Verity on ∞-Categories From Topology

February 12, 2008

Verity on ∞-Categories From Topology

Posted by John Baez

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Anyone who wants to keep up with the latest research on $n$ -categories needs to know what the Australians are doing. It helps to be on the Australian Category Seminar mailing list… which is free if you visit. (Maybe it’s free otherwise, too — I don’t know.)

Dominic Verity has been doing wonderful things lately, and here’s what he’s talking about tomorrow…

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Dominic Verity, Cobordisms and weak complicial sets, Australian Category Theory Seminar, February 13, 2008, Macquarie University, seminar room E7A 333.

Abstract: It is generally held that the totality of all $n$ -manifolds (with corners) should support a canonical weak $n$ -categorical structure, under which all composites of cells are describable as “gluings” of manifolds along common boundary components. At low dimensions, not only do we know that this is indeed the case, but we also know how to describe free weak 2- or 3-categories in terms of such higher categories of “cobordisms embedded in cubes”.

In this talk we describe how to build a weak complicial set (qua simplicial weak $\omega$ -category) $Cob^k$ , whose $n$ -simplices correspond to embeddings of manifolds of some fixed co-dimension $k$ into the geometric $n$ -simplex. Here we work in the PL-category in order to avoid some of the technicalities involved when working with manifolds with corners in the smooth setting. However, it is likely that our main results bear direct translation to that latter context.

From the point of view of the theory of weak complicial sets, $Cob^k$ is an interesting structure for a number of reasons. Firstly, its underlying simplicial set is a Kan complex. Secondly, it possess a stratification which makes it into a weak complicial set but which is non-trivially distinct from the “all simplices are thin” structure that comes with Kan-ness. Finally, we have strong reasons to conjecture that this non-trivial complicial stratification can be extended, by making thin all simplices which are “morally” equivalences, to a complicial stratification whose thin simplices bear a characterisation in terms of simple homotopy equivalences (and which are thus $s$ -cobordism-like).

If anyone reading this attends that talk, any comments on it would be much appreciated here!

Posted at February 12, 2008 9:45 PM UTC

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23 Comments & 3 Trackbacks

Re: Verity on ∞-Categories From Topology

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Sounds very much like this evolved from or was directly inspired by what Michael Hopkins was talking about here. (Or vice versa.)

Posted by: Urs Schreiber on February 12, 2008 10:10 PM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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Lately # I keep thinking to myself: “Well, hm, so it seems as if the definition of manifold is just wrong.”

Requiring something to be locally isomorphic to $\mathbb{R}^n$ is just asking for trouble. We wouldn’t scare any kid who isn’t scared by manifolds by saying: look, all we should really ask is that it can be probed by $\mathbb{R}^n$ s. So it’s maybe just a historical accident that we are talking about manifolds in the first place. We should be talking about smooth spaces (Chen, diffeological, or the “saturated Frölicher sheaves” which I am hoping will materialize when Andrew, Todd and maybe myself just keep conjuring them up.)

Anyway: did anyone ever try to consider the $\infty$ -category of cobordisms of such smooth spaces?

Posted by: Urs Schreiber on February 12, 2008 10:20 PM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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I would like to recall where this is headed, why in the context of quantum field theory one should care about the $\infty$ -category of abstract cobordisms – but will recall it from a certain perspective.

From my perspective, what’s going on is this:

we specify an $n$ -dimensional TFT (“of generalized $\sigma$ -model type” or “of a charged $n$ -particle”)

by specifying

- a domain $n$ -structure $\mathrm{tar}$ , sometimes known as target space

- a codomain $n$ -structure $\mathrm{phas}$

- a morphism between these

$\mathrm{tra} : par \to phas \,,$

sometimes known as the background field.

For instance, for Chern-Simons / Dijkgraaf-Witten theory with respect to the Lie / discrete group $G$ we’d take

- $par$ to be some model for $B G$

- $tra$ to be the corresponding model for the Chern-Simons 3-bundle (2-gerbe) with connection on $B G$ .

Next, we specify an $n$ -category $Par$ of $n$ -dimensional “parameter spaces” $par$ . This is, for instance, the cobordism category that Verity is after here.

In our example of Chern-Simons / Dijkgraaf-Witten, we’d take $Par$ to be some 3-category of 3-dimensional cobordisms.

