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I will brush-up the entry homotopy hypothesis. But not right now, right now I have to run and do something else. But here is some leftover discussion that was sitting there, and which I have now removed from the entry and reproduce here, in order that we go and use it to make the entry better, but not clutter it up.
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here the leftover discussion:
Tim: I would like to pose a question on the Homotopy Hypothesis. Playing devil’s advocate for the moment, since Kan complexes, and simplicially enriched categories, both satisfy the homotopy hypothesis, why bother to search for other models? That is a bit severe of course so a more ’constructive’ form of the question is what criteria should we be looking to be satisfied so as to say that a model of homotopy types is a good one? (Perhaps things like that the basic homotopy operations, such as Whitehead or Samelson products, should have clear formulations and clear interpretations. The higher operations could then be gradually required. The relation between obstructions to interchange laws (interchangeator!) and the low dimensional Whitehead products are ’clear’ even if not that immediately evident in most writing on the subject (and I include my own in that!), but in higher dimensions …. ?
Mike: My perspective is that we are looking for good notions of $\infty$-category such that the induced notion of $\infty$-groupoid satisfies the homotopy hypothesis. If all you want is to do homotopy theory, then homotopy theorists have gotten along quite well for decades with topological spaces, Kan complexes, and the like (although one might certainly hope, as you suggest, to get information about homotopy operations out as well). But as category theorists, we want a notion of $\infty$-category with not all cells necessarily invertible, which behaves well categorically rather than homotopically, but whose induced notion of $\infty$-groupoid still satisfies the homotopy hypothesis as a “check” or “anchor.” It’s hard to exhibit Kan complexes as the $(\infty,0)$-case of a notion of $\infty$-category; there are quasicategories for $(\infty,1)$-categories but beyond that it gets tricky (there is Street’s original definition, but it’s quite difficult to work with).
Urs: Lurie is now talking in (electronic) print about $(\infty,n)$-categories, here.
Mike: When I glanced at that, it looked as though he was actually using Segal categories for his $(\infty,n)$-categories rather than some generalization of quasicategories.
Tim: I think, Mike, you are missing my point. You use the word ’good’ with regard to notion. My point is that the higher homotopy operations ARE categorical structures that WILL BE there in the various models. For instance, there are formulae given in Curtis’s old survey article for the Samelson product in a simplicial group. (No proof is available in the literature.) This uses shuffle products and various combinatorial ideas that fit well from the categorical viewpoint. As category theorists we can hope to find new understanding of what makes the infinity groupoid models work and hence what the infinity category models should be. As these products DO use inverses the problem of interpreting their analogue in an infinity category is not easy. I feel it has, from my perspective, something to do with the basic homotopy coherent cube and the free simplicial cat resolution of a category, but I am not sure how to handle it.
Mike: I think we are talking past each other. I was trying to answer your question “why bother to search for other models?” by saying that from my perspective, what we’re doing isn’t just searching for new models for spaces, but rather searching for a good notion of $\infty$-category that still reduces to a model for spaces. I took your mention of higher homotopy operations as a proposed answer to the first question “why bother to search for other models?” So I wasn’t addressing homotopy operations specifically – I don’t have any disagreement with it – just pointing out that there are also other answers to the first question.
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Ronnie: What is wrong with strict $n$-fold groupoids as a model of homotopy $n$-types? This is a theorem, not an hypothesis! (Loday, Porter). One advantage of these is that one can do some calculations with them, using a higher homotopy van Kampen theorem, which Grothendieck interpreted as integration of homotopy types.
They are certainly a very rich structure, with other interpretations of special cases. My question is: what are the expected purposes of these other models? My aim since 1966 has been to find models which seem to give a natural extension of group theory, and hence to work with strict structures. My hypothesis is that this can always be done, if you can find the means to make the lax conditions part of the algebra.
What held me up for 9 years in constructing higher homotopy groupoids was to concentrate on spaces: when Philip Higgins and I started to look at pairs of spaces, then things quickly fell into place. Thus the clear functors from strict $n$-fold groupoids give rise to structured spaces. What is wrong with that?
