinite dimensional manifold. But if your basic point is that I want to do differential geometry on things that are almost but not quite manifolds, then you are correct.

I like the notion of a Frölicher $\infty$-stack, I think (I haven’t really grokked it yet, see my comment above about reading slowly). Reluctant as I am to confess to any drawbacks of the Frölicher stuff, for me then the main one is that we haven’t yet figured out the correct way to get non-set-based objects out of it. The obvious way, using Isbell duality, doesn’t work as I showed on the n-lab (by the way, has anyone read and checked that page to see if I’m right?).

However, this drawback boils down to the fact that if one starts with concrete test objects then everything that one defines from them is also concrete.

The obvious way round this is to start with a non-concrete site, but this is starting to sound suspiciously synthetic and while I’m not averse to that, I’m reluctant to go down that route just yet.

So maybe another way is to enrich the morphism sets in some fashion, maybe in the fashion suggested by the $\infty$-stacks/spaces.

So I like that idea, Urs!

Posted by: Andrew Stacey on May 29, 2009 8:57 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

I like the notion of a Frölicher $\infty$-stack

I should just add that the saturation condition itself is something I haven’t seen discussed in the context of those “structured $(\infty,1)$-topoi” as it is being called. But of course we are to be happy that there there still are some things not discussed in Lurie’s work. :-)

Posted by: Urs Schreiber on May 29, 2009 9:39 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Thanks for that!

Another basic question. In algebraic geometry there are a few different levels of smoothness-related things.

1. There is the overall category of algebraic spaces (of whatever kind). The point of this is that it allows you to say what it means for a map between spaces to be given by polynomials.

2. Then there is the concept of a smooth map, meaning essentially that the matrix of partial derivatives has the expected rank everywhere.

3. Last, there is the concept of a nonsingular variety (also called a smooth variety), which means that the map to the point is smooth.

I would think that these three different concepts should also exist in any version of smooth geometry, although the phrase “smooth structure on X” might reasonably mean the analogue of either 1 or 3. In algebraic geometry, there is a big difference between 1 and 3. You can often define differential invariants for any variety, i.e. in case 1, (this is simply because an algebraic structure is stronger than a differential structure), but they are often only well-behaved in case 3. (But it is also often possible to extend them to possibly singular varieties by doing some hard work.)

Once again, if someone feels like saying a few words about the extent to which this is similar to the different versions of smooth geometry, I would be grateful. For instance, if you want to put a smooth structure on the loop space of a smooth manifold, do you mean in the sense of 1 or 3?

Posted by: James on June 1, 2009 11:01 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

I think that there probably are similar levels in “smootheology”, however I think that one can’t make the distinction based on morphisms as we only really have the one notion of morphism. But this isn’t something I’ve thought overmuch about, nor heard much from from others, so there may well be other ways of formulating things that is closer to the picture you describe.

What you describe sounds like:

1. An overarching category containing everything.

2. A method for selecting “relatively nice” objects: in other words, nice morphisms.

3. A subcategory consisting of those objects that are objectively nice.

This fits in with the general “nice category containing some nice objects” philosophy that you’ll hear from time to time on this blog.

So, what do we have …

1. A nice category of smooth spaces. Well, actually we have a lot of these!

2. Nice morphisms. Hmmm, not so sure about these. In algebraic geometry you have “rank”. Our tangent spaces aren’t vector spaces so maps don’t necessarily have well-defined ranks.

3. Nice objects. Now this is the interesting part. At least, I think so. Others disagree. Obviously these should include manifolds, but also other things. Looking at work of Kriegl and Michor in the early 80s, the notion of parallel transport seems key. The idea being that just about everything that is truly differential about manifolds involves infinitesimal deformations and for that you need flows. Kriegl and Michor’s version of a “manifold” concentrates on that aspect. Other ideas on this line talk of “local models” but extend this a little beyond Euclidean spaces.

Loop spaces are extremely nice in that they have linear local models and so it only takes a little tweaking to the definition of a manifold to declare that loop spaces are indeed manifolds (they are sometimes called Fréchet manifolds). Other things are a little more exotic.

I think that your dichotomy probably still holds. One can define most things for general smooth spaces, but they are only well-behaved in the nice neighbourhood.

