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Let’s say that a strict n-category (including n=ω) X is weakly groupoidal if every cell of height > 0 is an equivalence. These appear to be quite a bit more homotopically interesting than strict n-groupoids, since a.) weak inverses are not unique, and b.) we can have nontrivial maps between loops formed by pairs of weak inverses (eg. fg, fgfg, fgfgfg, where f and g are weakly inverse to one another).
Can weakly groupoidal strict n-categories model more homotopy types than strict n-groupoids?
I should have looked in Dimitri Ara’s thesis before asking this question. He recalls some proofs (I think originally by other people) showing that the 2-suspended strict n-groupoids model products of Eilenberg-MacLane spaces. Moreover, he cites a result of Simpson that every quasi-strict 2-suspended 3-groupoid is weakly equivalent (in the folk model structure) to a strict 3-groupoid. Presumably this pattern (that we gain nothing by generalizing from strict to quasi-strict) continues for n>3.
I would bet that when we are looking at weak ω-groupoids (say Batanin-Leinster weak ω-categories with all cells being equivalences), every one will be weakly equivalent in the corresponding folk model structure to a weak ω-category whose cells are all strictly invertible. All of the richness we might expect from the cells between the loops (fg)^k can probably be accounted for by non-strictness in the units.
All of the richness we might expect from the cells between the loops (fg)^k can probably be accounted for by non-strictness in the units.
This is an existing conjecture, I just can’t recall who by.
Carlos Simpson might be one such person.
I have a related problem where the condition on the homotopy types is posed in terms of known invariants namely the Whitehead products. Suppose we have a homotopy type in which all Whitehead products vanish from dimension n onwards, what can be said about this from the k-categorical/groupoidal viewpoint. Strict omega categories and crossed complexes etc. give the simplest case of this it seems.
re #4 #5; yes: Simpson’s conjecture
Some comments:
(1) D. Ara shows definitively that simply connected strict n-groupoids with strict inverses model products of simply connected Eilenberg-Maclane spaces. He leaves open the case where the strict n-groupoids are allowed to have weak inverses, but Simpson shows that simply connected 3-groupoids with weak inverses also, up to equivalence, split as products of Eilenberg-Maclane objects.
(2) For finite n, I think it is more or less clear that Simpson’s argument could be adapted one layer at a time inducting from the top layer. So for finite n there should be no difference between the homotopy types realized by (simply connected) strict n-groupoids with strict inverses and those with weak inverses. For n=infinity, it is much less clear to me. One thing muddying the waters is that in the infinite case the notion of weakly invertible could mean different things (for example is it the inductive notion or the coinductive notion?).
(3) The non-simply connected setting is much more subtle. Even the case n=2 shows that you can’t expect to get a splitting into Eilenberg-Maclane spaces. However these previous results do tell you something. They tell you that the homotopy type realized by a strict n-groupoid has a universal cover which is a product of Eilenberg-Maclane spaces. It should be an interesting question as to what the possible k-invariants can be. Perhaps R. Brown’s (and others?) work sheds some light here?
D. Ara shows definitively that simply connected strict n-groupoids with strict inverses model products of simply connected Eilenberg-Maclane spaces.
I would have thought we knew this before: we knew that strict -groupoids are equivalently crossed complexes of groupoids and that if these are connected and simply connected they are equivalently just chain complexes.
What am I missing here?
And I’d think that connected strict groupoids, which we know are equivalently just crossed complexes (not “of groupoids”, due to connectivity) are equivalently the homotopy quotients (Borel constructions) of chain complexes (= EM-spaces) by strict 1-group actions if the crossed module in the lowest two dimensions is of semidirect product type.
While we’re here, a question:
There is a “realization” functor from strict -groupoids (lets say with weak inverses) to weak -groupoids. (Probably there are many; Simpson discusses several in his 3-types paper, but one presumes they are all more-or-less equivalent.)
