The n-Category Café

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.