### 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:

- Grothendieck’s conjecture (the homotopy hypothesis)
- The tangle hypothesis
- The definition of extended topological quantum field theory
- The stabilisation conjecture

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.

## Re: Questions on n-Categories and Topology

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

aremodelled 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)