May 21, 2007

Questions on n-Categories and Topology

Posted by John Baez

Here are some questions on n-categories and topology from Bruce Westbury. I’ll post a reply later — but why don’t some of you take a crack at them first?

– guest post by Bruce Westbury –

Now that we have several definitions of n-categories it seems to me that the next stage is to try and prove some results. The big projects that JB wrote about are:

If we are going to do anything with n-categories then a basic construction is that $n$-categories should be the objects of an ($n$+1)-category. This has been done by Simpson for the Tamsamani definition, and by Makkai for a variant of the Baez-Dolan definition (in the case $n = \infty$). Is this the current state-of-play?

Another basic idea is that we want to have a notion of equivalence. Do the experts know how to do this? and for which definitions of $n$-category?

Moving on and looking at the big projects; if we have positive answers to the above then the only big project that now has a precise statement is the Stabilisation Conjecture. The only other ingredient is suspension.

The other three big projects all have “$n$-categories with duals” in the statement. So it seems to me that $n$-categories with duals are more fundamental than $n$-categories. However as far as I am aware there has been no work done on the definition.

Each of the big projects (other than the Stabilisation Conjecture) also requires the construction of certain $n$-categories with duals. So my next question is whether there are constructions for any of the following $n$-categories:

• the fundamental $n$-category of a topological space
• the $n$-categories of tangles
• the $n$-categories of cobordisms
• the $n$-categories of $n$-Hilbert spaces

All of these should be $n$-categories with duals so it would help in thinking about what the definition might be if there were positive answers to any of the above. Of course it would be even better if a positive answer to any of the above also had a built in construction of duals.

Another line of thought is that $n$-groupoids are special $n$-categories with duals so if we had a definition of $n$-categories with duals then it should be clear what an $n$-groupoid is. We would just say that certain structure maps are equivalences. My next question is whether there is already a definition of an $n$-groupoid? and for which definitions of $n$-category?

Moving on again to the definition of an $n$-category with duals and about what is known in low dimensional cases: there is no difficulty with $n=1$. The case $n=2$ I believe I can do by combining the definition of a bicategory and the definition of a spherical category. However I have not written anything down. If anybody has any doubts about this then of course I have some incentive to try and write down a definition. For $n=3$ the only information I have is that JB in his paper with Langford on 2-tangles gave, in effect, a definition of a strict 3-category with duals. I say “in effect” because his semistrict 3-category had one object.

The other case that we should start with is the definition of a strict n-category with strict duals. Again I believe I know what this is but I have not written anything down and again if anybody has any doubts about this then of course I have some incentive to try and write down a definition.

This, as far as I know, reflects the current state of play. If you know better then I would be interested to hear about it.

Finally it seems to me there are two possible approaches to the definition of an $n$-category with duals. When I started thinking about this I was thinking of rewriting the definition of an $n$-category building in duals from the ground up. However thinking about JB’s definition of a strict 3-category with duals has led me down a different route. The idea of this approach is that there is an obvious forgetful “functor” from $n$-categories with duals to $n$-categories. Then the aim would be to construct a “left adjoint”. This would then give a “monad” on $n$-categories whose “algebras” would be n-categories with duals. The idea for constructing the “left adjoint” is to use tangles. If this approach succeeded then the Tangle Hypothesis would be true by construction.

Posted at May 21, 2007 4:37 AM UTC

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Re: Questions on n-Categories and Topology

So my next question is whether there are constructions for any of the following n-categories:

the fundamental n-category of a topological space

this is the only one I can really comment on, so here goes.

That fundamental $n$-category is going to be an $n$-groupoid (weak, of course), as I imagine you know. The best result I know about is that homotopy types are modelled by groupoids enriched in simplicial sets. There is not to my knowledge a construction $\Pi:Top \to sSet-Gpd.$

The question is, how do we relate $n$-groupoids as defined by the various methods and groupoids enriched in $n$-coskeletal simplicial spaces? (fingers crossed on the indexing there)

Posted by: David Roberts on May 21, 2007 7:04 AM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

How fundamental n-categories are defined may depend on whose definition of n-category you’re using, of course. In the style of definition of n-category attributed to me (as described for example in the guidebook by Eugenia Cheng and Aaron Lauda), the fundamental n-category functor is defined at more or less the same time as the notion of (n+1)-category, as part of an inductive process. I’m less familiar with how other people use their definitions toward this problem.