We want the TFT to be this: for each object $par$ of $Par$ there is supposed to be a notion of transgression $\mathrm{tg}_{par} tar$ of the background field $tra$ to $par$ , and we want to assign certain invariants of the transgressed thing to $\mathrm{par}$ .

This transgression is usually thought of as a 2-step process: first pull-back to a mapping space, then “integrate over the fiber”.

And here is a crucial insight, which is simple in retrospect, but which you may imagine me being fond of. So I’ll capitalize it:

Transgression is nothing but forming the inner hom: $\mathrm{tg}_{par} tra = hom(par,tra) : hom(par,tar) \to hom(par,phas) \,.$

I started saying this here, recently wrote a whole entry about it, Transgression (pdf, blog), aspects of which made it in a special example worked out in great detail in section 4 of Smooth 2-functors and differential forms (arXiv, blog), as well as in section 9 of $L_\infty$ -connections and applications (pdf, blog), which I expanded on for the example of Chern-Simons theory in States of Chern-Simons theory (I, II, III).

Sorry to drown you in links here. This is just to indicate that the capitalized statement is trustworthy. It has been checked.

If we accept it, it leads to a beautiful picture of quantum field theory:

Consequence. The extended functorial QFT defined by the background field $tra$ and the parameter space category $Par$ is the morphism

$QFT_{tar}^{Par} : hom(--,tra) : Par^{op} \to V \,,$

where $V$ is the thing we are enriching over.

Well, or maybe that’s just the “pre-QFT”, the full QFT being the image of that under some section-taking functor.

But I’ll leave it at that for the moment.

In any case, we will tend to want to set $Par = some \infty-category of cobordisms \,.$

Posted by: Urs Schreiber on February 12, 2008 11:30 PM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

(Maybe it’s free otherwise, too ; I don’t know.)

Yes, it is - you just have to ask.

As for Urs’ comments: Caveat Lector! Transgression has two meanings - exactly opposite to each other - one man’s (person’s)transgression is another person’s
suspension.

Posted by: jim stasheff on February 13, 2008 2:10 AM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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Next, we specify an n-category Par of n-dimensional “parameter spaces” par.

I think you must have made some typos. What happened to “ $tar$ ”? The background field $tra$ should go $tra : par \rightarrow \phas$ , right?

Posted by: Bruce Bartlett on February 14, 2008 9:11 AM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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Sorry, I just made the same error. I meant $tar : tra \rightarrow \phas$ .

Posted by: Bruce Bartlett on February 14, 2008 9:13 AM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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Bruce,

now we have all permutations tried out! :-)

It’s a pity that $par$ , $tar$ and $tra$ look so similar. You gotta read it out as “particle”/”parameter”, “target” and “transport”, then it works fine.

So the background field is denoted

$tra : tar \to phas$

(and, yes, I made a typo with this and then you made two other permutation typos ;-)

Then the transgression (or “suspension” as Jim points out, depending on your convention) of that background field to maps from $par$ to $tar$ is

$hom(par,tra) : hom(par,tar) \to hom(par,phas) \,.$

And we want to say “ $conf$ ” for $hom(par,tar)$ . And luckily, for once there is neither an “a” nor an “r” in this one.

Posted by: Urs Schreiber on February 14, 2008 12:20 PM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

Some centuries ago, a trend toward using symbols rather than (truncated) words took hold - back to the future!

Posted by: jim stasheff on February 14, 2008 1:37 PM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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Some centuries ago, a trend toward using symbols rather than (truncated) words took hold - back to the future!

I agree. Once everybody knows what this is supposed to be about I’ll talk about about background fields

$\nabla : X \to F$

and their transgression/suspension

$\nabla^\Sigma : X^\Sigma \to F^\Sigma \,.$

But this will take a bit, since it will require making people believe that it is a good idea to write $X$ for instance for the $\infty$ -path groupoid of a space, or the like.

Posted by: Urs Schreiber on February 14, 2008 2:37 PM | Permalink | Reply to this

Symplectic subgroups of exceptional lie groups

Does anyone know what the maximum symplectic subgroups of the exceptional lie groups are? I think Sp(8) is the maximum symplectic subgroup of E6 but thats about it. I’m sure E8 must have a larger symplectic subgroup than this but I can’t prove it. What about for F4 and G2; is USp(2) the maximum symplectic subgroup for these? What about E7? I can find no papers on this topic.

