The n-Category Café

A long time ago, Jim Dolan and I thought we’d figured out a nice answer, using iterated enrichment based on a recursively defined generalization of the Gray tensor product.

Right, we talked about that before here. I should have mentioned it. My question at the beginning here was meant more rethorically as an introduction to the next sentence, which advertised Carlos Simpson’s conjecture.

I should maybe elaborate: I am having a conversation with Ronnie Brown, whose various work on strict $\infty$-groupoids I am currently using a lot. I asked him something like: given that strict $\infty$-groupoids are insufficient to model all $n$-types (as recalled for instance at the very beginning of Cisinki’s Batanin higher groupoids and homotopy types), how insufficient precisely are they? Can we capture the essence of what it is they are missing?

In reply he kindly pointed me to Paoli’s article. I must have heard of Paoli’s work at the Fields Institue workshop but the punchline hadn’t stuck. Now I followed the referencesshe gives to Simpson’s conjecture and Joachim Kock’s work on it and was quite struck:

passing from strict $\infty$-categories to “$\infty-$snucategories” by just relaxing units seems to be so much more tracktable than anything else. It gives me the feeling that I am quite save with developing my application all in the context of strict $\infty$-categories, since it ought to be relatively straightforward to generalize everything to weak units one fine day.

So I fell in love with the idea and posted this entry to collect the citations and to fish for comments about it.

The main point I am wondering about: Kock and Paoli pass to Tamsamani-style $n$-categories to realize Simpson’s conjecture. But Simpson himself in his article sketches a much more “globular”-style of approach. On first sight this seemed very plausible and easy to make precise.

I am wondering if one can’t simply use this definition:

Let an $\omega$-semi-category be like an $\omega$-category but without any units. Then let an $\omega$-sesqui-category (or whatever) be an $\omega$-semi-category equipped with the data of one specified endo $(n+1)$-morphism $i_f$ on each $n$-morphism $f$ (going by induction on $n$) which satisfies $i_f \circ g \sim g$ for all compositions with all $g$, where $\sim$ is the usual notion of $\omega$-equivalence but with all occurences of identities replaced by these $i_f$.

I did try but never published a related method to get an infinity groupoid. The idea was only in the continuous case, not the smooth one. You start with the singular complex Sing(X) of the space, and then in addition form the simplicial monoid of filtered self maps of the spaces, Delta(n), or perhaps more clearly of the cosimplicial space, Delta. This acts on Sing(X) by precomposition, and its effect is to reparametrise the simplices. Then you can divide out by the action in the usual way by forming the equivalence relation generated by the action. The result is a simplicial set which is much near being a T-complex in Ronnie Brown’s sense than is the original Sing(X). I cannot remember the detailed filling structure as this was some 20 years ago (yes, these ideas were being considered in Bangor that long ago!) and Dominic Verities’ complicial sets were not yet available except in non-detailed versions. My feeling was that the result probably was a weak complicial set. Perhaps by adjusting the monoid of self maps being used one could get a nice theory that gave a semi-strict analogue. I never considered that.

It should be remembered that Hardie, Kamps and Keiboom,

(K.A. Hardie, K.H. Kamps and R.W. Kieboom, A homotopy 2-groupoid of a Hausdorff space, Appl. Categ. Struct. 8 (2000), 209-234;

and

K.A. Hardie, K.H. Kamps and R.W. Kieboom, A homotopy bigroupoid of a topological space, Appl. Categ. Struct. 9 (2001), 311-327,)

examined the 2-dimensional topological case in exhaustive detail. Their work suggests that a detailed treatment in the smooth case would be quite hard to do, or is the amount of detail that they give not strictly needed, although `good to have’? I do not know.

The other useful source that might help recalling is Martin Raussen’s work on reparametrisation already mentioned by John (possibly after the ATMCS conference in Paris in the summer).