Posted by: Andrew Stacey on June 2, 2009 11:59 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Andrew wrote:

I’m beginning to think that I’ve been approaching this issue from the wrong side. I’ve been trying to convince all of you that Frölicher spaces are the correct set-based generalisation of smooth manifolds and I don’t really feel that I’ve been getting anywhere.

You’ve gotten me to the point of thinking that Frölicher spaces are a nice idea, and that’s something.

But before I start working with them, I need them to help me prove a theorem I want to prove.

I got interested in this game because I wanted to make the path groupoid of a manifold into a smooth groupoid. I wanted to prove that a connection on a trivial $G$-bundle over this manifold was a smooth functor from the path groupoid to the Lie group $G$. And I wanted to do this because I wanted to categorify this result and thereby discover categorify the concept of ‘connection’. I eventually succeeded, with Urs.

It was clear right away that what I needed was a category of smooth spaces that contained the category of manifolds as a full subcategory and was also cartesian closed, with limits and colimits. Just this, by itself, does most of the job!

There is also another crucial thing: two 1-forms on a smooth space must be equal iff they’re equal on tangent vectors of smooth curves. That mainly requires choosing the right definition of 1-form.

Chen spaces and diffeological spaces make all this stuff work, so I am happy. At least I’m happy now that Alex and I wrote a paper working out the details. I would have been even happier if someone else had already done it for me!

Now you come along and say that Frölicher spaces are even more beautiful, and also have all the properties needed to make my theorems true. (Is that right? I think you told me they’re cartesian closed, with all limits and colimits.)

And so I say “Good!” But I don’t suddenly switch to working with Frölicher spaces. Instead, I just say “Good! Now my results probably hold in the category of Frölicher spaces, too!”

I could of course have axiomatized the properties I want from a category of smooth spaces, and proved my theorems starting from these axioms, to make my results ‘platform-independent’. Then as soon as you checked that these axioms hold for Frölicher spaces, I’d know my results apply in that context.

That would have been the gold-plated approach.

But I was too lazy to do that. Instead, I was being a typical mathematician, who is content to find one context in which their results hold. It took enough work checking that there one category that had all the properties I needed; I was not sufficiently interested in comparative smootheology to study the 2-category of all such categories, as you have begun to do.

However, maybe it’s not my job to convince you but your job to convince me! After all, I’m the differential topologist here and you are the category enthusiasts. So you should be trying to convince me that I need to use your category theory in my work.

No, I’m not a category theorist who needs differential topologists as customers to make my living. So you’ll get no ‘hard sell’ from me. I was doing some differential topology of my own, and I needed a convenient context in which to do it. Now I have one — in fact many — so I’m satisfied.

Saying that diffeological spaces form a quasitopos is like saying that Australia has sheep. Good for them, but so what? How does that help me in my study of Norwegian moose? - I mean, manifolds.

Perhaps the good properties of diffeological spaces don’t help you. All I really know is that they helped me.

More precisely, they helped Urs Schreiber and Konrad Waldorf and I figure out the theory of categorified connections. In the end, this theory can be described using just manifolds and differential forms. But we needed a nice category of more general smooth spaces to reach this goal.

Frölicher spaces would probably work too. But diffeological spaces did the job just fine.

This means that I have, for each $n$, a complete copy of the category of ordinary manifolds wherein I essentially discard the diffeology above dimension $n$ (obviously I can’t discard it completely, but insist that everything higher locally factors through something lower). By Boman’s result, all of these are exact copies of the category of manifolds - even manifolds with corners, thanks to Kriegl and Michor. Why should I accept a category with so much junk in it?

If you are trying to prove theorems that don’t hold for these weird diffeological spaces, then indeed you should switch to some other category! It just so happens that most of the results I was trying to prove are so general that they hold even for these weird spaces. So, why shouldn’t I prove them at a high level of generality?

In general, I’m perfectly happy to work in a well-behaved category that has bad objects in it — as long as I can prove the theorems I want, the bigger and better-behaved the category, the happier I am.

I don’t go searching for anthills in my back yard, dig them up, and then say “Ooh, yuck! My back yard is full of ants!” And similarly, I don’t go around looking for nasty spaces and say “Ooh, yuck! My category has nasty spaces in it!” As long as the ants stay out of my kitchen, I’m content. And as long as my category lets me easily and efficiently prove the theorems I’m interested in, and contains all the objects I care about, I’m content.

