`Pursuing Stacks'

In 1983 a new Manuscript by Alexander Grothendieck entitled `Pursuing Stacks' (referred to below as [PS]) began to be distributed from Bangor, UK, and it eventually ran to almost 600 pages. Many publications refer to [PS], which is now being edited by G. Maltsiniotis , for publication by the Société Mathématique de France. This is the story of how this Manuscript came about.

In the 1970's, Philip Higgins and I had been developing a generalisation to higher dimensions of the van Kampen theorem on the fundamental group. This was published, with related material, in four papers in 1981, and required various notions of higher categories and groupoids.

In 1982 I came across a paper by Jack Duskin in which he referred to
Grothendieck's interest in *n*-categories. In the summer of that year
I was going to a conference in Marseille-Luminy, `Homotopie algébrique
et algebre locale', and so I wrote to Grothendieck sending him some offprints
and preprints, referring to Jack's comment, and asking if I could visit him
for discussions. He replied that he was out of things for some time, but
that, if I was interested, he could send copies of three letters he had written
in 1975 to Larry Breen. Of course I replied `yes!'. The letters were in French,
and two were typed and easy to read. The third was handwritten and not so
easy to decifer, especially as some of the mathematical terms were unfamiliar
to me. So I made a draft translation and took this typed version to
Marseille-Luminy, where it was corrected by Jean-Louis Loday, and separately
by Larry Breen.

This version I sent to Grothendieck with some pages of extra comments in relation to our work. I explained that .-groupoids as defined by us did not model all homotopy types, but only those which fibred over a K(G,1) with fibre a topological abelian group. Also, .-groupoids came in a variety of geometric forms, corresponding to discs, globes, simplices and cubes, but all algebraically, though highly non trivially, equivalent.

He was surprised and delighted by my interest. He also reacted to the limitations of the structures we considered, and returned to develop the ideas outlined in the letters to Breen, on weak ¥-categories as models of all homotopy types, and on the intended applications. The result, over time, was [PS], written, as he explains, in English as a response to a correspondence in English, although [PS] develops principally mathematical themes other than that of multiple categories.

Our sympathy developed with the correspondence, which also included Tim Porter, and he early insisted on using first names. I had asked if I could duplicate anything he sent, and he agreed to this. Consequently, as parts of [PS] arrived (they were also sent to Larry Breen), I sent them to a few people. As the MS circulated, others joined in corresponding with Grothendieck. Thus, through correspondence, he returned to public mathematical life, and [PS] has become increasingly influential. For some aspects on homotopy theory, see the home page of Maltsiniotis; the 2002 thesis of his student Cisinski was strongly influenced by [PS].

[PS] is sometimes referred to as `a letter to Quillen'. This `letter' part is actually a kind of prelude to Pursuing Stacks proper; it refers to the Bangor group as being prepared to give time to foundations, but lacking in geometry; for a later modification to his views on the Bangor work, see letter 17, dated 09/06/03, of the 69 letters between us.

I also sent Grothendieck a copy of a failed Research Grant Proposal for the
then Science Research Council (proposers Tim Porter and me): we wanted to
develop his general programme, as we then understood it, in a wide ranging
way and with the help of a number of international collaborators. Perhaps
this suggested his preparing for the CNRS in 1984 his own proposal,
`Esquisse d'un
programme'. This also has had a wide influence, as shown for example
by publications of
Leila
Schneps and Pierre Lochak. In Esquisse, Grothendieck refers to our
correspondence as `*a baton rompu*'. This
translates literally to `as a bent stick', and colloquially means a conversation
`ranging over this and that', with wayward changes of direction.

I visited him at his house in 1986, and on a drive from there to Montpellier
I finally convinced him that I had been explaining in letters that
*n*-fold groupoids modelled all weak homotopy *n*-types (Loday's
theorem). He exclaimed: `That is absolutely beautiful!'. (My work with Loday
started in 1981 and took a lot of my attention from then on, which was one
reason why I did not follow up many of the ideas in [PS].)

In 1991, I had a short and engaging letter in response to a postcard sent to him from Iona, and this mentions a serious reconsideration of some of the issues in `Pursuing stacks'.

He could be totally absorbed in mathematical ideas. One letter remarks that he is thinking of a new theory of form, and has to remind himself to eat and to sleep.

He had a strong interest in detail and small things. He was against `*le
snobisme*', and so was delighted with a comment of Henry Whitehead: `It
is the snobbishness of the young to suppose that a theorem is trivial because
the proof is trivial.' He wrote that mathematics was held up for centuries
for lack of the `trivial' concept of zero. One of his striking phrases was
`the difficulty of bringing new concepts out of the dark'. He thought
speculation to be an essential creative activity
(see letter 14).

