The n-Category Café

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)

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?

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.

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:

• The homotopy hypothesis
• The tangle hypothesis
• The stabilization hypothesis
• The TQFT hypothesis

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:

• Goerss and Jardine, Simplicial Homotopy Theory.
• Mark Hovey, Model Categories.

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:

• Dominic Verity, Weak complicial sets, a simplicial weak omega-category theory. Part I: basic homotopy theory.
• Dominic Verity, Weak complicial sets, a simplicial weak omega-category theory. Part II: nerves of complicial Gray-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:

• André Joyal, Graduate course on quasicategories.

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

• Eugenia Cheng, Graduate course on $n$-categories with duals and TQFT.
• Eugenia Cheng, An $\omega$-category with all duals is an $\omega$-groupoid.

(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:

• Denis-Charles Cisinski, Batanin higher groupoids and homotopy types.
• Eugenia Cheng, Batanin $\omega$-groupoids and the homotopy hypothesis.

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.