October 8, 2008

Semistrict Infinity-Categories and ω-Semi-Categories

Posted by Urs Schreiber

What is a semistrict $\infty$-category? The strictest possible version of weak $\infty$-category which is still general enough to satisfy the homotopy hypothesis?

Carlos Simpson in Homotopy types of strict 3-groupoids conjectured (on p. 27) that one remarkable answer is: $\infty$-categories with strict composition but possibly weak identities – “$n$-snucategories” (strict, non-unital).

Joachim Kock in Weak identity arrows in higher categories remarked that

The conjecture in its strong form has startling consequences, defying all trends in higher category theory: every weak higher category should be equivalent to one with strict composition!

and set out to formulate the conjecture in terms of the notion he calls a fair $n$-category inspired by Tamsamani’s $n$-categories. With Joyal he then proved the conjecture up to $n=3$ (with some restrictions) in Weak units and homotopy 3-types.

A little later Simona Paoli in Semistrict Tamsamani $n$-groupoids and connected $n$-types proposed notions of semistrict Tamsamani $n$-categories, showed (as mentioned in TWF 245) that they do satisfy the homotopy hypothesis, and conjectured (p. 69) that one of these versions “corresponds” to Kock’s “fair $n$-categories”.

Of course there may not be the semistrict version of $\infty$-categories, and different choices may be useful for different purposes, as remarked in the very last two paragraphs on p. 22,23 of E. Cheng and N. Gurski’s The periodic table of n-categories for low dimensions I.

The two conjectures that Carlos Simpson stated are (p. 27)

Conjecture 1: There are functors $\Pi_n$ and $R$ between the categories of $n$-snugroupoids and $n$-truncated spaces (going in the usual direction) together with adjunction morphisms inducing an equivalence between the localization of $n$-snugroupoids by equivalences, and $n$-truncated spaces by weak equivalences.

Conjecture 2: The localization of the category of $n$-snucategories by equivalences is equivalent to the localization of the categories of weak $n$-categories of Tamsamani and/or Baez-Dolan and/or Batanin by equivalences.

Posted at October 8, 2008 6:18 PM UTC

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Re: Semistrict Infinity-Categories and ω-Semi-Categories

Urs wrote:

What is a semistrict $\infty$-category? The strictest possible version of weak $\infty$-category which is still general enough to satisfy the homotopy hypothesis?

Answer this question and become famous!

Everyone who’s worked hard on $n$-categories has thought about this. 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. Then Jim found a mistake — the construction broke down when it got to 4-categories, which is precisely when it was just starting to do something new.

Later Brian Day and Ross Street tried a very similar idea, which broke down for the same reason at the same point.

Later Sjoerd Crans invented ‘teisi’, which are somewhat similar. I think he had to define them one dimension at a time, and did it up to dimension 6 or so.

The idea of all these approaches is to keep associativity and unit laws strict but weaken the interchange laws. The Joyal–Kock approach based on weakening only unit laws is also promising. So, as you note, there’s probably not be a unique nice answer to your question. Making some things weak may allow you to keep other things strict.

So, weakness is probably a bit like the lump in the carpet or the twist in the Möbius strip: you can push it around, but you can’t make it go away. When we understand this in detail, we’ll know a lot about $n$-categories.

Avoid this problem, unless you wish to spend a lot of time on it!

Posted by: John Baez on October 9, 2008 5:54 AM | Permalink | Reply to this

Re: Semistrict Infinity-Categories and ω-Semi-Categories

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 $i_f$.

Posted by: Urs Schreiber on October 9, 2008 9:54 AM | Permalink | Reply to this

Re: Semistrict Infinity-Categories and ω-Semi-Categories

In discussion of the above, Thomas Nikolaus, a young colleage of mine, asked which notion of path $\infty$-groupoid of topological spaces would have this property of being a strict $\infty$-category with ony identities being weak.

He suggested to take 1-morphisms to be the continuous maps from the standard interval into the space, modulo homeomorphism from the interval to itself.

This makes composition indeed strictly associative. But (as long as we do not demand paths to have “sitting instants”, i.e. to be conmstant in a neighbourhood of the boundary of the interval) this does not make the constant paths be units under composition.

But for any reasonable notion of 2-morphisms in this context, these identities should indeed be weak identities.

One should probably build a strict $\infty$-groupoid in this vein by iteratively taking k-morphisms to be paths, modulo homeomorphisms, in the space of (k-1)-morphisms. One needs to check that all exchange laws are strictly satisfied, but I think that will be true.

My question:

a) has this construction been considered anywhere?

b) Since it seems of every space I can also always form an entirely strict $\infty$-groupoid by demanding all paths to have sitting instants: how can one understand the fact that paths with sitting instant “forget” (apparently) topological information?

Posted by: Urs Schreiber on October 14, 2008 9:00 AM | Permalink | Reply to this

Re: Semistrict Infinity-Categories and ω-Semi-Categories

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).

Posted by: Tim Porter on October 26, 2008 8:23 AM | Permalink | Reply to this

Re: Semistrict Infinity-Categories and ω-Semi-Categories

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.

That sounds very much to me like the simplicial version of what I was sketching #! I supoose the filtered self-maps of $\Delta^n$, being filtered, will be homeomorphisms (?).

So I suppose in the spirit of your description the construction I described would be the globular space $Glob(X)$ whose $k$-cells are maps from the standard $k$-globe modulo filtered self-maps of the standard $k$-globe.

If we are after seeing a semistrict $\infty$-groupoid that would maybe be a useful point of view. I think I was essentially suggesting that $Glob(X)$ would be a strict $\omega$-category without units in the ordinary sense, but woth weak units.

Thanks for the pointer to Hardie-Kamps-Kieboom, I wasn’t aware of that work.

Posted by: Urs Schreiber on October 27, 2008 7:13 PM | Permalink | Reply to this

Re: Semistrict Infinity-Categories and ω-Semi-Categories

Urs said:

I suppose the filtered self-maps of Δ n, being filtered, will be homeomorphisms (?).

No just think of all the maps from the unit interval to itself that take 0 to 0 and 1 to 1. Once you have that behaviour on the boundary of a 2-simplex life gets really fun!

There are other papers by Hardie-Kamps and Kieboom and two with subsets of them together with Ronnie and myself:

http://www.tac.mta.ca/tac/volumes/10/2/10-02abs.html

and

http://www.tac.mta.ca/tac/volumes/14/9/14-09abs.html

that may be useful.

Posted by: Tim Porter on October 27, 2008 7:40 PM | Permalink | Reply to this

Re: Semistrict Infinity-Categories and ω-Semi-Categories

No just think of all the maps from the unit interval to itself that take 0 to 0 and 1 to 1.

Right, sure. I wasn’t thinking properly.

Once you have that behaviour on the boundary of a 2-simplex life gets really fun!

Hm, I’d need to think about this. I feel more like just dividing out homeomorphisms, but maybe I am not seeing clearly enough.

Posted by: Urs Schreiber on October 27, 2008 7:49 PM | Permalink | Reply to this

Re: Semistrict Infinity-Categories and ω-Semi-Categories

I feel more like just dividing out homeomorphisms, but maybe I am not seeing clearly enough.

It depends what you need. The monoid is the allowable reparametrisations of the singular simplices so you may not need the process of reversing maps or having constant maps as identities, so can get away without certain of the types of map.

Posted by: Tim Porter on October 27, 2008 9:03 PM | Permalink | Reply to this

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