Mike: I don’t think anyone is saying that there is anything wrong with strict $n$-fold groupoids as a model of homotopy $n$-types or structured spaces. I certainly don’t think there is. I think that probably the feeling is that having a variety of different combinatorial models means having a variety of different tools, each of which may have its advantages and disadvantages. Ordinary topological spaces and simplicial sets also have advantages and disadvantages. Tim already mentioned some of the hopes that some people have for $n$-groupoids as a model of $n$-types: more insight into homotopy operations such as Whitehead products. I’m not a calculational homotopy theorist myself, but it seems plausible.
And, as I said before, I think another purpose of the homotopy hypothesis is to use modeling of $n$-types by groupoids as a litmus test for a good notion of $n$-category, irrespective of whether this modeling gives us any new information about the $n$-types. And I do personally feel that strict $n$-fold categories have something to tell us in this picture that has not been properly investigated (or, at least, if it has, I’m not aware of it).
Ronnie: I entirely agree with Mike on horses for courses. It is helpful also to analyse what are and should be the courses!. There are and will be many of them. There is already a description of Whitehead products $\pi_2 \times \pi_2 \to \pi_3$ in the classifying space of a crossed square, using the $h$ map.
John Baez: Just to reiterate some of Mike Shulman’s points:
1) Weak $\infty$-categories are important in themselves, for a myriad of reasons.
2) Any notion of weak $\infty$-category for which the corresponding weak $\infty$-groupoids don’t model homotopy types is wrong.
3) Therefore, for every proposed definition of weak $\infty$-category, we must check that the corresponding weak $\infty$-groupoids model homotopy types.
Similar remarks apply to all proposed concepts of $n$-category and $(\infty,n)$-category.
Urs: I looked at Ronnie Brown’s review article above, but didn’t find Loday’s ’81 article yet. Maybe somebody can help me: what is the map from spaces to $cat^n$-groups which comes from the equivalence in Loday’s theorem? I am asking because the natural guess would be that we send a space to its $n$-fold fundamental groupoid $\Pi_{n-fold}(X)$ in what should be an obvious way. But that cubical $n$-groupoid $\Pi_{n-fold}(X)$ will have thin fillers (“connections”) etc and hence be equivalent to a strict $n$-groupoid, which is known not to model all homotopy types. How is this resolved?
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John Baez: See Section 5 of Simona Paoli’s paper.
Ronnie: This is a good question. The argument of Loday (1982) for getting a cubical resolution of spaces was developed in
R. Steiner, Resolutions of spaces by $n$-cubes of fibrations. J. London Math. Soc. (2) 34 (1986), 169-176.
It does not lead to an easily analysed functor $\Pi$!
However since the models usually discussed have some kind of structure such as a filtration one would expect the spaces modelled to have an analogous structure. In particular, $n$-fold categories (groupoids) have a multiple filtered structure. It may not be unexpected that there are problems in possible inverse functors or constructions to the forgetful functor (structured space) $\to$ (spaces).
Analogously, squashing from a multiple strict structure to a globular structure may necessitate certain non-strictness; this has been analysed in some cases by Simona Paoli.
It is interesting that Urs thinks there should be obvious functors from spaces to the algebraic model. It was for us quite a sweat to produce the full structure of cubical $\omega$-groupoid with connections from a filtered space, the main problem being to prove the compositions are well defined on the appropriate homotopy classes. Even the existence of the functor
$\Pi: (n-cubes of spaces) \to (cat^n-groups)$encapsulates a lot of information, as is shown by the complications of the Ellis-Steiner crossed $n$-cubes of groups, and their nontrivial equivalence with cat-n-groups.
Mike Shulman: I’m guessing that what Urs had in mind is more like “intuitively obvious.” Taking the example of Batanin ω-categories with which I’m a little more familiar, the underlying globular set of the fundamental $\omega$-groupoid is certainly “obvious” but some work is required to produce the globular operad and its action. (Perhaps the amount of work required is roughly proportional to the “algebraicity” of the definition of $\omega$-category in use?)
I now have to ask a possibly-controversial question that I’ve been sitting on for a while, namely “Are $cat^n$-groups really an instance of the homotopy hypothesis?” (Or more precisely, a relative of it, since the homotopy hypothesis proper deals only with $n$-groupoids.) To me one of the essential aspects of the homotopy hypothesis is that $n$-groupoids are considered up to categorical equivalence when modeling homotopy $n$-types. The Thomason model structure whereby 1-categories model all homotopy types is not, in my view, an instance of the homotopy hypothesis, because the notion of “equivalence” used in $Cat$ is not the categorical one. (Of course, this detracts nothing from the importance or usefulness of modeling homotopy types in this way; it is just a question of terminology and conceptual clarity.) Thus, in order to consider Loday’s result a relative of the homotopy hypothesis, I would like to know that the notion of “equivalence” of $cat^n$-groups that it uses is in some sense “categorical.” So is it? I don’t mean to imply that I think the answer is no; I really don’t know the answer. In fact, it’s not even clear to me how to make the question precise.
