Conjectures in Grothendieck's “Pursuing stacks”

I read on the nLab that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to this question,

Grothendieck conjectured, and Cisinski proved, that the class of weak equivalences in the Thomason model structure was the smallest basic localizer.

I am interesting in knowing what other conjectures made in PS have turned out to be true, or other "ideas" that have been successfully realized/formalized. Ideally it would be nice to include references to the relevant papers.

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I wish you'd called your question 'Pursuing "Pursuing stacks".' –  Tom Leinster Dec 5 '12 at 23:59
Nice one, Tom. :-) –  David Roberts Dec 6 '12 at 0:14
may be tag "open-problem" ? PS Nice question (and answer ) ! –  Alexander Chervov Dec 6 '12 at 5:33
@Tom 'Pursuing "Pursuing stacks"' is a shameless plug from B.Toën who wrote an article with this exact title. –  Niels Dec 6 '12 at 13:40
Niels: really? What genius! –  Tom Leinster Dec 6 '12 at 20:13

To dash off a quick answer, Pursuing Stacks is composed of (if memory serves correctly) three themes. The first was homotopy types as higher (non-strict) groupoids. This part was first considered in Grothendieck's letters to Larry Breen from 1975, and is mostly contained in the letter to Quillen which makes up the first part of PS (about 12 pages or so). Maltsiniotis has extracted Grothendieck's proposed definition for a weak $\infty$-groupoid, and there is work by Ara towards showing that this definition satisfies the homotopy hypothesis.

The other parts (not entirely inseparable) are the first thoughts on derivators, which were later taken up in great detail in Grothendieck's 1990-91 notes (see there for extensive literature relating to derivators, the first 15 of 19 chapters of Les Dérivateurs are themselves available), and the 'schematisation of homotopy types', which is covered by work of Toën, Vezzosi and others on homotopical algebraic geometry (e.g. HAG I, HAG II) using simplicial sheaves on schemes. This has taken off with work of Lurie, Rezk and others dealing with derived algebraic geometry, which is going far ahead of what I believe Grothendieck envisaged.

During correspondence with Grothendieck in the 80s, 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.

Edit: I forgot something that is in PS, and that is the theory of localisers and modelisers, Grothendieck's conception of homotopy theory which you mention, which is covered in the work of Cisinski.

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Dear David, thanks a lot for this answer! –  Adeel Dec 7 '12 at 9:49

I agree with Tim that calling Pursuing Stacks a "letter to Quillen" is erroneous, especially as Quillen never replied. Grothendieck also wrote: "This is written in English in response to a correspondence in English." At one stage he planned more volumes in French but it seems got diverted from this. I hope the following will be of help, in addition to David Robert's answer, to put the situation with regard to models for homotopy theory in context.

In a letter dated 02/05/1983 Alexander Grothendieck wrote to me: "Don't be surprised by my supposed efficiency in digging out the right kind of notions--I have just been following, rather let myself be pulled ahead, by that very strong thread (roughly: understand non commutative cohomology of topoi!) which I kept trying to sell for about ten or twenty years now, without anyone ready to buy'' it, namely to do the work. So finally I got mad and decided to work out at least an outline by myself."

But this question is about the homotopy theory of categories and the related question of "Why simplicial sets?", for which see also the good answers in March 2011 to Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?.

Dan Kan's first contribution to combinatorial homotopy was in terms of cubical sets. When he went to Princeton the disadvantages of cubical sets were found: cubical groups did not satisfy the extension property, and the geometric realisation of the cartesian product of cubical sets had the wrong homotopy type, while as shown by Moore and Milnor respectively, the situation was fine for simplicial sets. So they did not attempt to refine the cubical theory. In my doctoral studies at Oxford, 1956-59, the exposition was all simplicial, especially when Michael Barratt came back from Princeton in 1957.