Posted by: Steve Wheeler on February 12, 2008 11:32 PM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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There’s something that I wonder about whenever anyone talks about the $\infty$ -category of cobordisms. Verity’s abstract says that $\mathrm{Cob}^k$ is a Kan complex; in other words, an $\infty$ -groupoid rather than just an $\infty$ -category. This seems like an example of the general fact that any $\infty$ -category with duals at all levels is actually an $\infty$ -groupoid—the difference between duals and inverses goes away when you push things all the way up to infinity. (Eugenia gave a talk about this at Chicago once, but I think she said that John first suggested it to her.)

Anyway, the thing I really wonder about is this: an $\infty$ -groupoid is the same as a topological space (qua homotopy type), and topologists are already familiar with a space (or spectrum) whose homotopy groups are cobordism groups of manifolds; it (or one version of it) is called $MU$ . Is it naive of me to expect the $\infty$ -groupoid of cobordisms to be somehow related to $MU$ ? Have people written about this somewhere and I’ve just missed it completely?

Posted by: Mike Shulman on February 17, 2008 6:15 AM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

MU is for complex manifolds. Which cobordism theory should correspond??

Posted by: jim stasheff on February 17, 2008 1:36 PM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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When Jim Dolan and I first speculated about the $\infty$ -category of cobordisms, we conjectured that the $\infty$ -category of framed cobordisms was the free stable $\infty$ -category with duals on one object (see page 28 here). We argued that if one took the duals in this $\infty$ -category and promoted them to inverses, one would get the free stable $\infty$ -groupoid on one object, namely the sphere spectrum. As evidence for this, we pointed out that the sphere spectrum represents framed cobordism theory.

Later Eugenia Cheng noticed that it’s very hard — maybe impossible — to tell the difference between an $\infty$ -category with duals and an $\infty$ -groupoid. The difference only seems to show up when you work with $n$ -categories, and then the difference kicks in at the $n$ -morphism level. In fact, she has proved a theorem to this effect: an $\omega$ -category with all duals is an $\omega$ -groupoid.

There’s still some wiggle room to try to weasel out of this conclusion — for example, by adopting a definition of ‘ $\omega$ -category with duals’ that violates the hypotheses of her theorem.

But, at present it looks like the $\omega$ -category of framed cobordisms should be an $\omega$ -groupoid, and that it should be the free stable $\omega$ -groupoid on one object, and that this should be the sphere spectrum — or if you prefer, the infinite loop space $\Omega^\infty S^\infty$ .

Is it naive of me to expect the $\infty$ -groupoid of cobordisms to be somehow related to $MU$ ? Have people written about this somewhere and I’ve just missed it completely?

I think that $MU$ should correspond to the $\infty$ -category of complex cobordisms (i.e., cobordisms with stable normal bundle equipped with a complex structure).

In fact, I think the spectrum for any cobordism theory should correspond to an $\infty$ -category of cobordisms in this sort of way. For example, the spectrum $MSO$ should correspond to the $\infty$ -category of oriented cobordisms.

We’ve talked about this idea a lot on this blog. Briefly: one should be able to get a ‘fundamental $n$ -category with duals’ from any stratified space, and the Thom–Pontryagin construction gives a bunch of stratified spaces…

Posted by: John Baez on February 17, 2008 7:06 PM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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But, at present it looks like the $\omega$ -category of framed cobordisms should be an $\omega$ -groupoid, and that it should be the free stable $\omega$ -groupoid on one object, and that this should be the sphere spectrum — or if you prefer, the infinite loop space $\Omega^\infty S^\infty$ .

Ah, this seems to be the answer to the question Bruce, Eugenia, Simon and myself were trying to understand here.

How can I understand that the free stable $\omega$ -groupoid on one object “is” the sphere spectrum?

Possibly you more or less already told me once, but don’t really understand it yet.

Posted by: Urs Schreiber on February 17, 2008 10:28 PM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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There should be some way to explicitly define the Kan complex that corresponds to the “the free stable $\infty$ -groupoid on one object”. That definition should be pretty simple/tautological, I suppose. Does it fit into a blog comment?

Posted by: Urs Schreiber on February 17, 2008 10:39 PM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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Urs wrote:

There should be some way to explicitly define the Kan complex that corresponds to the “the free stable $\infty$ -groupoid on one object”. That definition should be pretty simple/tautological, I suppose.

What’s simple and tautological to see using Kan complexes is that ‘the fundamental $\infty$ -groupoid of $S^k$ is the free $\infty$ -groupoid on a $k$ -automorphism’. The rest is just abstract nonsense, as explained in the comment below.