That is an excellent question… by which I mean that I have not an easy answer. I will have to think about it. There is a sense in which an equivalence can be realised by a span or anafunctor type of structure, and this is what higher order butterflies are I expect, but I have not yet seen any draft from Ettore and Behrang so cannot be sure.
I am now through with a first go at reworking the entry homotopy hypothesis.
I have
rewritten the Idea-section, aiming to use all the relevant statements that were scattered over the earlier entry, but trying to bring it all into a more focused and more comprehensive discussion
created a bunch of subsections, one for each model of $n$-groupoid for which we have a discussion of a homotopy hypothesis. This list may very well have to be expanded, but it captures all that was in the entry before.
made stub subsections for the 1-, 2- and Gray-groupoid case. Here the statements should at least be written out in full words (I am in a haste now, can do that later)
polished the section on the HoHo for Kan complelxes
added a section for the HoHo of algebraic Kan complexes
left the section of $cat^n$-groups untouched for the moment.
I have now spelled out all the details of the proof of the homotopy hypothesis for algebraic Kan complexes: see the section homotopy hypothesis – for algebraic Kan complexes.
@Tim,
so Behrang and Ettore are working on higher butterflies?? Cool!
But perhaps I should be able to work something up too (if I had time), because butterflies are the same as (right principal) bibundles between groupoids internal to groups, which are the same as saturated anafunctors between groupoids internal to groups. Then one could look at groupoids internal to Gpd(Grp) - with strict morphisms - and consider saturated anafunctors there. And so on. (I so easily throw out that line, whereas I’m certain it’s a lot of work)
Tim,
I ended up drastically shortening the subsection on the homotopy hypothesis for $cat^n$-groups. Because of this:
I kept the statement of Loday’s result – but made it a formal theorem (would be great if we could get hold of the statement of what the weak equivalences of $cat^n$-groups are with respect to which the homotopy category in the statement is formed)
I moved the references to Brown and Paoli given there to the References-section. They seemed to be not directly related to the statement about $cat^n$-groups (?)
I also removed the pointer to Pelissier’s article, because I wanted to put it in a separate subsection, as the entry claimed that he proves a homotopy hypothesis theorem for Segal groupoids. But does he? I have not yet found that statement in his text. Could you please check and point me to the page? I will then create a separate subsection for that.
I removed some rough statements about the higher van Kampen theorem. I am not sure where these originated from and what their purpose was. If you tell me what you (or was it Ronnie?) intended to get at concerning the homotopy hypothesis, I’ll gladly help with writing up some paragraph.
And so on. (I so easily throw out that line, whereas I’m certain it’s a lot of work)
It seems that Christian Blohmann and Chenchang Zhu have solved the “and so on” here in a nice fashion. I might have mentioned this before.
But does he?
Oh, I found it: it’s in section 6.3.4. I’ll read it and then add something on this to the entry.
okay, I have added a brief section on the homotopy hypothesis for Segal groupoids to homotopy hypothesis.
This page looks very nice now. I clarified in the intro that “topological spaces” are being considered modulo weak homotopy equivalence.
@Urs. I think the right way to handle Loday’s theorem and Catn stuff is to create a new page where it can be discussed more easily. It is useful stuff to have around, but may be slightly ’adjacent’ to the homotopy hypothesis. A discussion-type page with more details of the result and the proof might go well here.
I seem to have not much time at the moment so would not be able to do that much work on it in the near future.
I patched the reference to catn groups and linked to the ’discussion’ at cat-n-group. (The dimensions were wrong as happens often. Crossed modules model 2-types but are cat^1 groups so cat n groups model n+1 types.)
The link between catn-groups and n+1-groupoids is one that could be explored more (I do not think it is adequately examined in the literature.) I have seen some papers related to it but they are not that detailed and do not really satisfy the (vague) criteria that I have in mind, e.g. direct constructions both ways with explicit relations between the notions of both homotopy and weak homotopy modulo those constructions.)