However the 1960 book by Hilton and Wylie on algebraic topology was cubical, as were some 1962 notes of Federer from Brown University, a later book by Massey, and cubical sets continued to be found useful in various places. Our 2011 book on "Nonabelian algebraic topology" is almost entirely cubical, because of its emphasis on the use of higher homotopy Seifert-van Kampen theorems.

My 1965 intuition for using cubical sets was based on generalising the van Kampen theorem to higher dimensions. It seemed entirely reasonable that the above diagram could be expressed as: the big square is the composition of the little squares. C. Ehresmann's 1965 book on "Categories structuree" gave a definition of double categories which expressed this nicely. Indeed, I have answered this on mathoverflow as using matrix notation where $(a_{ij})$ denotes a composable array and $[a_{ij}]$ denotes the composite. So one has an easy definition of the $n$-fold cubes of the nerve of an $n$-fold category, except that there seems currently no name for the cubical type geometry underlying an $n$-fold category. Note that composable sequences of morphisms in a category are used in describing the nerve of a category, but it seems more difficult, at least for me, to define multiple compositions in simplicial or globular terms, although the simplicial nerve of an $n$-fold category is easily defined as an $n$-fold simplicial set.

By contrast, the singular cubical complex of a space, or filtered space, is ideally suited for the description of multiple compositions, using an array notation. I have already explained this in answer to this mathoverflow question.

So in considering what category in which to work the question of "what should be adequacy and convenience?" is crucial. It is as reasonable to ask this for combinatorial models of homotopy theory as it was in 1963 to ask it for categories for topology in my paper Ten topologies.

A property that was also required for the conjectured proof of a putative higher van Kampen theorem using homotopy classes of maps was the notion of "commuting cube", and that "any composition of commuting cubes is commutative". Chris Spencer and I found that the notion of "connection" for a double groupoid was good for this, and that it allowed an equivalence between crossed modules and single pointed edge symmetric double groupoids with connections. Then Philip Higgins and I found in 1974 the construction of the homotopy double groupoid of a pair $(X,A,x)$ of pointed spaces using homotopy classes of maps $I^2 \to X$ which take the edges to $A$ and the vertices to $x$. This gave the first homotopy fundamental double groupoid, which enabled the proof of a 2-dimensional van Kampen theorem, including the usual theorem for the fundamental group as a special case, not just an implication.

There are grounds for suggesting that simplicial sets are convenient, but are not entirely adequate, since they cannot easily express multiple compositions. On the other hand, cubical sets with connections are adequate for this test, but not entirely convenient! Andy Tonks proved that cubical groups with connection are Kan complexes. One reason for inconvenience is that although they have been shown to form a strict test category in the sense of Grothendieck, in the paper given here, the geometric realisation of the categorical product is only of the homotopy type of the product of the realisations, not actually homeomorphic to the product as in the case of simplicial sets. The latter homeomorphism property implies that in the right convenient category, the geometric realisation of a simplicial group is a topological group.

The cubical setup is also not sufficient to describe the geometry underlying $n$-fold categories, and indeed there seems currently no name for such a structure in which cubes have different types of faces in different directions. Yet Grothendieck remarked to me on my affirming Loday's theorem, that (strict) $n$-fold groupoids model weak homotopy $n$-types: "That is absolutely beautiful!"

So it as well not to assume we have the final story, and to investigate options!

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This is more a comment than an answer, but its length makes me post it as an answer. I want to react to what I have just read, for the first time, about "Pursuing Stacks" at the nLab, and the words used there as well as in your question. I find it extremely irritating when people only use words such as "ideas" and "conjectures" to describe the content of "Pursuing Stacks". It makes me wonder whether they have read it or not. The words "rigorous", "results" or "theorems" are not used unless they describe other people's work. This is perfectly true that you would find ideas and conjectures in "Pursuing Stacks", but it does not prevent many notions and results regarding them to be "rigorously worked". It absolutely blows my mind that, nearly thirty years after the writing of this text, people still talk about it in that disrespectful way. (Those who do not think this is disrespectful to credit Grothendieck with "ideas" and "conjectures", no matter how "deep" or "beautiful", while others get the credit for the rigorous results, can read "Récoltes et semailles", where this question is precisely addressed at length.)