How I would like the debate to go is to start by discussing: “What is the essence of a manifold? What is it that makes a manifold so special?” This is, to me, the most interesting aspect of the whole debate.

This makes it clear that you have goals very different than mine. So, you shouldn’t expect our discussion to help you very much.

In fact I even doubt the existence of a ‘debate’. It’s not as if I think everyone in the universe should use diffeological spaces.

John wrote:

Since diffeological spaces are concrete sheaves on a concrete site, a vast arsenal of weaponry springs into play whenever you use them.

So what? Just because it’s readily available doesn’t mean that I should use it.

True. All I can say is that I need this vast arsenal of tools when I’m doing differential topology.

John wrote:

But if you want a topos of smooth spaces, you might as well join Kock and Lawvere and Reyes and work on synthetic differential geometry.

I clearly don’t know enough about topoi to understand this paragraph. If I want a topos, why wouldn’t I choose the route of synthetic differential geometry?

No reason not to — synthetic differential geometry is great stuff! That’s what I was trying to say: “if you want a topos of smooth spaces, use synthetic differential geometry!”

If diffeological spaces really have advantanges over Frölicher spaces then I suspect that they are merely in the realms of extremely abstract and abstruse category theory that, in day-to-day differential topology, is extremely unlike to bother me. So there should be some differential-geometric theorems that show that diffeological spaces, and not Frölicher spaces, are necessary to understand ordinary smooth objects.

Again, I have no desire to prove there’s something one can do with diffeological spaces that one can’t do with Frölicher spaces. In fact I would like nothing better than for you to write a big book full of theorems about Frölicher spaces, which I could then use.

I suspect that at some point I will want some extra properties of the category of diffeological spaces that the category of Frölicher spaces lacks, but I don’t think I’ve hit that point yet. For example, I like local cartesian closedness, and I might need it someday, but I haven’t needed it yet. And it’s quite possible that when I get to needing a better category I’ll want a topos. Luckily, there are already some choices waiting for me.

Posted by: John Baez on May 27, 2009 4:17 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

John, thanks for the reply. I shan’t comment on all of it, though all of it was extremely interesting.

Just a couple of things:

No, I’m not a category theorist who needs differential topologists as customers to make my living. So you’ll get no ‘hard sell’ from me.

I deliberately said “category enthusiast” not “category theorist”. I sometimes feel as though category theory is to pure maths as pure maths is to the rest of the world. The standard response to “I’m a category theorist” is “Oh, I was never much good at category theory in [grad] school.”. Maybe category theory needs more enthusiasts such as yourself who can tell the rest of us why we should care about categories.

What I meant by my comment was that I’m not a real category enthusiast, but I’m probably a little more open to category theory than the average pure mathematician. So you could use me a bit like a test case for the categorical view of things. I should be an easy case (apart from my general state of busyness and slow reading ability, of course). I wasn’t particularly referring to diffeological spaces but to the whole categorical set-up.

(my warning about exaggerating to make a point should probably be re-read here! Maybe I should have a permanent link to that phrase)

As I said to Urs above, I got into this game because I was trying to understand a specific smooth space. My introduction to Frölicher spaces was via Kriegl and Michor which is extremely analytic, almost the exact opposite of category theory. I want to do analysis on these spaces and for that I need really tight control.

I don’t go searching for anthills in my back yard, dig them up, and then say “Ooh, yuck! My back yard is full of ants!” And similarly, I don’t go around looking for nasty spaces and say “Ooh, yuck! My category has nasty spaces in it!” As long as the ants stay out of my kitchen, I’m content.

I do! That’s the fun of backyards. You never know what’s in it. And you also never know what’s going to sneak into it. After a (very mild) earthquake in the Bay Area, a small crack appeared in our wall. We weren’t bothered until the ants found it. And now, we have to keep a close guard on our doors because someone in our family has a fascination with snails and wants to bring them in to the house to study them better! So I keep a close eye on the ants in the backyard to make sure that they stay there.

This makes it clear that you have goals very different than mine. So, you shouldn’t expect our discussion to help you very much.