It seems to me that such a realization should come with an -categorical left adjoint. Note that is a (weak) -category. There ought to be a “strictification” functor , which is an -categorical left adjoint to . The proper question seems to me to be, not which homotopy types come from strict -groupoids, but rather, what is the nature of this adjoint pair?
Has anybody studied such a strictification? (I haven’t found anything on this.)
What about Kapranov and Voevodsky’s paper -groupoids and homotopy types? They prove that (some version of) weakly groupoidal strict -categories model all homotopy types, although I’m not sure I understand their main result correctly.
Re #9, Urs, yes we basically knew this already. I don’t think Ara claims originality. He refers to it as a folklore result. Though it is nice to have a clear honest reference for this fact.
A comment re #6, which I forgot to mention earlier. There are some spaces with vanishing Whitehead products which cannot be realized by strict n-groupoids, yes? The easiest one I know is the simply connected 3-type with pi_2=Z and pi_3 = Z/2, with the non-trivial k-invariant. This is a stable k-invariant and in fact this is the stable 1-type of QS^0, the sphere spectrum. As an infinite loop space it has vanishing whitehead products. Simpson’s argument for the 3-type of S^2 applies verbatim to this space as well. More generally just about any non-trivial loop space should also give an example.
@Charles: I think it is just a reflective Quillen embedding if we reduce the set of cofibrations (and increasing the set of fibrations) in the homotopical theory of weak omega groupoids (based on theta-tilde) enough for all of the strict omega groupoids.
A lot of the weirdness in the generators of the weak equivalences in homotopical theories of weak n cats comes from having to make corrections for the overly strong conditions for contractibility imposed by the presence of all those extra cofibrations. If the homotopy hypothesis holds, the answer was worked out by Ronnie Brown.
As it so happens, I think that practically everything I noted is an obvious conjecture raised by your very paper!
@Chris (13): Probably we should declare Hopf maps to be honorary Whitehead products for the purposes of this discussion: “”.
@Harry (14): I’m skeptical that the adjunction I attempted to describe is some kind of Quillen embedding. Possibly I just don’t understand the distinction between various kinds of “strict n-groupoids” (by which I always mean: strict n-categories with some kind of invertibility conditions), but I don’t see how that will make a difference. Certainly if I consider truly strict omega-groupoids (isomorphisms at all levels), then this can’t be a Quillen embedding, as noted above (e.g., Ara’s paper).
The case of 3-types is a little deceptive. If I understand things (and perhaps I don’t), the -category of simply connected strict 3-groupoids (even with weak inverses allowed, a la Simpson), is equivalent to a full subcategory of the -category of simply connected 3-types (namely, that consisting of 3-types with trivial Whitehead products and Hopf maps.) Therefore, the “3-strictification functor” should be easy to describe on simply connected objects (just kill the products).
On the other hand, there is no reason to think that the -strictification functor, , when applied to a simply connected -type, should give you back something equivalent to a strict 3-groupoid. (For instance, I’d expect the -strictification of to have .)
@Charles: Two things: As someone who proved that result of Ara independently (I even mentioned it on the nforum a pretty long time ago), I am pleased to let you know that there is a very important condition on that result, which should be clear when it is restated this way:
Let (C,W,F) be a model structure on Psh(Θ). If C is the class of all monomorphisms, then the map D_1[J] -> D_1 (where D_1 = Δ[1] and D_1[X] is the suspension of X) is not a weak equivalence if D_1[J] and D_1 are fibrant (Ara proves it for D_n, n>0, and his proof is different than mine but is also very nice (The special-case counterexample I gave gives rise to the same solution (suspending Js) as Ara’s family of counterexamples (and obviously both of these solutions are clearly inspired by your own solution)).
My proof uses the fact that a weak equivalence between cofibrant-fibrant objects is a homotopy equivalence together with the existence of the functorial cylinder object (-)×J, and then a quick proof that it simply is not a J-homotopy equivalence. (They are fibrant in the Cisinski-Joyal conjecture precisely because they are strict ω-categories), while Ara’s proof, if I remember correctly, shows that it is not a trivial fibration and then shows that it is a fibration, therefore proving that it cannot be a weak equivalence (and he does this for all n>0).)