I felt the entry on the homotopy hypothesis was still lacking the general abstract statement (that there is an equivalence of $(\infty,1)$-categories $Top \simeq \infty Grpd$) right up front. So I added it in a section Abstract statement.
I added Jardine’s proof of the homotopy hypothesis for cubical sets to a new section homotopy hypothesis – cubical sets.
I added a reference to A stratified homotopy hypothesis to homotopy hypothesis.
I found it very strange that the entry on the homotopy hypothesis did not mention Grothendieck, nor his letter to Quillen. It does now!
Hm, now the paragraph seems to claim that the concept is due to Grothendieck in 1983 but has been established by Quillen earlier in 1967. The entry should say what it is that Grothendieck added 16 years after Quillen.
The term ’homotopy hyypothesis’ was, I believe, first used by John Baez when he read what Grothendieck had discussed in his ‘letter to Quillen’ and earlier in letters to Larry Breen. These had discussed the idea that some form of infinity groupoid should form a model for a homotopy theory of spaces that should extend the links with groupoids and covering spaces that was inherent in the SGA1 description of the fundamental group(oid). There is little or no mention at this stage (i.e. in the early pages of PS) of Kan complexes and the results from Quillen’s homotopical algebra. The fact that Kan complexes gave an equivalent model to spaces was, of course, known to AG, but he was not thinking of Kan complexes as infinity groupoids as far as I remember. It therefore seems to me that the discussion that AG launches into (and which later gave rise to John’s use of the term Homotopy Hypothesis) does not really add something to Quillen’s result.
As far as I know the first mention of the idea that Kan complexes could be thought of as a potential notion of infinity groupoid was by me in a letter to AG dated 16/6/1983. I wrote: “I believe that, in fact, the ultimate in non-strict or lax ∞-groupoid structures is already essentially well known (even well loved) although not by that name. The objects to which I am referring are Kan complexes (in either simplicial or cubical languages).” The entry here as it was written before my changes, did not mention AG at all and read as if the HH was an idea of Quillen. (NB. Grothendieck does not seem to say if he ever found out if Quillen had received his letter, but does not seem to have got any reply.)
I reread what I had written and I take your point that there was still possible confusion so have altered things to try to get rid of that, but AG did not add anything to Quillen as the “HH” as he discussed was really thought of as being more important for non-abelian cohomology and the application of that to algebraic geometry. In fact the term Homotopy hypothesis is a misnomer as it is not a hypothesis, rather it is a test: someone comes up with a notion of lax/weak infinity groupoid, and the test is ‘does it model all homotopy types?’
I think there is a good case to say that in the 1970s and early 80s, Dan Kan had had a very similar intuition to Grothendieck, but coming from another direction. That might be worth while exploring sometime, but the direction of development that Kan took does not always make it easy to be sure what his intuitions were. Of course, the two threads are now well and truly merged but they were not to start with.
Yes, I know, Grothendieck was after the homotopy hypothesis for algebraic models of higher groupoids. I am just saying that the wording of your paragraph didn’t get that across. I have edited a bit more, check it out here.
I have added a bit more, and changed some parts that, to me, were anachronistic or, in fact, inaccurate. I also changed the title of the section to read References and some history. I think and hope that my rewording takes up all the points you were making. I feel that the HH is important mainly because it provides the bridge between classical ideas about algebraic topology on the one hand and higher category theory (which to my mind did not really exist as such until after the stimulus provided by PS, hence my use of the term ’anachronistic’ above.)
Looks good. I made “Breen” point to Larry Breen.
Glad it works. I like to get the ’historical’ aspect reasonably correct, otherwise someone using the lab to ’dip their toe’ into the area can get unnecessarily confused when they look for a modern bit of terminology in a paper from the 1960s, say.
Would it be worth adding your quote
I believe that, in fact, the ultimate in non-strict or lax ∞-groupoid structures is already essentially well known (even well loved) although not by that name. The objects to which I am referring are Kan complexes (in either simplicial or cubical languages)
to the entry, along with Grothendieck’s reply that those aren’t what he had in mind (do I remember that correctly?)?