Well, maybe I won’t find any collaborators here to study this particular issue, but it is and has been extremely useful to refine my ideas and find some interesting questions to work on (once I have the time to work on them …). So I hope you don’t mind me carrying on about it, and trying to divert various discussions to my own ends.

In fact I would like nothing better than for you to write a big book full of theorems about Frölicher spaces, which I could then use.

Would you settle for an n-lab entry?

Posted by: Andrew Stacey on May 29, 2009 9:18 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

If diffeological spaces really have advantanges over Frölicher spaces then I suspect that they are merely in the realms of extremely abstract and abstruse category theory that, in day-to-day differential topology, is extremely unlike to bother me.

I don’t think local cartesian closure is “extremely abstract and abstruse.” For instance, I would venture that the failure of local cartesian closedness is the principal defect of the category of compactly generated topological spaces. It isn’t only local cartesian closure itself, although this tends to get important whenever you start talking about families of objects parametrized by some other object. But local cartesian closure implies the very useful fact that pullbacks preserve colimits (and, in good cases where the adjoint functor theorem applies, is equivalent to it).

I tend to get lost in all the different types of smooth spaces, but the niceness of subsequential spaces for algebraic topology, in which continuity is detected only by sequences, suggests to me that detecting smoothness by maps out of curves is quite reasonable.

Posted by: Mike Shulman on May 27, 2009 11:15 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Mike wrote:

I tend to get lost in all the different types of smooth spaces, but the niceness of subsequential spaces for algebraic topology, in which continuity is detected only by sequences, suggests to me that detecting smoothness by maps out of curves is quite reasonable.

There’s a theorem saying that if $C$ is a closed convex subset of $\mathbb{R}^n$, a function $f : C \to \mathbb{R}$ is smooth iff $f \circ \gamma$ is smooth for all smooth curves $\gamma : \mathbb{R} \to C$.

But try to prove it! Even when $C$ is the unit interval it takes some thought, thanks to the endpoints. After you try to prove this theorem for a while you may decide that instead of being ‘quite reasonable’, it’s almost miraculous.

Nonetheless it’s true, and that’s part of Andrew’s case for Frölicher spaces, where smoothness is detected by maps out of $\mathbb{R}$ and also, equally well, by maps into $\mathbb{R}$.

Posted by: John Baez on May 28, 2009 5:28 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

I should have said, detecting smoothness by maps out of curves, once you know that it suffices for ordinary manifolds, seems like a promising approach to generalized smooth spaces. I was, indeed, quite surprised when I was first told that it does so suffice, and I must admit to not ever having actually tried to prove it myself.

One of the nice things about subsequential spaces is that the embedding of (sequential) topological spaces preserves not only all limits (since it has a left adjoint), but many colimits. The corresponding question for smooth spaces doesn’t seem to have received a lot of attention (by which I mean it hasn’t been mentioned on the nCafe, so far as I can tell, or in Andrew’s or John and Alex’s papers). Namely, given some nice category $Sm$ of smooth spaces, what limits and colimits are preserved by the embedding $Mfd \hookrightarrow Sm$?

Of course, one of the main reasons we want a good $Sm$ is that $Mfd$ lacks a bunch of limits and colimits, but surely we want to keep some of the limits and colimits it does have. For instance, a transverse pair of smooth maps have a pullback in $Mfd$; is this pullback preserved in $Sm$? Presumably so in categories that involve “maps in,” but not as obviously in “maps out” categories. Presumably the colimit coming from an open cover is preserved in categories like Chen spaces and diffeological spaces which are (concrete) sheaves for some coverage including open covers? What about non-open covers?

Posted by: Mike Shulman on May 28, 2009 8:50 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

[…] keep some of the limits […]

By the way, the whole motivation for [[derived smooth manifolds]] is to get a category of smooth spaces not just with all limits, but also with the right behaviour under limits.

The point here is maybe more familiar in the dual picture: diffeological spaces aka sheaves of sets on $Diff$ have all quotients, but their cohomology doesn’t go along nicely with taking quotients. To get that, one needs to pass to sheaves of groupoids on $Diff$.

For limits it’s dual to that: we want groupoid-valued co-probes to remember correctly what happens under fiber products.

David Spivak discusses this nicely in his introduction, following the introduction of Lurie’s [[structured space]].