I think Eugenia and Nick Gurski were trying to modify this type of definition to define n-categories of cobordisms – see here for one version of their working draft (I think there are others).

Posted by: Todd Trimble on May 21, 2007 1:38 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

But isn’t the “fundamental $n$-category” actually an $n$-groupoid, and aren’t for $n$-groupoids things much clearer. Or even: clear?

Posted by: urs on May 21, 2007 3:52 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

Yes, it’s an n-groupoid. Sorry, I’m not following your question – are what things clearer or clear?

Posted by: Todd Trimble on May 21, 2007 4:39 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

Sorry, probably I didn’t get the point. What I wanted to say, though, is:

Isn’t it clear that and how the fundamental $n$-groupoid of a space is an $\omega$-groupoid?

Posted by: urs on May 21, 2007 5:20 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

I’m not sure it’s clear to everyone (including myself) – I do note that it was one of the questions on Bruce Westbury’s list, unless I’ve misinterpreted his meaning.

Let me see if I understand what you’re saying (and maybe put it stupidly to elicit a response) – are you saying that for each of the dozen or so definitions of n-category, it’s clear how to go about defining the corresponding notion of fundamental n-groupoid? If so, has someone written down details? If no one has (but it’s clear anyway), can someone be troubled to write them down?

A follow-up question: comparing the definitions of n-category is hard work. Has there been much work on comparing the corresponding notions of n-groupoid? (Is the ‘the’ in ‘the fundamental n-groupoid’ rock-solid?)

Posted by: Todd Trimble on May 21, 2007 6:04 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

I’m not sure it’s clear to everyone

Okay, in case of doubt here, you should assume that I am misremembering something.

I went back to John’s slides on the homotopy hypotheses.

I seemed to recall that this included the statement that we can think of the fundamental $n$-groupoid as a Kan complex and that this is also called an $\omega$-groupoid.

But possibly I am mixing things up.

Posted by: urs on May 21, 2007 6:20 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

No, it’s all straightened out now. I’m sure you were using the word in a standard sense, but obviously I had something else in mind. Thanks for clarifying!

Posted by: Todd Trimble on May 21, 2007 8:55 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

Maybe Urs is saying something sort of like this:

If we only wish to study $n$-groupoids, as opposed to more general $n$-categories, we should really use the simplicial approach, because a vast amount has been worked out using this approach, and sophisticated tools are available. So, we can define $\infty$-groupoids to be Kan complexes, and define $n$-groupoids to be Kan complexes with a special property.

If you’re wondering what a Kan complex is, and what this ‘special property’ is, try pages 16–18 here. For a more thorough introduction to Kan complexes, start with May’s old book Simplicial Methods in Algebraic Topology, then tackle Goerss and Jardine’s Simplicial Homotopy Theory… and by the time you’re done, maybe Joyal and Tierney’s book on the subject will be out!

Similarly, if we only want to study $(n,1)$-categories, the simplicial approach is also very well-developed. An $(n,1)$-category is an $n$-category where all $j$-morphisms have weak inverses for $j$ > $1$. Joyal calls simplicial $(\infty,1)$-categories ‘quasicategories’. A quasicategory with the same special property I just alluded to is an $(n,1)$-category.

Joyal’s book on quasicategories should eventually appear as part of the proceedings of the IMA workshop on $n$-Categories: Foundations and Applications. This book will be a bit like Quasicategories for the Working $\infty$-Mathematician. For now, you can listen to Joyal’s lectures on quasicategories, or read about them in Lurie’s paper on $\infty$-topoi.

The fully general simplicial $\infty$-categories are being studied by Street, Verity and others — see this and this. So far, I find this theory much scarier than Kan complexes or quasicategories. Eventually it should be very nice.

In particular, just as Kan complexes are special quasicategories, quasicategories are special simplicial $\infty$-categories! So, we have three nested theories, with more results about the simpler and historically earlier ones — but all three are part of a compatible package! And, with this package it should be very easy to prove the Homotopy Hypothesis: in a sense it’s already done.