I wondered about that. In any case I would not have time for the next few weeks. AG’s reply on 28-06-83 would be good to include abut it exists as a scanned poorly typed (his famous typewriter!) and glancing at it this morning it is hard to pin down what parts would be most useful. (It will be included in the ’forthcoming’ volume of AG’s letters, but that has been ‘forthcoming’ for many years. I did put a brief summary in the chapter for Mathieu and Gabriel, but I have yet to get feedback on my latest rewrite. Perhaps this sort of thing should be put in a new entry or in the one on Pursuing Stacks.
I think Quillen was utterly amazing. About 14 years before “Pursuing stacks” he had, without much ado, already presented the homotopy theory of infinitesimal algebraic infinity-groupoids. About 10 years before “Pursuing stacks”, he had given his student Kennth Brown the right idea for the homotopy categories of infinity-topoi. If Quillen had also picked up some trait of more interest to the tabloids, it might have helped him become more iconic, which might have helped avoid the big dead-lock in the development of higher category theory (lasting from “Pursuing stacks” to about 2006).
I always felt that somewhere along the line, people have forgotten Quillen’s background. His thesis on partial differential equations Formal Properties of Over-Determined Systems of Linear Partial Differential Equations. His work on simplicial objects grew out of his need for a cohomology for commutative algebras in deformation theory as well as various other applications, so this explains, in part, the link with methods relevant to infinitesimal ideas.
I just glanced at the St. Andrews history entry for him and it is worth reading. I had not remembered that he had spent time at the IHES with Grothendieck and his group in that interesting period 1968-69.
In fact, without trying to lessen Quillen’s achievement at all, Grothendieck had much of idea that infinity stacks should be developed and how that could be done, by the time of Illusie’s thesis, (1970), which of course derives a lot of its methods from Quillen’s cohomology theory, so probably from Quillen’s visit to Bures in 68-69. I think one opportunity that was missed at about that time was that the homological algebraists did not take Barr-Beck cohomology seriously enough, as they were opposed to category theory as such. Another, slightly later, was that Boardman-Vogt was thought of as being unreadable, and not much use to the ‘real mathematics’ that mattered. (That sort of value judgement is still a bane on new developments.) The interaction between homotopy theory and category theory was being encouraged by a small group of people on both sides, but equally well being impeded by others… I will not suggest names! The problem may be that the methods in the two subject areas are closely related, but the motivations are not so closely related.
I am not sure that ‘deadlock’ is the right word, as the slow progress during that period provided us, now, with many different approaches and links, not just one. I recall a referee’s report on the last of my papers with Cordier. It said, more or less, that people working with homotopy coherence did not use the simplicial categorical methods as they were getting on fine using ad hoc ideas! (We submitted it to a different journal and got a warm, helpful and positive report… such is life.)
The relationship between Grothendieck and Quillen is extremely interesting from a historical point of view. Both clearly had a high respect for the other; in Grothendieck’s case, I have, in his writings, only seen expressions of great respect and admiration for Quillen’s work, in a way that I have not seen for anyone else (even someone like Serre). There are also some interesting oral records pertaining to the time that Quillen was at IHÉS, especially the uncut version of the Illusie et al tape; basically, they were complete opposites as characters, it seems, but somehow gelled, mathematically at least, anyway.
In particular, my impression is that Quillen’s work and that of Artin-Mazur were more or less the only significance pieces of homotopical algebra that Grothendieck was acquainted with when writing Pursuing Stacks and Les Dérivateurs. I think the influence of Quillen on Grothendieck’s thought is highly under-estimated, as is the converse direction.
However, I wouldn’t go so far as to say that Quillen anticipated the homotopy hypothesis. The homotopy hypothesis is about algebraic notions of infinity-groupoid, and, as Tim wrote, the earliest historical evidence of this idea is certainly Grothendieck’s letters to Breen, from the mid-1970s. I am not aware of any evidence whatsoever that Quillen had any thoughts along these lines. How Quillen’s results may be interpreted in hindsight in the light of a certain vision is a very different matter from coming up with that vision.
Quillen was without doubt a magnificent mathematician. But I believe that Kan had an extremely strong influence on his homotopical work; I would say that the historical evidence points to Quillen’s work on homotopical algebra as being influenced strongly both by Kan and by Grothendieck (as well of course by motivations coming out of his own earlier work). In particular, my feeling is that the ’visionary’ aspects of Quillen’s work, such as they are, must be considered firmly in the context of Kan and Grothendieck’s work. Kan’s influence is especially under-rated. And when it comes to the homotopy hypothesis, I see only Grothendieck’s hand.
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