Posted by: John Baez on February 18, 2008 4:20 AM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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Urs wrote:

How can I understand that the free stable $\omega$ -groupoid on one object “is” the sphere spectrum?

Of course this hasn’t been proved yet — it hasn’t even been made fully precise. But, I imagine you’re more interested in trying to understand why it ‘should’ be true.

It’s probably the coolest idea I’ve ever run across… mainly due to James Dolan.

In essence, it’s very simple: draw a sphere, but think of it as an $n$ -category. You get the free $n$ -category on what you’ve drawn.

I gave a popularized explanation in week102.

I gave a more detailed explanation this on page 28-29 of Higher-Dimensional Algebra and Topological Quantum Field Theory.

I gave an even more detailed explanation around page 25 of Categorification: start with the chart, then go forwards and backwards until you understand it.

But, here’s a sketch of how to understand this stuff.

It really helps to start by understanding why the fundamental $(n+k)$ -groupoid of $S^k$ should be the free $(n+k)$ -groupoid on a $k$ -automorphism.

A 1-automorphism is just an automorphism of some object. If you draw an object and an automorphism of this object, you’ll see it looks like a circle.

A $k$ -automorphism is a $k$ -morphism that’s an automorphism of the identity of the identity … of some object. If you draw this, you’ll see a $k$ -automorphism looks like a $k$ -sphere!

That’s the basic idea. There’s also a lot of other evidence, mainly coming from the Thom–Pontryagin construction and the role of the sphere spectrum in topology… and also coming from explicit calculations in low dimensions. The case $k = 2$ , $n = 1$ is incredibly illuminating, and so is $k = 3$ , $n = 1$ .

Anyway, suppose it’s true: ‘the fundamental $(n+k)$ -groupoid of $S^k$ is the free $(n+k)$ -groupoid on a $k$ -automorphism’.

Then, take the $k$ -fold loop space. This is just reindexes our $k$ -morphism so it becomes an object. So, we get: ‘the fundamental $n$ -groupoid of $\Omega^k S^k$ is the free $k$ -tuply groupal $n$ -groupoid on one object’.

Then, take the limit $k \to \infty$ and get ‘the fundamental $n$ -groupoid of $\Omega^\infty S^\infty$ is the free stable $n$ -groupoid on one object’.

In case you’re worried, $\Omega^\infty S^\infty$ is something topologists think about all the time. It’s a perfectly nice space — in fact, an infinite loop space! An infinite loop space is essentially the same thing as a spectrum with vanishing negative homotopy groups… and $\Omega^\infty S^\infty$ is essentially the same thing as the sphere spectrum!

Then, take the limit $n \to \infty$ and get ‘the fundamental $\infty$ -groupoid of $\Omega^\infty S^\infty$ is the free stable $\infty$ -groupoid on one object’.

By homotopy hypothesis, we don’t need to worry much about the difference between a space and its fundamental $\infty$ -groupoid, at least if we’re just doing homotopy theory. So, we can just say ‘ $\Omega^\infty S^\infty$ is the free stable $\infty$ -groupoid on one object’.

Or: ‘the sphere spectrum is the free stable $\infty$ -groupoid on one object’.

If I had a few hours to explain this with the help of a blackboard, you’d walk out feeling convinced and also very happy. It’s incredibly beautiful.

Posted by: John Baez on February 18, 2008 4:12 AM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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If I had a few hours to explain this with the help of a blackboard, you’d walk out feeling convinced and also very happy. It’s incredibly beautiful.

At least I already feel incredibly interested in this! I want to understand this truly, eventually.

What you say makes very good intuitive sense to me. I’ll try to find the time to have another close look at the references you provided to see if I can also get a better technical understanding of some aspects.

Posted by: Urs Schreiber on February 18, 2008 11:38 PM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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Here’s a really good exercise to see the power of these ideas: use them to compute the homotopy group $\pi_3(S^2)$ and show

$\pi_3(S^2) \cong \mathbb{Z}$

You can do it! To do it, just take $n = 1$ , $k = 2$ in what I wrote above.

So: accept the fact that the fundamental 3-groupoid of $S^2$ is the free 3-groupoid on a 2-automorphism.

In other words: accept the fact that the fundamental groupoid of $\Omega^2(S^2)$ is the free braided 2-group on one object $x$ .