The correspondence has been deposited with Maltsiniotis, who intends it as an Appendix to the published [PS]. About half has been typed, up to the end of 1983, and comes to about 88 pages. Hans Baues remarked once after he came out with an idea: `I would not have thought of that if I had not been reading Grothendieck!'. He was referring to Pursuing Stacks! I hope these publications will encourage many.

For more on the life of AG, see the AMS Notices articles by Allyn Jackson Part I Part II Nov 2004.

Ronnie Brown

Last changed February, 2005

Additional note (April 19, 2005):

Some minor corrections have been made. On the suggestion of Allyn Jackson,
this article was submitted to the Notices of the American Mathematical Society,
but they decided not to use it.

Additional note: (June 17, 2005)

A brief extract on `snobbisme' and triviality will appear in the Math.
Intelligencer.

A short email explaining the origin of `Pursuing stacks' will appear
in the AMS Notices for September, 2005.

Additional note: (March 23, 2006)

I asked Mikhail Kapranov about the influence of Pursuing Stacks and Esquisse d'un Programme on Soviet mathematics in the 1980s. He replied:

March 21, 2006

Dear Ronnie,

From what I remember, Gelfand advocated reading both Esquisse and Pursuing stacks. Voevodsky was very interested in both anabelian geometry and higher stacks. Drinfeld was influenced by Esquisse in his paper on "Drinfeld associator" and a version of the Grothendieck-Teichmueller group appearing in the theory of quasi-Hopf algebras. This is probably the most serious influence on Soviet mathematics of the period.

Best,

Mikhail

May 26, 2006

Two further quotes from the correspondence are in the article `Analogy, concepts, ...', by Ronnie Brown and Tim Porter, one on zero, and another on working towards understanding, without worrying about its impact.

Additional note: October 6, 2006 You can read extracts from two letters to Ronnie Brown: one on speculation , and one on his later reaction to geometry at Bangor. Pursuing Stacks is due to be published in Documents Mathematiques, and G. Maltsiniotis' book `La theorie de l'homotopie de Grothendieck' Asterisque, 301, 2005, develops the homotopical ideas in Pursuing Stacks.

Additional note: November 27, 2006

One comment of AG was:

......people are accustomed to work with fundamental
groups and generators and relations for these and stick to it, even in contexts
when this is wholly inadequate, namely when you get a clear description by
generators and relations only when working simultaneously with a whole bunch
of base-points chosen with care-or equivalently working in the algebraic
context of *groupoids*, rather than groups. Choosing paths for connecting
the base points natural to the situation to one among them, and reducing
the groupoid to a single group, will then hopelessly destroy the structure
and inner symmetries of the situation, and result in a mess of generators
and relations no one dares to write down, because everyone feels they won't
be of any use whatever, and just confuse the picture rather than clarify
it. I have known such perplexity myself a long time ago, namely in Van Kampen
type situations, whose only understandable formulation is in terms of
(amalgamated sums of) groupoids.

This use of the fundamental groupoid on a set of base points was a main message of my book on Topology published in 1968, republished and expanded in 1988, and now available as Topology and Groupoids (2006). A survey on groupoids is also downloadable, and this leads to Higher Dimensional Group Theory.

Additional note, 23 September, 2007

I have put some other Grothendieck quotes in the article `Promoting mathematics' (also as pdf file of printed article for MSOR).

I first heard of Grothendieck when I was a raw DPhil student under Henry
Whitehead in 1958, and was working with Dick Swan on writing notes of Dick's
lectures on sheaves, which were based on Grothendieck's Tohoku paper. At
the 1958 ICM at Edinburgh, I met Raoul Bott informally. He said that AG was
extraordinary as he could play with concepts, and also worked very hard to
make things almost tautological! Later I heard this amazing person, only
7 years older than I, give a forceful long talk, arguing with Serre! 25 years
later I found myself encouraging him to write more of his thoughts on
*n*-categories and homotopy theory.

6 January, 2008

Another nice article is: Piotr Pragacz, `The Life and Work of Alexander Grothendieck', American Mathematical Monthly, November, 2006, 831-846. He does not mention `Pursuing Stacks', nor `Esquisse d'un programme', which are likely to be the most influential of Grothendieck's works of the 1980s.