Posted by: Urs Schreiber on May 28, 2009 9:45 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Of course, one of the main reasons we want a good Sm is that Mfd lacks a bunch of limits and colimits, but surely we want to keep some of the limits and colimits it does have. For instance, a transverse pair of smooth maps have a pullback in Mfd; is this pullback preserved in Sm? Presumably so in categories that involve “maps in,” but not as obviously in “maps out” categories. Presumably the colimit coming from an open cover is preserved in categories like Chen spaces and diffeological spaces which are (concrete) sheaves for some coverage including open covers? What about non-open covers?

That’s a very good question, Mike. I’m glad you asked that. I wonder what the answer is for Frolicher spaces.

Posted by: Andrew Stacey on June 1, 2009 10:47 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Looks like I might be giving the Comparative Smootheology talk a time or two more. If anyone has any suggestions on how to improve it then please let me know.

More jokes, maybe?

Posted by: Andrew Stacey on May 29, 2009 9:20 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

You may want to soften your claim about orbifolds as Froelicher spaces (slide 4 on the printout). The problem, as I see it, is this. If you think of orbifolds as sets with extra structure then all sorts of things either go wrong or become artificial. For instance there is no convincing way to make the loop space of an orbifold into an infinite dimensional orbifold. Also, you have no twisted sectors and you don’t get the correct fundamental group. I can go on like this, but I should probably get off my hobby horse. My bottom line is that whatever orbifolds are (and I don’t pretend to know all the answers), they should be at least a 2-category.

Posted by: Eugene Lerman on May 29, 2009 4:57 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Yes, Eugene’s point here is an important one.

As I said just yesterday or so in another thread (paaroting what greater minds than me keep saying, but its true nevertheless):

its not enough to have all limits and colimits. We want limits and colimits to be the correct.

To get the correct colimits, we need sheaves with values in $\infty$-groupoids.( aka stacks)

To get the correctlimits, we need sheaves on $\infty$-groupoids (aka derived stacks)

Posted by: Urs Schreiber on May 29, 2009 6:41 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Im in a haste, as you can tell, being on the train where connection breaks down any second.

But one quick followuo:

what I just said doesn’t mean that diffological spaces/Froelichr things etc are somehow obsolete. They are still very useful for describing rectified $\infty$-stacks, namely $\infty$-groupoids internal to these categories of sheaves.

It’s a big useful theorem that rectified $\infty$-stacks are sufficient: the $\infty,1$-cat presented by the model structure on simplicial sheaves is (equivalent to) the $(\infty,1)$-category of non-recitiied $\infty$-stacks.

Ah, have to run

Posted by: Urs Schreiber on May 29, 2009 6:49 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

That’s a fair point. Orbifolds is in that list because I wanted something that everyone would immediately think “That’s something important I’ve heard lots about” and which meant that I could say that we wanted a co-complete category. In my original list, all the other things we either limits or exponential objects. One of the advantages of speaking last is that you get to say “As … was saying yesterday, we want …” so I was able to add the smooth path groupoid from Konrad’s talk which gave me another construction that used quotients. However as orbifolds also got mentioned I let that stand but, as you say, I probably shouldn’t.

So I wasn’t really trying to claim that orbifolds should be thought of as Frölicher spaces, I was trying to motivate the desire for a $C^5$ category, but I guess I can be accused of making it look like I was trying to claim that orbifolds should be thought of as Frölicher spaces.

Now that I have the path groupoid example I can safely remove orbifold from the list and shall do so.

Posted by: Andrew Stacey on May 30, 2009 10:13 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Now that I have the path groupoid example I can safely remove orbifold from the list and shall do so.

Now this last sentence I didn’t quite follow. Maybe I am misunderstading what you are saying here. Here is how I think about this:

The [[path groupoid]] and an orbifold is a smooth groupoid both happen to be groupoids internal to diffeological spaces, and also to Frölicher spaces, I’d think.

The usual notion of orbifold is even more tame than the path groupoid, as it is even what is usually called a [[Lie groupoid]], i.e. internal to ordinary manifolds.