Anyway, the absolutely key point here is that the result requires that the class of cofibrations is exactly the class of monomorphisms. However, rather than enlarging the weak equivalences at the detriment of the fibrations, it is at least plausible to make the case that we can enlarge the class of weak equivalences instead at the detriment to the class of cofibrations. Now I don’t want to say that this is definitely something that is possible, since to be honest, I was willing to give up the Quillen embedding as a cost of having all of the monomorphisms be cofibrations, and I haven’t had the time to actually come up with a worked-out model structure where this is true, but I could maybe speculate on what the generating cofibrations could be: The set of generating cofibrations could be something like the union of the set of globular sphere inclusions D_n[ø] -> D_{n+1} together with the spine inclusions (which happen to generate the anodynes in the sense of Cisinski)
I have no idea if those generators work, I haven’t checked them.
I have really only discussed this idea in detail with Cisinski, but I did write an e-mail to Mike Shulman on October 19, 2011 discussing my results as well as this idea of an “injective” and “projective” model structure for categories Θ_n\wr A (also some ideas involving those nasty tensor products)
You can take a look at it (it’s pretty detailed) here.
My original e-mail to Cisinski on this idea of an injective and projective model structure seems to be lost on my harddrive, although I may have been fortunate enough to keep a copy of it somewhere on my gmail webmail. However, I’m not motivated enough to look for it right now, and the e-mail I sent Mike discusses the idea in a bit more depth.
Anyway, the point is that if we can restrict the class of cofibrations in a way that turns the “anomalous” “should-be-trivial-fibrations” weak equivalences in the repaired model structure into honest trivial fibrations, we should get the correct model category target. This is the model structure that behaves like the Joyal model structure (which ought to be both the “projective” and “injective” in this sense).
Also, regarding the “strictification” functor spitting back a quasi-strict groupoid from a weak groupid, I think that this is exactly what we should expect to see. As we know from quasi-categories, the “strictification” is useful for coming up with definitions like “subcategory” and determining when a cell is an equivalence. This is exactly what we can do with the strictification functor.
@ Charles (15)
On the other hand, there is no reason to think that the -strictification functor, , when applied to a simply connected -type, should give you back something equivalent to a strict 3-groupoid. (For instance, I’d expect the -strictification of to have .)
But has a model as a strict n-groupoid (n=2, even). I would expect your functor not to change n-groupoids which are already strict?
@Chris (17):
Why? also has a model as a strict n-groupoid (n=4). But there are maps (e.g., cup square) which do not come from maps of strict 4-groupoids.
@Harry (16):
I’m not following. Which result of Ara are you generalizing?
re#12:
Hey Marc, thanks for that reference! I had missed that. That seems to be at least pretty close to a full answer to Harry’s question. I made a quick note on this here in the page on homotopy hypothesis. (Everybody feel free to expand there.)
re #13:
Chris, what was the subtlety (the previously non-“honest” part)? Of course I can ask Dimitri myself…
Urs, Marc: I think Simpson is saying (in “Homotopy Types of Strict n-groupoids”, see especially Corollary 5.2), that the Kapranov-Voevodsky paper is wrong.
Charles, ah, thanks. I had missed that, too.
@Charles (20): Thanks! I was wondering why no one ever mentions that paper in discussions of the homotopy hypothesis… Now I know.
@Charles: I was referring to this:
I’m skeptical that the adjunction I attempted to describe is some kind of Quillen embedding. Possibly I just don’t understand the distinction between various kinds of “strict n-groupoids” (by which I always mean: strict n-categories with some kind of invertibility conditions), but I don’t see how that will make a difference. Certainly if I consider truly strict omega-groupoids (isomorphisms at all levels), then this can’t be a Quillen embedding, as noted above (e.g., Ara’s paper).