Posted by: John Baez on May 21, 2007 7:18 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

I was very struck by Lurie’s comments that rather than considering an inductive definition of $n$-categories (which ends up being fundamentally circular), one can consider an inductive definition of $(\infty,n)$ categories given that we can define an $\infty$-category to be a simplicial set. He attributes this idea to Tamsamani. The question, then, is there any particular reason we’d ever need an $n$-category as opposed to an $(\infty,n)$-category?

Posted by: Aaron Bergman on May 21, 2007 7:41 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

Aaron wrote:

The question, then, is there any particular reason we’d ever need an $n$-category as opposed to an $(\infty,n)$-category?

Since an $n$-category is a special sort of $(\infty,n)$-category, the answer to this question is clearly:

Posted by: John Baez on May 21, 2007 9:34 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

Perhaps you’ll like this formulation better, then. Do you think it is likely that $(\infty,n)$ categories are a better behaved notion than n-categories?

Posted by: Aaron Bergman on May 21, 2007 9:52 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

I’ve never quite been able to understand that comment in Lurie’s paper. That is to say, I know what it means to enrich a category over say vector spaces. I even think I have some idea what it means to enrich an $(\infty, 1)$-category over something like vector spaces. However, I have no idea what it means to enrich an $(\infty, 1)$-category over another $(\infty, 1)$-category.

Posted by: Noah Snyder on May 22, 2007 6:51 AM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

Maybe Urs is saying something sort of like this:

If we only wish to study $n$-groupoids, as opposed to more general n-categories, we should really use the simplicial approach

I think the trouble was that what I was really thinking was that if we wish to study $n$-groupoids only, then there is the simplicial approach and just that.

Which is wrong, it seems. But might be right “for practical purposes”. :-)

Posted by: urs on May 21, 2007 7:56 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

David Roberts wrote:

The best result I know about is that homotopy types are modelled by groupoids enriched in simplicial sets. There is not to my knowledge a construction $\Pi: Top \to sSet-Gpd$

How could we know that homotopy types are modelled by groupoids enriched in simplicial sets without knowing a Quillen equivalence between spaces and groupoids enriched in simplicial sets?

There should be some such Quillen equivalence…

First, we have a functor

$Sing: Top \to s Set$

sending each space $X$ to its ‘singular simplicial set’ $Sing(X)$. Going back we have ‘geometric realization’

$|\cdot| : sSet \to Top$

and these two functors form a Quillen equivalence.

Next, I hope any simplicial set has a kind of ‘path groupoid’ which is a groupoid enriched over simplicial sets. I think I’ve seen people do this, but I don’t know quite how it goes. I have some guesses, but I won’t bore you with them. I hope this gives a functor

$P : s Set \to s Set-Gpd$

which for some model structure on $s Set-Gpd$ extends to a Quillen equivalence.

So, composing these I’d hope to get a functor

$\Pi: Top \to s Set-Gpd$

which is part of a Quillen equivalence.

Do any experts lurking out there know how to fill in the holes in this idea?

$\Pi: Top \to Top-Gpd$

and somehow apply the Quillen equivalence between $Top$ and $sSet$ to get

$\Pi: s Set \to s Set-Gpd$

But, I don’t know how to do this, either.

Posted by: John Baez on May 23, 2007 4:20 AM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

We had a long discussion about the fundamental $n$-category (with duals) of a stratified space in the comments of this post.

Should we expect an adjunction between $n$-categories with duals and $n$-groupoids?

If so, would this have something to do with a left adjoint of the forgetful functor from the category of spaces to the category of stratified spaces?

Posted by: David Corfield on May 21, 2007 9:06 AM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

left adjoint of the forgetful functor from the category of spaces to the category of stratified spaces

Whoops, I meant of course

left adjoint of the forgetful functor from the category of stratified spaces to the category of spaces

I suppose a ‘trivial stratification’ functor might do.

Posted by: David Corfield on May 21, 2007 6:48 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

Moving on again to the definition of an n-category with duals…The case n=2 I believe I can do by combining the definition of a bicategory and the definition of a spherical category. However I have not written anything down. If anybody has any doubts about this then of course I have some incentive to try and write down a definition.

I have thought a bit about 2-categories with duals, ever since I tried to prove that eg. 2Hilb is a 2-category with duals, and it invited me to adopt a slightly different viewpoint than the one presented by John and Laurel in their 2-tangles paper.