29 January, 2008

Here are his intentions for `A la Pursuite des Champs', extracted from his much broader Esquisses d'un Programme'. Perhaps this will help others to follow through these ideas.

He finds the structures we considered up to that time inadequate for all homotopy types, as I had explained early on to him, and it was only later in 1985 or 1986 on the occasion of a visit to him that I convinced him that `strict n-fold groupoids model homotopy n-types' at which he exclaimed `That is absolutely beautiful!'. On the other hand there is a lot to say about a `linear' model as one step towards coefficients for a nonabelian cohomology. This is the subject of a book in preparation, titled Nonabelian algebraic topology, in accordance with the idea that cohomology (at least of groups) is a special case of cohomology of spaces. There is a lot to do in this area, with these tools.

Here is his response (12/04/83) to a question:

The question you raise ``how can such a formulation lead to computations'' doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand -- and it always turned out that understanding was all that mattered.

24 February, 2008

It may also be useful to refer to John Baez TWF 35 and compare it with our book plans on "Nonabelian algebraic topology ". Also do a web search on "Nonabelian cohomology", and look at recent papers on the arXiv by J. Faria Martins (some with collaborators) who is really using the crossed complex technology, including the Higher Homotopy van Kampen theorems. I continue to be puzzled by the general lack of reference to these theorems in algebraic topology! Grothendieck saw these theorems as `integration of homotopy types', though never used them. I suppose many workers in algebraic topology find it difficult to believe in `higher homotopy groupoids', as not part of the canon. Also the theory is substantial, fitting together in an intricate way, which is not so easy to get into. It was certainly not so easy to work it out, and to make real the intuitions of 1967! I have elsewhere acknowleged the contributions of those who made this possible.

23 December, 2008

Georges Maltsiniotis writes on 21 December,

"As you probably know there is a conference in January at the IHES celebrating the 80th birthday of Alexandre Grothendieck, where I am invited speaker. I plan to speak about the work of Grothendieck on homotopy theory based on "Pursuing Stacks" and the "Dérivateurs", and I will announce the forthcoming publication of "Pursuing Stacks" and the correspondence by the SMF."

He was definitely interested in n-categories. When I was in Paris in 1967 and Giraud was advising me on my post-doc, Giraud gave me the full definition in the strict case. I remember asking Grothendieck in 1973 when we he was in Buffalo if the non-strict n-dimensional case would be needed. He laughed and replied, "I hope not!". In my 1966 post-doc Grothendieck got me to Strassburg and Verdier where I learned some algebraic geometry but I was most interested in categories. I got to know him quite well when we got him to Buffalo in 1973 and 1974. The non strict 2-dimensional case is due to Benabou.

Jack Duskin , 23/12/08

February 11, 2009

The area of higher dimensional category theory has now being given a wiki type exposition at the ncatlab.

I would also like to point out that the notion of orbit groupoid and the determination of the fundamental group of an orbit space under a discontinuous action of a discrete group is given a full exposition in Topology and Groupoids. This implies earlier work of Armstrong and Rhodes. Perhaps someone would like to look at versions of this theory for pro-groups and for toposes, seeing as work has been done on van Kampen theorems in these contexts (search on the arXiv).

February 15, 2009

The wikipedia entry on Grothendieck is also helpful.

March 25, 2009

See a video of G. Maltsiniotis' lecture Grothendieck et l'algèbre homotopique, where he talks on 'Pursuing stacks' and 'Les dérivateurs', at the conference devoted to "L'héritage de Grothendieck" held at IHES at the end of January, 2009.

February 13. 2010

In view of a letter from Grothendieck dated Jan 3, 2010, in which he asks for publication of all material written by him to be stopped (full details of the letter and discussion may be found at

http://sbseminar.wordpress.com/2010/02/09/grothendiecks-letter/#comment-7842 )

I want to emphasise that he gave full permission for anything he sent to me to be circulated, as stated above. The letter gives no reason why he wishes this stopping of publication.

February 16, 2010

It should be remembered that Alexander applied for, was granted and accepted a 2 year Research position from the CNRS on the basis of the proposal in Esquisse d'un Programme, and this carried a good salary and I believe pension, this position thus not being available to others. Various further works were promised in this proposal, in particular continuing the programme of `Pursuing Stacks'. Since such further work by him never appeared, the legal and moral obligations of his research proposal and of his contractual periods of university and scientific employment should be considered; from my own point of view, I will thus make freely available as much of his writings from those periods as is possible, especially those for which he gave express permission to distribute.

Return to `Higher dimensional group theory'

Return to Ronnie Brown's home page.