(As you know, a 2-group is a monoidal category where every morphism has an inverse for composition and every object has an inverse for tensor product. It make sense for such a thing to be braided.)

Then, show that $Aut(x) \cong \mathbb{Z}$ .

This means that

$\pi_1(\Omega^2 S^2) \cong \mathbb{Z}$

and thus, switching back to the old viewpoint,

$\pi_3(S^2) \cong \mathbb{Z} !!!$

It’s a lot of fun to realize you can compute some homotopy groups of spheres this way.

The other really fun example is

$\pi_4(S^3) \cong \mathbb{Z}/2$

This involves the free symmetric 2-group on one object.

Posted by: John Baez on February 19, 2008 12:23 AM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

Can I comment very quickly with regard to the structures I am working with here. A weak complicial set consists of an underlying simplicial set equipped with a specified set of simplices, called a stratification. The simplices in this stratification are said to be thin, and they might be thought of as the set of “equivalences” up to which we would like to classify the elements of the complicial structure under consideration.

With the aid of a modified horn filler condition, these thin simplices also determine composition operations on our complicial set which makes it behave like a weak ∞-Category.

Now, in the case of the simplicial set Cob_k, as a simplicial set it is actually a Kan complex, so it possesses an “all simplices are thin” stratification giving it a weak $infin;-groupoid structure. This, however, is not quite what we want here since this then classifies the simplices of Cob_k upto cobordism (or at least “ambient cobordism).

Instead, we would like only to classify things only up to ambient isotopy. To do this we choose a smaller stratification on the same underlying simplicial set under which we restrict ourselves to regarding a simplex as thin iff it is some suitable kind of “track of an ambient isotopy”.

Now in this new complicial structure it is not the case that all cells (or simplices in our case) behave like equivalences. Furthermore the compositional structures here are now defined up to ambient isotopy rather than upto ambient cobordism.

We might now ask, if we regard Cob_k as a weak complicial set under this new ambient isotopy stratification, is it the case that every simplex that is “morally” an equivalence in there (by dint of having an equivalence inverse) is also thin. The answer to this is most probably no - I suspect that you can construct families of counter examples related to the well known examples of non-trivial 4-dimensional s-cobordisms.

So the next question is, what happens when I make all of these “morally” thin simplices thin? Is it possible that we end up with the stratification we started with - under which all simplices are thin?

The answer to the latter question is - no we don’t. It is possible to directly construct a complicial stratification on the same simplicial set which is strictly smaller than the “all simplices are thin” one but which contains the “ambient isotopy” one and which is replete under equivalences (that is closed under “moral” equivalences).

Indeed, with more work, it is highly likely that we can precisely characterise the thin simplices of the equivalence repletion of the ambient isotopy stratification in terms of some relatively straightforward simple homotopy (s-cobordism like) conditions. This result however, relies upon a slight generalisation of the classical s-cobordism theorem that I have yet to completely verify.

Notice also that we do not assume any kind of “duals” structure on Cob_k. Indeed one of the primary points of our choice of stratification is to ensure that the the cobordisms which mediate this kind of duals structure do not get “mixed into” our definitions of thinness and composition.

Posted by: Dominic Verity on February 20, 2008 10:04 AM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

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Isar Stubbe is at Macquarie University now, and he sent an email saying that Verity was sick in bed and did not deliver his lecture as planned! Instead, Richard Wood spoke about Frobenius objects in cartesian bicategories.

Posted by: John Baez on February 17, 2008 9:48 PM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

May I say that I am now fully recovered and gave my first talk on this work at the Sydney Category Seminar today.

I am told by my PhD student, Micah McCurdy, that when he spoke about coming to study with me last year at least one person expressed the opinion that he thought I was already dead. The good news is that my sickness of last week has in no way hastened me to that end.

Posted by: Dominic Verity on February 20, 2008 10:10 AM | Permalink | Reply to this

Re: Verity on ∞-Categories From Topology

Thanks for the summary of your talk — who could possibly do it better than you?

I’m glad you’re fully recovered, and I’m glad the news of your demise was greatly exaggerated.

Posted by: John Baez on February 21, 2008 6:52 PM | Permalink | Reply to this

Read the post What I learned from Urs
Weblog: The n-Category Café
Excerpt: Bruce Bartlett talks about some aspects of the program of systematically understanding the quantization of Sigma-models in terms of sending parallel transport n-functors to the cobordism representations which encode the quantum field theory of the n-pa...
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February 12, 2008