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I was reading an answer of Cisinski on MO, and I got carried off to a paper. I have only skimmed the paper, but Cisinski’s comment is really interesting (and contains some context for my post that is not actually in the paper)
I was skimming through Dimitri Ara’s recent thesis, and I wasn’t sure if you’d all seen it, but it seems to give a definition of a weak $\infty$-category (that is, $(infty,infty)$, but the distinction is nonexistent).
I wasn’t sure if you guys had seen it, but here’s a link. It seems like an extremely nice definition/construction/theorem. I’m not sure that I’ve got it totally understood, but it seems like it takes the Batanin-Leinster globular operad things, unwinds them, and then gives a “coherator” in terms of them. Then he somehow relates this to Joyal’s theta-categories and also quasicategories, among other things.
If anything, it seems exciting only because it seems like it unifies all of these disparate approaches into one coherent picture of higher category theory. I think it also addresses Mike’s concern from a while ago about quasicategories having the higher morphisms act as both coherence data and higher morphisms.
But a question for you: Do these types of higher categories still suffer from the dearth of functors that we have for Batanin-Leinster higher categories? That is, do we still have to apply a cofibrant replacement like in Richard Garner’s paper to get the proper functors between these higher categories? Further, given any two Ara-style weak $\infty$-categories, do we have a weak $\infty$-category of functors between them (do we have an Ara $\infty$-category of Ara $infty$-categories?)
Harry, I think that for any theory of (infty,infty)-categories (e.g. Batanin’s and Joyal’s) it is an open problem how to define the (infty,infty)-category of functors; I mean what is beyond noninvertible (infty,infty)-transformations, modifications and so on. For (infty,1)- of course in some models one has (infty,1)-category of functors.
@Zoran:
Has anybody thought about it at all?
As you know, I posted about this definition at the Cafe a while back. I haven’t read through Ara’s thesis yet, though, because I have a lot of stuff to do and reading French is very time-consuming for me. Insofar as the definition is algebraic (comes with specified composition operations), I would expect that yes, one needs a cofibrant replacement to “get all the functors.” And yes, some people have definitely thought about how to define noninvertible higher transfors (myself included), but I don’t know of any really satisfactory answer that has yet emerged.
@Mike: Even if a really satisfactory answer hasn’t emerged, what kinds of things have people (again, yourself included) come up with (I guess in the context of Batanin/Leinster stuff, since you haven’t really worked with Ara/Maltsiniotis/Grothendieck categories)?
One idea which just presented itself to me (and is really a question to the more expert among us) is what about weakening one level of transfors at a time? For example, moving to weak functors, and having strict higher arrows. Then have weak transformations, and everything strict above that. This depends on whether these higher categories actually exist, which we can check for $n$-categories with $n=2,3$ (although, perhaps not the 4-category of 3-categories - has anyone done this?).
That (i) seems a good plan, but also (ii) is linked to an idea that I have aired on several occasions. :-) and so far has not got any response. :-(
My idea was in a more special instance, (i.e. in simplicial groups or S-groupoids), but corresponds to the passage from 2-crossed modules to 2-crossed complexes. The former ’are’ the Moore complexes of simplicial groups that satisfy the condition that the Moore complex NG_n is trivial in dimensions greater than 2. (So for instance the ordinary interchange law is not necessarily satisfied and we have Gray tensors/Peiffer lifting etc. see the n-Lab entry.) In a 2-crossed complex, the Moore complex in those dimensions need not be trivial, but contains no non trivial thin elements. (In this context thin elements are the products of degeneracies). This means that higher Whitehead products in the corresponding homotopy types must be trivial and ’morally’ say things are as strict as they can be in those dimensions.
There is clearly a corresponding strictness in the homotopy coherence model for these things or rather for the non-groupoid versions. The interchange law corresponds to a square face of the universal homotopy coherent 4-simplex (!) remember homotopy coherent diagram that S[4] is the free S-category on the 4-simplex, and the hom from 0 to 4 in it is a cube so has one more face than ’expected’. That face is an ’interchanger’. In S[5](0,5} there are two interchangers corresponding to (012)x(2345) and (0123)x(345) in the arcane notation that I tend to use! These are a (2,3)-whiskering and a (3,2)-whiskering and in general need not be trivial. I think that David’s idea corresponds (in this context) to looking at h.c. things where these are trivial (i.e., are really ’identities’.)
To a limited extent this was explored in a thesis (Florence Marty: Approche en dimension superieure des 3-categories augmentées d’Olivier Roy, Montpellier 1999), in which the low dimensional cases are examined in great detail and a notation and general approach is given for higher dimensions with some comparison with the approaches of Batanin and Penon. (Penon was on the Jury as were Ross Street and Clemens Berger.)
Even though the morphisms of algebras in the Batanin/Leinster definition are the “wrong ones”, do they at least form a cartesian-closed category?
@Harry - you could check this using bicategories and the two sorts of morphisms of those - weak and strict. We know Bicat is cartesian closed for the Gray tensor product and weak 2-functors, but I doubt it is so for strict 2-functors.
@Tim
This would be interesting from an anafunctor/butterfly point of view. If we consider the 2- (or even I suppose eventually 3-category) of 2-crossed complexes with strict maps - and then localise at the ’essential equivalences’, and then see what 2-arrows there are, and see if there is some way to localise those 2-arrows which are ’essential equivalences’, if there are any that are not invertible. I suspect we may run into the same problem that says there isn’t a decent bicategory of bicategories, but that is being pessimistic.
And I should say that the link to the thesis I gave is only a ’part II’ - part I not being available electronically.
I have a hard copy :-)
We know Bicat is cartesian closed for the Gray tensor product and weak 2-functors
Actually, what we know is that the category of bicategories and strict 2-functors is closed monoidal with the Gray tensor product. I think it very unlikely that it is cartesian closed, but I don’t remember checking it.
One natural way to try to define higher transfors is: if you have a notion of weak functor, then define a k-transfor to be a weak functor $A\times D^k \to B$, where $D^k$ is the free-living $k$-morphism. The problem then is trying to compose them. IIRC if you try this with A and B algebras for some contractible globluar operad, and cofibrant-replacement weak-functors, then it looks like there should be a “functor $\omega$-category” $[A,B]$ which is an algebra for a different contractible globular operad. The point being that in order to compose (say) pseudonatural transformations (1-transfors), you need to compose 1-morphisms and also 2-morphisms in B (and higher morphisms too, as high up as the weakness goes); thus the 1-dimensional composition operations that act on $[A,B]$ are built out of all-dimensional composition operations that act on $B$.
@Mike: I think that might be substantially easier to do with this definition than the Batanin/Leinster one since there is a lot of explicit stuff that invokes those higher pasting diagrams that looks very similar.
@Harry: Perhaps; let me know if you succeed! (-:
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