I have some explanation below for why this isn’t the case, but if you want to cut to the chase, just skip to the last paragraph of this post.
If there is a “projective” model structure on the category of n-cellular sets (where the usual “injective” model structure is given e.g. as the one in my arXiv paper (which is canonically Quillen equivalent to your model category of Theta_n-spaces for finite n (the case n = ω is different for some complicated reasons involving this coinductive stuff)), then the strict n-nerve is exactly a Quillen embedding of the folk model structure on strict n-categories into the model category of weak n-categories.
I’ll try to explain this “projective”/”injective” idea one more time. Dimitri Ara and I both have papers on the arXiv giving constructions of model structures on the category of Theta_n-sets (my paper actually only covers the case where n=ω, but only because it is the only case that is not equivalent to a version of your very own Theta_n-spaces). These model structures that we’ve defined are on-the-nose equal for finite n anyhow. However, they both have the property that the cofibrations are exactly the monomorphisms, but while this is a blessing in one way, it’s a definite defect in another.
In your paper, you realized that it was necessary to Bousfield localize not only at the “spine inclusions” (I forget the term that you used for them), but also at the family of maps given by discnerve(*->E) and its suspensions, where E is the freestanding isomorphism. However, this should only be true in one of the model structures on Theta_n-sets. In the other hypothetical model structure with fewer cofibrations, those maps would already be strong deformation-retracts of trivial fibrations. That is, if we restrict the class of cofibrations, then we really end up with a quasi-category-like characterization of fibrancy. The localization of the “complete segal” maps is penance for being too “greedy” with the cofibrations… of course, not to cast aspersions =).
Anyway, what we should have is Psh(Theta_n) with the model structure constructed by Ara and myself independently (and equivalent to your model structure for weak n-categories on sPsh(Theta_n) by simplicial completion) should be Quillen-equivalent by the identity functor to another model structure on Psh(Theta_n) with the same weak equivalences but fewer cofibrations and more fibrations, and the Theta-n nerve from strict n-categories to that model structure is a Quillen embedding on the nose. So the Theta-n-nerve is ∞-categorically the embedding of the ∞-category of strict n-categories (represented by the folk model structure) as a homotopically full subcategory of the ∞-category of weak n-categories. My point is that there is no result by Ara that proves this impossible. The statement for quasi-strict n-groupoids would just be a corollary. The Quillen embedding in question is the n-categorical analogue of the Quillen embedding of Cat into quasicat. However, whether or not this is true hinges on whether or not we can give a construction of this “projective” model structure on Theta_n-sets, which I have not proven either way (and nor has Dimitri, to my knowledge).
@Harry:
If there’s an Quillen embedding , then there should exist strict -groupoids and , such that and . Since it’s a full embedding, the map representing the cup square should, (up to homotopy) come from some map of strict 4-groupoids.
Let in be the homotopy fiber of . Then should be the homotopy fiber of . The -type is very similar to the -sphere; in fact, it is equivalent to the -truncation of . According to Simpson, this -truncation can’t be realized from a strict -groupoid.
This seems like a contradiction to me. What am I missing?
@Charles: Hm… I have one stupid idea (pardon it), but can’t the 3-truncation of S^2 be equivalent to the realization of a non-truncated 4-type? Like, does the fact that we’re looking at 4-types give us the needed flexibility to glue in some 4-cells and save the day? I think that if that is the case, we have no contradiction, since the point is that we could never actually form the original holim in St3Gpd, so the right Quillen functor need not preserve it.
I mean, if there’s not a way out, I’d be extremely surprised, since I would certainly have expected that strict n-categories with pseudofunctors between them should embed fully and faithfully into weak-n-cat (being the full subcategory spanned by the strict subobjects). Indeed, this is the whole idea behind Richard Garner’s definition of the category of weak n-categories and weak functors (construct a cofibrant replacement comonad Q on the category of Batanin-Leinster weak n-categories and define Hom(X,Y) to be Hom(QX,Y) (something like this that is equivalent to it.. it’s actually supposed to be maps in the category of coalgebras for that comonad?)), so if what you say is true, I think it proves the homotopy hypothesis false, at least for what seems to be the most promising algebraic definition of weak n-functors between weak n-categories.
