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@Todd. Thanks for correcting my atrocious English!
Does anyone have any ideas as to how we could provide a bit more for this entry?
I have attempted to incorporate David Roberts' answer from here, along with some other rewriting and reorganization. I should mention that I still haven't actually looked at PS so I hope experts will correct my mistakes.
Welcome to the nLab and nForum, Adeel!
Thanks for incorporating the material from David’s response, and thanks for dropping us a line here. If I may just say one thing: I see that you removed some material that had been there. The general custom here these days is to discuss, in case an author edits out material that had been there, why those changes were made (because other authors may wish to defend what they had written). There is no problem retrieving material that was once there, so don’t worry about it, but perhaps we should discuss those changes a little (e.g., if you feel the former version was in some way unfair to Grothendieck).
Sorry about that, are you referring to this part from the former version?
Due to the almost free-association style of authorship, there are many back-trackings, corrections and erroneous trains of thought, erroneous in the sense that the author discusses a proposal, and after reflection, corrects his original definition or development. Despite this, the rich ideas lead to a large body of theory, rigorously developed, and to conjectures of considerable weight, some of which have been proved by Denis-Charles Cisinski.
I felt that the consensus from the MO thread was that this part was unfair to Grothendieck. I think that the description "a sort of research diary of about 600 pages, including many back-trackings and corrections" sufficiently gives a sense of the style without dwelling on it too much. I also wanted to state clearly that PS isn't just ideas and conjectures but also rigorous results.
No problem! Anyway, I’m not sure about the former version being unfair – but, like you, I haven’t actually looked at Pursuing Stacks. I’d like to hear from others who are more familiar (and were more involved with former versions). Could be that everyone will in the end agree with the present version.
I read PS when it was first produced, as a blog! The style was exactly that of a blog. He comments on the birth of a grandchild, on cooking vegetables in the Korean style, and lots of other things, and there is the record of his false starts, analysis of what goes wrong with the ’obvious’ approach, what he hopes should be true, and occasionally a reasonably solid conjecture. The bulk of the material is very much his working out the basic ideas of the ’modeliser story’. Discussing and comparing various models for homotopy and applying them to his view of what is needed. There is a lot of meat in there (as well as the vegetables).
In some parts he is reacting to material that Ronnie or myself and others were sending him. For instance, to start with he did not know of Thomason’s ideas on categories as modelling homotopy types. In that case, I gave him the reference and he reacted, I remember, by first being surprised, then commenting on the form of the cofibrations, and finally (and rapidly) incorporating the insights that that had given him into the next few entries in the ’blog’. It is thus a research diary, but much more. He mentions all the scratchwork he was doing before typing the next page of PS.
As to the edit, I have no objection to this. What was omitted was not that informative, and may be gave the wrong idea. It is a difficult work to describe. I should look out my (slightly incomplete) copy and write some more.
I have updated the links at Pursuing Stacks a lttile, for instance including links to Grothendieck-Maltsiniotis omega-groupoid, to Dimitri Ara’s work on that (beware the hypen bug, which is still in effect-…), also to homotopical alegbraic geometry etc.
Gave the paragraph on downloadable file versions a section headline, such as not to hide this information in the bulk text.
Gave this entry an actual reference-statement, so that it’s less mysterious which document it’s all about:
À la poursuite des Champs
1983
Duplicate
(Repeated post)
added the publication data (here):
Georges Maltsiniotis (ed.)
Documents mathématiques 20
Société mathématique de France (2022)
Revision 15 by Adeel Khan says:
During correspondence with Grothendieck in the 80s, André Joyal constructed what we now call the Joyal model structure on the category of simplicial sets to give a basis to some of the ideas being tossed around at the time.
Can we say more here? What correspondence is this, and is it publicly available? Can we provide an independent source on this?
The letter by AJ to AG in which the model structure is described is referenced at model structure on simplicial sheaves: here.
Unfortunately, the link says “pdf scan” while it really links to a LaTeXed transcript. I used to have the actual pdf scan of the original. Am trying to see if I can re-discover it in my inbox…
Oh, I am only now seeing what that quote really claims. I have added (here) a warning:
Check: This may rather be referring to the model structure on simplicial sheaves due to the letter Joyal 1984
I’ll check with Adeel right now.
It can also be the case that it is in some other letter from Joyal to Grothendieck. Is their whole correspondence available online?
That is all copied from David Roberts’s answer here, see post #3 above. (Not sure why I didn’t explicitly cite the MO link, but I guess it was my first time editing the nLab.) Anyway, we should ask David about the source for this.
I asked on MathOverflow, and David Roberts doesn’t know a reference.
Interestingly enough, a similar claim is repeated on Wikipedia, prior to David Roberts’s answer:
https://en.wikipedia.org/w/index.php?title=Andr%C3%A9_Joyal&oldid=384566085
He did the first real work on quasi-categories, after their invention by Boardman and Vogt, in particular conjecturing[2]. and proving the existence of a Quillen model structure on sSet whose weak equivalences generalize both equivalence of categories and Kan equivalence of spaces.