If one follows the lines you suggest, that is, if we take the definition of a spherical category and “2-categorify” it (2-category=bicategory), you’ll end up saying that a 2-category with duals is a 2-category $C$ equipped with a weak 2-functor $* : C^op \rightarrow C$, with a monoidal structure, and some other things satisfying a whole bunch of identities.

I don’t like that way of thinking, because in the examples where our objects are categories of some kind, and the morphisms are functors - like 2Hilb - the $*$-operation on morphisms is going to be “take the adjoint”.

And that’s where I differ perhaps with many here at the n-cafe, and hold a philosophical objection : although in the case of 2Hilb an adjoint (a) does exist and (b) is unique even up to unique natural isomorphism, there is no canonical definition for it.

And so I am very hesitant indeed to think of the $*$-operation as (even a weak) 2-functor, because I would like to think that a 2-functor must at least have a meaningful defintion and not be defined by making arbitrary choices. Even if you insisted on making arbitrary choices for the adjoints, you’d run into a set-theoretic difficulty : you couldn’t use the axiom of choice for example, because the collection of all 2-Hilbert spaces doesn’t even form a set!

Here’s the way I would prefer to think of a 2-category with duals (I mentioned this here). I would prefer to think of the $*$-operation on the 2-morphisms as a structure , the $*$-operation on the 1-morphisms as a property , and I’m unsure about the $*$-operation on objects :-)

In other words : I would prefer to work locally . You only say that for every morphism $\sigma : A \rightarrow B$, there exists a morphism $\sigma^* : B \rightarrow A$ satisfying etc. etc. This doesn’t take away the whole point of 2-categories with duals either : they would still be just as useful for 2-tangles, because if you think about it, one only ever needs to work locally . There is no need to use brute force and choose adjoints for every possible morphism, right before we even begin, if in our applications we’re only going to work with some small region of our 2-category.

Posted by: Bruce Bartlett on May 21, 2007 12:07 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

Bruce suggested a

a slightly different viewpoint

on $n$-categories with duals: don’t specify duals, but just assert their existence.

I haven’t thought enough about this issue to make a concrete comment, but I notice that in the world of $n$-categories we run again and again into this very dichotomy of two possible viewpoints:

either we assert that things exist (for instance composites of morphisms), or we choose particular representations (a particular composite) and then are left with these choices and a bunch of coherence laws satisfied by them.

Posted by: urs on May 21, 2007 1:27 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

Yes, I agree this dichotomy comes up a lot when one thinks about n-categories. It’s basically the contrast between a completely algebraic approach and approaches which have some form of non-algebraic flavour to them. For what it’s worth (and admitting that I don’t know half as much as I pretend to about n-categories), I would tend to vote for an approach which is as algebraic as possible… up to the tipping point where this principle becomes untenable, and one is forced to compromise :-)

I think the example of adjoints is a good testing-ground to contrast these various approaches. How are we to think of them? One advantage of thinking of them as a a property, and not as a structure, is that it allows one to choose the appropriate adjoint for any given problem at hand, rather than being forced to use the one someone arbitrarily chose for you right in the beginning.

Posted by: Bruce Bartlett on May 21, 2007 1:52 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

Bruce Bartlett writes:

For what it’s worth (and admitting that I don’t know half as much as I pretend to about $n$-categories), I would tend to vote for an approach which is as algebraic as possible… up to the tipping point where this principle becomes untenable, and one is forced to compromise :-)

The algebraic approach is very tempting for low-dimensional calculations. It tends to get tiring as you move to higher dimensions, since it tends to make you explicitly keep track of coherence laws. Eventually this becomes too exhausting. Perhaps if all the coherence laws were hidden inside some sophisticated black box one could feel happy knowing they’re there, but not needing to peek inside very often.

For what it’s worth, all the simplicial approaches I mentioned are completely non-algebraic — unless you go ahead and choose operations that supply fillers for all the relevant horns.

The homotopy theorists, who have proved more and harder theorems than any of us $n$-category theorists, have demonstrated the practicality of simplicial, non-algebraic approaches.