This is nevertheless true that "Pursuing Stacks" contain many ideas and conjectures, and David Roberts's answer gives, to my very partial knowledge, an accurate rough overview of what has been developed since then. I would just mention the fact that Grothendieck did not view simplicial sets as more homotopically relevant than other test categories. As regards $\infty$-groupoids, his approach is purely algebraic. Therefore, while I certainly do not claim that one of the approaches is better than the others, the prevalent simplicial approaches do not seem to me to be the one advocated by Grothendieck.

EDIT 1 (10 December 2012): This "answer" has got one downvote and someone has canceled their upvote. Perhaps I should have explained more. I find the question and the answers given so far quite sad because they all point out the fact that almost nobody has read "Pursuing Stacks", and people mostly talk about it by word of mouth. If some people had read it carefully, then sure enough they would have cited accurately some of Grothendieck's results. What are these precises references in the literature? I only know that there was a working group organized by Bénabou in the eighties, devoted to Pursuing Stacks, soon after its sending, but it quickly came to an end, and that someone in the Netherlands asked a student, whose name nobody seems to recall, to work on test categories, which he did not pursue very far. There certainly has been a lot of work in "categorical homotopy theory" since the writing of "Pursuing Stacks", but have Lurie, Rezk, Toën, Vezzosi (all cited by David Roberts) read "Pursuing Stacks" (I certainly do not claim they should have)? I think one should be very careful before claiming that one's own work or other people's work is indeed related to the content of "Pursuing Stacks" and even more before stating that it "realize" (?) or "formalize" some "ideas" of "Pursuing Stacks". To some extent, this text is like the Bible: eveybody talks about it but almost nobody knows what is written there.

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I have just gone into the n-Lab and have tried to change the wording to reflect Jonathan's very valid points. I have only added a little and changed a little, but would suggest that the entry does need some more work. Does any one have typed out a table of contents of PS? Then linked comments with the future development would of the main themes might be useful. –  Tim Porter Dec 6 '12 at 15:45
I would like to remind Jonathan (or anyone else) that the nLab (ncatlab.org/nlab/show/HomePage) is a wiki, and that you should feel encouraged to modify anything that you find wrong or objectionable. (But) speaking on behalf of nLab regulars, I would like to ask those who work on nLab entries to make a note of their efforts at the nForum nforum.mathforge.org/discussions/?CategoryID=0, which also serves as a discussion board. Alternatively, you can initiate a discussion at the nForum of proposed changes to nLab entries, before you actually make those changes. –  Todd Trimble Dec 6 '12 at 16:23
@Jonathan: You can bet your last dollar that Ronnie Brown has read Pursuing Stacks. I believe Tim Porter has as well. –  Todd Trimble Dec 10 '12 at 19:15
I should say I have read the entirety of (the electronically available copy of) Pursuing Stacks, so apart from possibly faulty memory, I know what's in there. Toën mentions here ens.math.univ-montp2.fr/~toen/msri2002-2.pdf (on page 1) that what he describes is his understanding of the problem of schematisation of homotopy types as presented in PS. This note by Toën is just an outline of other work he has done. Maltsiniotis and Cisinski have both certainly read PS. I don't know if Lurie has, but he is in effect 'second generation' after Toën-Vezzosi. –  David Roberts Dec 11 '12 at 1:09
David, thanks for your comments. I hope you did not understand I was claiming nobody had read "Pursuing Stacks" and I am glad you have. But how many precise references to this text could one point out in the literature? It is true that some people present (I am not qualified to judge whether they are right or not) their work as solutions to questions raised by Grothendieck, but what has been written about the solutions brought by Grothendieck himself? This is something which I find very sad, but for many years most people seem to have been thinking that Grothendieck could not have (ctd) –  Jonathan Chiche Dec 11 '12 at 9:41