[2] A. Joyal, A letter to Grothendieck, April 1983 (contains a Quillen model structure on simplicial presheaves)
This was written by Zoran Škoda on September 13, 2010. It appears to have been borrowed from the nLab article André Joyal, where Revision 3 (December 28, 2009, also by Zoran) says
In 1980-s Joyal has invented a Quillen model category structure on the category of simplicial sets (and categories of simplicial presheaves).
Perhaps Zoran can clarify where in Joyal’s letter to Grothendieck is this mentioned?
To close the loop, I’ve edited my answer to specifically refer to the model structure on simplicial presheaves (edit: actually just simplicial sheaves, per Urs’ comment below), which is the actually relevant point (the stacks that AG was pursuing!), as well as link to the source letter on Maltsiniotis’ website.
I’ll note in passing that the WP page with the letter citation dates the letter as 1983, whereas the simplicial presheaves letter is really 1984. So there was some fuzziness even then. I agree that Zoran is the best person to ask next.
Joyal seems to be citing his own work “in preparation” in the ’00s for the source of the model structure whose fibrant objects are the quasicategories, at latest in 2006, based on the citation of Theory of quasi-categories I in the 2006 Quasi-categories vs Segal spaces. In the (published in) 2002 paper “Quasi-categories and Kan complexes” Joyal cites Theory of quasi-categories (not Theory of quasi-categories I), but doesn’t say anything about the model structure yet. However, in the CRM notes from 2008 (partly based on a manuscript—the IMA lectures—from 2004, plus the then draft of the book Theory of quasi-categories; there is also a 2007 version of the notes with the model structure) Joyal says “The results presented here are the fruits of a long term research project which began around thirty years ago.”
Verity, in his (arXived 2006) paper Weak Complicial Sets A Simplicial Weak ω-Category Theory Part I: Basic Homotopy Theory writes
we round out our presentation by localising our model structure and transporting it to the category of simplicial sets itself, in order to provide an independent construction of a model category structure on that latter category whose fibrant objects are Joyal’s quasi-categories [10].
where [10] is Joyal’s 2002 paper, so that the model structure was known to experts at least by 2006, even if not announced in 2002.
So I’m tempted to guess that the whole 1980s origin of the Joyal model structure for quasi-categories might be an urban myth.
Added: I couldn’t find a mention of a model structure for quasicategories in the 2004 slides from IMA for the talk of Joyal/May/Porter, except for the closing sentence:
Baby camparison should give that the hammock localizations of all models for weak categories have equivalent hammock localizations. Model category theory shows how.
Tim’s notes likewise don’t seem to mention the model structure. So perhaps the date for the model structure can be pinned down to between 2004 and 2006, at least as far as going by Joyal’s public statements. One point in favour of this is that Tim’s notes include the open question
In general, what is the precise relationship between quasi categories (a weakening of categories) and Segal categories (also a weakening of categories)? (This question is vague, of course, and would lead to many interpretations)
which is what Joyal and Tierney’s 2006 paper pins down, in terms of a Quillen equivalence of model categories. If the question in 2004 had been merely one of trying to match up existing model structures (the Segal category one existed in 1998), I doubt Tim would have called it a vague question!
Thanks everyone for digging into this. Since the claim in the entry about the origin of the model structure for quasi-categories looks increasingly unlikely and in any case not backed up (on the contrary, likely an absent-minded confusion with the model structure on simplicial sheaves) I have deleted the sentence (the one quoted in #15). When anyone has evidence that it was true after all, we can easily add the remark back in.
Instead, I have added (now here) a more proper mentioning of the exchange that is known to have taken place:
In conversation with Grothendieck on these matters, André Joyal, 1984 described the model structure on simplicial sheaves which, in hindsight, is an early model (along with that by Brown 1973) for $\infty$-stacks as now in use in modern higher topos theory.
I also asked a question about this here: https://mathoverflow.net/questions/425082/when-did-the-joyal-model-structure-on-simplicial-sets-originate
Just a minor side point regarding #20-#22 above:
Joyal’s 1984 letter defines the model structure on simplicial sheaves, not that on pre-sheaves. The latter was highlighted and developed (if not introduced) by Jardine.
Jardine 1987 writes, after recalling Joyal’s 1984 letter:
A point that all authors (including myself) seemed to miss up to now is that, in the proofs of the results above, it is not so much the ambient topos that is creating the homotopy theory as it is the topology of the underlying site. These proofs may be generalized to produce local and global homotopy theories for simplicial pre-sheaves on a Grothendieck site which depend on nothing but the axioms for the site. This paper presents these results.
Good point, thanks, Urs.
Incidentally, I used to speculate (and still do) that the phenomemon at play here is the cause of the delay of part III (“Cohomology”) of Sketches of an Elephant: Contrary to a wide-spread hope in the early category theory community, cohomology is just not a topic to be tacked on to 1-topos theory, but is by nature $\infty$-topos-theoretic. While cohomology theory can be cast in terms of 1-toposes equipped with extra (model-)structure, here the 1-topos-theory gets in the way more than it propels the theory.
I wish Johnstone would emulate Knuth and release smaller units of part 3 so we can at least get the two chapters (SDG and realizability toposes) that are finished.
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