But, I think people should develop all sorts of different philosophies and try all sorts of different approaches! Only time and experimentation will show which are the most fruitful. It’s way too early to start weeding out what might be promising candidates.

Posted by: John Baez on May 21, 2007 7:48 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

For what it’s worth, all the simplicial approaches I mentioned are completely non-algebraic — unless you go ahead and choose operations that supply fillers for all the relevant horns.

I’ve been under the impression that this was sort of morally wrong. There isn’t a unique thing that fills in the various operations, and to choose one is to impose too much structure.

Posted by: Aaron Bergman on May 21, 2007 9:57 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

Aaron wrote:

I’ve been under the impression that this was sort of morally wrong. There isn’t a unique thing that fills in the various operations, and to choose one is to impose too much structure.

I agree completely! The price you pay for equipping your gadgets with all this structure is demanding that your morphisms between gadgets preserve all this structure only up to something — where this something needs to satisfy a bunch of complicated laws of its own. And then you have to do the same for 2-morphisms between morphisms of gadgets, and so on.

So, Bruce Bartlett should explain why he likes the algebraic approach. I know one thing: it comfortably resembles ‘traditional algebra’, where we have operations satisfying equational laws. I don’t know if he has a stronger reason.

Posted by: John Baez on May 21, 2007 11:16 PM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

One reason for an algebraic approach might be to shed light on algebraic models for homotopy n-types, among other things.

For example, even if Joyal and Tierney hadn’t given their characterization of homotopy 3-types in terms of Gray groupoids, it might have been possible to discover this anyway by applying coherence of algebraic 3-categories (their 3-equivalence to Gray categories) to the algebraic fundamental 3-groupoid. It’s possible that further development of coherence theory for algebraic n-categories would yield associated insights into homotopy n-types, and vice-versa.

As we at the n-category cafe know, there has always been a dialectic between homotopy theory and higher-dimensional algebra, beginning with Stasheff’s work on homotopy types of loop spaces, continuing more recently with iterated loop spaces and iterated monoidal categories, and still more recently with Batanin’s investigations into higher Eckmann-Hilton laws.

Here’s to a future where the inner logic of coherence theory and homotopy theory play off one another, with rich payoff for both!

Posted by: Todd Trimble on May 22, 2007 12:25 AM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

So, Bruce Bartlett should explain why he likes the algebraic approach.

Perhaps I prefer a more algebraic approach… but the point of my post was really to argue the opposite! Namely, in the context of defining a “2-category with duals”, a hard-line algebraic approach seems the wrong one to me. I explained the reason already : good examples of 2-categories with duals are supposed to be things like 2Hilb where where the objects are categories of some sort, and the morphisms are functors of some sort. And the duality, on the level of morphisms, sends a functor to its adjoint.

But oftentimes, the adjoints aren’t naturally defined (they only ‘exist’ up to unique natural isomorphism), so to me, it just wouldn’t be cricket to think of the duality operation as a 2-functor

(1)$* : C \rightarrow C$

where $C$ is our candidate ‘2-category with duals’. It’s the philosophical objection that ‘one shouldn’t have to make arbitrary choices when defining something’.

I was saying this stuff because it seemed precisely the opposite approach compared to

(a) Bruce Westbury’s viewpoint above, where he would define a 2-category with duals by “combining the definition of a bicategory and the definition of a spherical category”, and

(b) John’s approach in the 2-tangles paper, where the duality on the 2-category is explicitly regarded as a structure and not as a property . Recall Definition 10,

A monoidal 2-category with duals is a monoidal 2-category equipped with the following structures : …

2. For every morphism $f : A \rightarrow B$ there is a morphism $f^* : B \rightarrow A$ called the dual of $f$, and 2-morphisms $i_f : 1_A \Rightarrow f f^*$ and $e_f : f^* f \Rightarrow 1_B$ called the unit and counit respectively

The dramatic boldface is added by me ;-) The point is that one is specifying, right at the very beginning, the adjoint of a morphism, etc. etc. It is an algebraic approach. One could get away with it there because one was only dealing with a very strict situation. But I’m expressing my doubts whether this is the right notion of a general “monoidal 2-category with duals”.

