Thanks, and no rush!

]]>I have been hoping that the Grothendieck letters would become available via Maltsiniotis & Co as they were typed up. I included a quote from that letter in the New Spaces chapter (and do not know when that will come out either!) I will try to add a scanned version of the letter to that page shortly, but do have a few things to do today.

]]>There is a reference to a letter from yourself to Grothendieck, Tim (dated 16 June ’83). It would be good to have a link to this on the page.

]]>added a link to the chapter in New Spaces on this.

]]>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.

]]>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.)

]]>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 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.

]]>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?)?

]]>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.

]]>Looks good. I made “Breen” point to *Larry Breen*.

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.)

]]>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.

]]>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.

]]>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.

]]>I found it very strange that the entry on the homotopy hypothesis did not mention Grothendieck, nor his letter to Quillen. It does now!

]]>I added a reference to A stratified homotopy hypothesis to homotopy hypothesis.

]]>I added Jardine’s proof of the homotopy hypothesis for cubical sets to a new section homotopy hypothesis – cubical sets.

]]>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 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.)

]]>@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.

]]>This page looks very nice now. I clarified in the intro that “topological spaces” are being considered modulo weak homotopy equivalence.

]]>okay, I have added a brief section on the homotopy hypothesis for Segal groupoids to homotopy hypothesis.

]]>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.

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