Posted by: Urs Schreiber on May 31, 2009 1:51 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

The purpose of the slide in question was to give a quick explanation of why the quest for a complete, cocomplete, cartiesian closed category is interesting. Thus it was intended to establish a connection in the listeners’ minds between this topic and other topics that they are likely to have already heard about, and hopefully some of which they already think of as important. Someone who hasn’t already heard of Frölicher spaces, diffeological spaces, or Chen spaces may wonder what the point of it all is, but if I tell them that it’s something to do with loop spaces, orbifolds, embedding spaces, and things like that then these are things that they will have already heard about and so they will hopefully be prepared to listen to me for an hour.

(Notice that I’m sort of assuming that my audience is more differential than categorical here, which wasn’t the case in Ottawa. However, I would expect that a categorical audience would be much more open to hearing of a talk on this theme without overmuch motivation than a differential audience, so my motivation was much more aimed at the few differential topologists who had snuck in without telling anyone).

So I only wanted to quickly establish a vague connection in people’s minds. I didn’t want to go into any great depth about exactly what that connection was. If anyone had asked, then I would have been more honest - to the extent that my knowledge allowed!

However, I should be more careful that I don’t explicitly say anything false and looking at the slide again I see that I do write “To contain …” all these things whereas, as Eugene and Urs point out, it’s not strictly true. So I should change “To contain” to “Motivation”. Maybe, “Motivation: to study:” is the most truthful way to put it.

Posted by: Andrew Stacey on June 1, 2009 8:30 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

I have a question about the dictum of Grothendieck’s quoted above: “It’s better to have a good category with bad objects than a bad category with good objects.”

This has come up many times over the years here, but I haven’t ever seen a citation. Does anyone know one? If not, can anyone at least identify an early point where it was attributed to him?

Posted by: James on September 17, 2009 1:35 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

In response to Thomas Nikolaus’ question over here, I thought I’d summarize the idea of a ‘Fermat theory’, as explained in Anders Kock’s talk ‘Kähler differentials for Fermat theories’.

Observation: If $f : \mathbb{R} \to \mathbb{R}$ is a smooth function,

$f (x+y) = f(x) + y \tilde{f}(x,y)$

for some smooth function $\tilde{f} : \mathbb{R}^2 \to \mathbb{R}$. Note also that

$f'(x) = \tilde{f}(x,0)$

Fermat noted this for $f$ a polynomial; Hadamard generalized it to smooth $f$. All the usual rules for derivatives can efficiently be derived from this formula — try it and see!

This is also true if $f$ is parametrized, that is, a smooth function from $\mathbb{R} \times \mathbb{R}^n$ to $\mathbb{R}$.

This suggests defining the following kind of algebraic theory. A Fermat theory is an extension of the algebraic theory of commutative rings, such that for any $(n+1)$-ary operation $f$ there is a unique $(n+2)$-ary operation $\tilde{f}$ such that

$f(x + y, \vec{z}) = f(x, \vec{z}) + y \tilde{f}(x,y,\vec{z})$

where $\vec{z}$ is a list of $n$ variables, and I’m writing the operations $f$ and $\tilde{f}$ as if they were functions, to avoid commutative diagrams.

This lets us define an operation

$\frac{\partial f}{\partial x} (x, \vec{z})$

by

$\frac{\partial f}{\partial x} (x, \vec{z}) = \tilde{f}(x,0,\vec{z})$

With a bit of more work we get for each $n$-ary operation $F$ a list of $n$-ary operations $\partial_i F$. So, if $T(n)$ denotes the set of $n$-ary operations in our algebraic theory $T$, we get maps

$\partial_i : T(n) \to T(n)$

For any algebraic theory, $T(n)$ is an algebra of that theory — the free $T$-algebra on $n$ generators. So, when $T$ is a Fermat theory, $T(n)$ is automatically a commutative ring. One can check that each map

$\partial_i : T(n) \to T(n)$

is a derivation of this ring — this is really just the chain rule.

Theorem: The map

$\langle \partial_1, \dots, \partial_n \rangle : T(n) \to \prod_{i = 1}^n T(n)$

is the universal $T$-derivation of $T(n,1)$.

Alas, I’m not quite sure what this theorem means. But I think it’s something like this: if $M$ is a module of $T(n)$ and $\delta : T(n) \to M$ is a derivation, then $\delta$ factors through the map $\langle \partial_1, \dots, \partial_n \rangle$.

So, this theorem gives us a concept of Kähler differentials for $T(n)$.