Also, this very new preprint by Dimitri Ara seems to suggest that we do dodge the bullet: http://www.normalesup.org/~ara/files/strweak.pdf
Harry: I wouldn’t think so. I mean, I have no reason to think that the 3-truncation of S^2 can be realized as a strict 4-groupoid, if it can’t be realized by a strict 3-groupoid.
Basically, I’m making assumptions about the functors , e.g., they preserve homotopy limits, that they are compatible with the inclusions , etc. I don’t actually know enough to prove those kinds of statements.
Fully faithful embeddings at the category level don’t need to give fully faithful embeddings at the the -category level.
Fully faithful embeddings at the category level don’t need to give fully faithful embeddings at the the (∞,1)-category level.
In other words, at the -category level, “being strict” may not be a property of an object, but rather structure that can be placed on it.
Well, I guess that my point is that there seems to be something surprising going on! If the pseudofunctors between strict n cats are not weak n-functors, then what are they?! Ara’s embedding must preserve fibrations, and I should hope that it preserves weak equivalences, but I guess that maybe it fails to be homotopy-full?
@ Charles. I was naively assuming you were thinking that strict n-groupoids should be considered as a full-sub--category of weak groupoids, meaning that we have strict n-groupoids but all weak maps between them. Then the cup-square map from to should be allowed.
But I was wrong to assume this.
Your nice example from (24) shows in particular that the full sub--category of strict n-groupoids, thought of in this way, is not closed under homotopy limits. So it can’t be a reflective subcategory.
I think Mike in (27) sums the hope up nicely. That there is some structure that we can add to n-groupoids which will force them to be strict. Let me just sketch out a guess about this. After writing it, I see it is mostly just saying things that we already know.
In the simply connected case we get exactly those spaces which admit the structure of being an abelian group, i.e. simply connected simplicial abelian groups. This is a structure we can place on simplicial sets. Is there a good description of the left adjoint of the forgetfull map for these? (Maybe assuming the simple connectivity throughout). I guess there is. Given a simplicial set, you just get the free simplicial abelian group generated by the simplicial set, a.k.a. its homology chain complex.
In the general case, instead of a simplicial abelian group we get a “crossed complex”. I think I would like to think of this as a functor from the fundamental groupoid to a simply connected abelian group objects. Namely for each point in the space X, we get the univeral cover of (the component of) X based at that point. This assignment is functorial for paths. If X is a strict -groupoid, then these universal covers will be strict abelian group objects.
So I guess the left adjoint you might be looking for takes a simplicial set, looks at all the univeral covers at all base-points (viewed as a functor of the fundamental groupoid) and forms the free simplical abelian group out of these. The result is a functor from the same fundamental groupoid to simply connected simplcial abelian groups.
Does that sound like the sort of description you were after? I would hesitate to call this left adjoint a “strictification” functor, though I guess in some sense that is exactly what it is.
@Chris: Yes, that is exactly the sort of description I would like.
Actually, I’ve been looking at explicit models of the sort of simplicial construction for “strictification” that you describe. (There is a bit of complication: abelianizing a space gives you something with ; you have to get rid of the in the right way.) Although it is easy for me to fiddle around with simplicial constructions like this, I have not yet had success in relating this kind of simplicial construction to strict -groupoids, but that’s what I’d hope for.
There is an very appealing consequence that comes out of all this, if it’s true: Strictification of -groupoids is conservative. That is, a map of spaces (aka, -groupoids) is a weak equivalence if and only if the map of strict -groupoids is an equivalence. Proof: Use the “homology Whitehead theorem”.
Optimistically, one could even ask if is comonadic. That would be a kind of optimal form of Mandell’s theorem.
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