Practically all I’ve learnt about higher categories has come from John’s writings… and indeed I had the impression that in fact it was John who harboured his own secret algebraic inclinations, from which I subsequently followed his cue :-) After all, in the last section of John’s excellent Fields talk (which somehow I managed to miss at the time , though I watched and listened to the slides twice, once with a friend!), John says:

— So, why not just use simplicial methods, and forget about ‘globular’ n-categories?

Better answer : globular methods clarify the structure of $\infty$-categories, and thus $\infty$-groupoids, and thus homotopy types - given the homotopy hypothesis. —

Admittedly, that’s just a pro-globular paragraph and not necessarily a pro-algebraic paragraph. (By the way, I’m just poking fun with these “pro-globular” and “pro-algebraic” catchwords. I know that John is not “pro-anything”!)

In the remaining slides, John talks about how, in the globular approach, one obtains all sorts of algebraic operations, like ‘composition, ‘whiskering’, ‘braiding’, etc., which might shed light on the combinatorics of homotopy types. Perhaps I missed the point, but I took this to mean that John was saying that it can be worthwhile to think of things in a more algebraic way… seeing as we already understand them quite well in a non-algebraic (for instance, simplicial) way.

Posted by: Bruce Bartlett on May 22, 2007 12:39 AM | Permalink | Reply to this

Re: Questions on n-Categories and Topology

I’ve been putting off replying to Bruce’s questions, since it’s an intimidating task: summarizing the work of dozens of mathematicians on some highly technical projects that haven’t reached any sort of completion. It’s like describing an enormous fractal coastline.

But let me give it a try. Instead of trying to review all existing approaches to these hypotheses:

I’ll just sketch some promising work so far, and mention two projects one might try.

To formalize and prove any one of these hypotheses, we need an approach to $n$-categories and $\infty$-categories — or at least $n$-groupoids and $\infty$-groupoids, or $n$-categories and $\infty$-categories with duals.

Here I’ll focus on the simplicial and the globular approaches. I won’t attempt to cover the promising multisimplicial approach of Tamsamani and Simpson, or the opetopic approach.

In the simplicial approach, a version of the homotopy hypothesis has already been proved, since simplicial $\infty$-groupoids are just Kan complexes, and the category of Kan complexes is known to be Quillen equivalent to the category of topological spaces. For details, try:

The technology of model categories always comes as a rude shock to people who enter this subject by becoming interested in $n$-categories — even the very definition of a model category takes a while to digest — but it’s important. One should think of a model category as a very nice way of presenting a simplicially enriched category — which you can think of as an $\infty$-category with all its $j$-morphisms for $j > 1$ being invertible. For details, try:

• William G. Dwyer and Daniel M. Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980), 267-284.
• William G. Dwyer and Daniel M. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18 (1980), 17-35.
• William G. Dwyer and Daniel M. Kan, Function complexes in homotopical algebra, Topology 19 (1980), 427-440.

Starting from the homotopy hypothesis, the best approach to the tangle hypothesis seems to be generalizing ‘the fundamental $n$-groupoid of a space’ to ‘the fundamental $n$-groupoid with duals of a stratified space’. The reason is that there’s a deep relation between tangles and certain stratified spaces. For details, try our earlier discussion here.

Given all this, it might be a nice project to invent a simplicial concept of $n$-category with duals!

There’s already a simplicial approach to $\infty$-categories:

It’s been much more developed in the case of $\infty$-categories where all $j$-morphisms are invertible for $j > 1$; these are called ‘quasicategories’ by Joyal:

However, $n$-categories with duals seem quite undeveloped in the simplicial approach. Here the globular approach seems to shine:

(The latter paper formalizes an important issue that has vexed Jim Dolan and I for a long time: it seems that only with a dimensional cutoff is the concept of ‘$n$-category with duals’ different from that of an $n$-groupoid!)

Batanin’s approach to globular $n$-categories has a lot of momentum going for it. Unfortunately, the theory of globular $n$-groupoids is less developed than that of simplicial $n$-groupoids. Some good work has been done, but more needs to be done. For the state of the art, try these:

This suggests another nice project: finish proving the homotopy hypothesis for globular $\infty$-groupoids, and then define a fundamental $n$-category with duals for a stratified space!

There’s a lot more to say, but I hope this helps a bit.

Posted by: John Baez on May 28, 2007 5:25 PM | Permalink | Reply to this

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