Posted by: John Baez on January 2, 2010 10:01 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

First of all, thanks for this detailed answer John!

I think you meant that a Fermat Theory is a Lawvere algebraic theory with extra properties. A Lawvere algebraic theory is a category with finite products such that every object is a product of the generic object T, right?

An example for an algebraic theory is provided by the full subcategory of smooth manifolds consisting only of $\mathbb{R}^n$ for each $n$. The property of being a Fermat theory is for this example satisfied as you mentioned at the beginning of your post. The free algebra on $n$ generators $T(n)$ is here exactly $C^\infty(\mathbb{R}^n)$ (the n-ary operations) and the derivations are the standard partial derivatives. What I called $C^\infty$-rings in my last post are algebras for this theory in the category of sets (so as I said, $C^\infty(\mathbb{R}^n)$ is the free algebra on $n$ generators).

Now you said, that $\prod_{i=1}^n T(n)$ should be the universal derivation of $T(n)$? And the modules you mention are just modules over the underlying ring of T(n)? In our example this would mean, that $\prod_{i=1}^n C^\infty(\mathbb{R}^n)$ is the module of Kaehler-differentials of $C^\infty(\mathbb{R}^n)$ which is not true as proved in the other blog entry for the case $n=1$.

But in the abstract of his talk, Anders Kock says:
“There is a notion of a module over a Fermat algebra, and of a derivation into such. For a given algebra A, there is a universal module Ω A receiving a derivation from it. This is the module of Kaehler differentials.”
So maybe the notion of module over a Fermat algebra is more then just a module on the underlying ring, as I alreay expected in my last post in the other entry (sorry I can’t link to it).

Posted by: Thomas Nikolaus on January 3, 2010 12:38 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Thomas wrote:

I think you meant that a Fermat Theory is a Lawvere algebraic theory with extra properties. A Lawvere algebraic theory is a category with finite products such that every object is a product of the generic object T, right?

Right. That’s what I meant by the phrase ‘algebraic theory’.

I have expanded my comment into a short $n$Lab entry on Fermat theories. There I mention that the theory of $C^\infty$-rings is a Fermat theory.

… maybe the notion of module over a Fermat algebra is more then just a module on the underlying ring,

Yes, quite possibly. I know what a module for an algebra of any operad is. So maybe I should know what a module for an algebra of any algebraic theory is. And maybe that’s what Kock meant. But I’m not sure — my notes don’t say.

Similarly, from what you wrote, I now suspect that a ‘$T$-derivation’ is more than a mere derivation. But again, my notes don’t say what a ‘$T$-derivation’ is.

I have not found anything written by Kock on this subject, but there might be something.

Posted by: John Baez on January 3, 2010 3:23 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

The following may also be relevant in this context:

Posted by: Bas Spitters on January 4, 2010 10:48 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Posted by: Toby Bartels on January 5, 2010 4:48 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

John wrote:

I know what a module for an algebra of any operad is. So maybe I should know what a module for an algebra of any algebraic theory is.

What is a module for an algebra of any operad?

Posted by: Thomas Nikolaus on January 3, 2010 8:17 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Try page 16 of Kriz and May’s Operads, Algebras, Modules and Motives. Better yet: think about how it works for modules of associative algebras and modules of Lie algebras, and invent the generalization yourself!

By the way: Kriz and May’s Definition 4.1 deals with modules of algebras of linear operads, but I don’t think that feature is important. More importantly, now that I look, their definition seems to generalize the notion of ‘bimodule’ for an associative algebra. But it’s easy to change their definition to a definition of ‘left module’ or ‘right module’. Maybe they do that later…

Posted by: John Baez on January 3, 2010 7:28 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Hm, expected something like that, but lets see what that definition of module spits out for operads in style of Lawvere theories:

1) Lets start with the case ordinary rings, because it’s simpler than $C^\infty$-rings. The Lawvere theory of commutative rings is given by the operad which has as $n$-ary Operations the set of polynomials in $n$-variables $\mathbb{R}[x_1,\ldots,x_n]$ and the composition is given by insertion of polynomials. Then an algebra in Set (with cartesian monoidal structure) for this operad is given by a Set $A$ for which any polynomial $p \in \mathbb{R}[x_1,\ldots,x_n]$ provides a map $A^n \to A$ which is compatible with insertion of polynomials. That is a commutative ring.

Note that we did not assume that $A$ is an abelian group but this is part of the algebra structure. It’s provided by applying by the polynomial $q(x_1,x_2) := x_1 + x_2$.

Now lets see what a module in Set for such an algebra $A$ is: We have a set $M$ and for each Polynomial $p \in \mathbb{R}[x_1,\ldots,x_n]$ a map $A^{n-1} \times M \to M$which is compatible with the algebra structure on $A$ in the obvious way (I hope thats clear, otherwise I could write down the diagramm…). Now I have the feeling that this is a strange thing. For example the polynomial $q(x_1,x_2) := x_1 + x_2$ provides a way of ADDING elements of the Algebra $A$ to elements of the Module…

2) The operad for $C^\infty$ rings is analogous to the operad for ordinary rings, except that the $n$-ary operations are not only polynomials in $n$-variables but all smooth maps in $n$-variables: $C^\infty(\mathbb{R}^n, \mathbb{R})$. Each $C^\infty$-ring has an underlying ring by just forgetting that we could apply more than polynomials. So the modules are equally courious (or even more because there are way more strange operations).

Posted by: Thomas Nikolaus on January 4, 2010 1:26 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Yesterday I tried to find out what Anders Kock told in his talk about modules and derivations for Fermat-theories. In doing so I came across the paper On 1-form classifiers by A. Kock and E.J.Dubuc. This seems to be the source of his talk in Ottawa and I could’t believe that it was written in 1984!!!

However, let me explain what they say about modules and derivations. Let $\mathbb{T}$ be a Fermat theory with the set of $n$-ary operations denoted by $T(n)$. Remember from John’s nLab entry that we have partial derivatives $\partial_i : T(n) \to T(n)$for $1 \leq i \leq n$.

Examples are the theory of $C^\infty$-rings where $T(n) = C^\infty(\mathbb{R}^n,\mathbb{R})$ or the theory of commutative $\mathbb{R}$-algebras where $T(n) = \mathbb{R}[x_1,\ldots,x_n]$ and the partial derivatives are in both cases given by the ordinary partial derivatives.

Let $A$ be a $\mathbb{T}$-algebra. An $A$-module is simply a module for the underlying ring of $A$. But the notion of derivation $A \to M$ has to be changed. Let $A$ be an ordinary algebra, then a map $\delta: A \to M$ is a ordinary derivation if the relations

1) $\delta(a + b) = \delta(a) + \delta(b)$

2) $\delta(\lambda a) = \lambda \delta(a)$

3) $\delta(a \cdot b) = a \delta(b) + b \delta(a)$

Those three identities imply that for any polynomial $p \in \mathbb{R}[x_1,\ldots,x_n]$ and algebra elments $a_i$ we have

$(*) \quad \delta \Big( p(a_1,\ldots,a_n)\Big) = \sum_{i=1}^n \frac {\partial p} {\partial x_i}\Big(a_1,\ldots,a_n\Big) d(a_i)$And in fact 1, 2 and 3 are equivalent to (*) what can easily be seen by taking the polynomials $p_1(x,y) = x + y, p_2(x) = \lambda x$ and $p_3(x,y) = xy$.

That observation motivates the

Definition: Let $A$ be a $\mathbb{T}$-algebra and $M$ be an $A$-module. A derivation $\delta: A \to M$ is a map such that for each relation $f \in T(n)$ and elements $a_i$ in $A$ we have$\quad \delta \Big( f(a_1,\ldots,a_n)\Big) = \sum_{i=1}^n \frac {\partial f} {\partial x_i}\Big(a_1,\ldots,a_n\Big) d(a_i)$

Note that each derivation $\delta : A \to M$ is especially a derivation on the underlying ring, which follows again by taking the above introduced polynomials $p_1,p_2$ and $p_3$ (and because a Fermat theory is by definition an extension of the theory of rings, so $T(n)$ contains all polynomials).

Posted by: Thomas Nikolaus on January 5, 2010 1:36 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa II

Thanks, Thomas. I moved this stuff to here.

Posted by: Urs Schreiber on January 11, 2010 4:10 PM | Permalink | Reply to this

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