Alexander Grothendieck: some recollections

I first saw Alexander Grothendieck (AG) when as a raw research student I attended his lecture at the 1958 Edinburgh ICM.

I knew of his Tohoku paper as this was a source for Dick Swan’s 1958 lectures at Oxford on “The Theory of Sheaves”, where I was charged with working with Dick on writing up the notes. Now I saw this amazing figure telling Serre and others what was what! I also heard a comment of Raoul Bott: “Grothendieck was extraordinary as he could play with concepts, and also was prepared to work very hard to make arguments almost tautological.”; which sounded good to me.

Of course over the years I heard much more, and indeed organised at Bangor a Meeting on Toposes in 1972. I knew of his surprising departure from IHES in 1970.

In early 1982 I read a paper by Jack Duskin which referred to Grothendieck’s interest in \(n\)-categories. I was at that time concerned about very negative views expressed in the UK on my work with Philip Higgins on higher groupoids in algebraic topology, and on plans to develop this work with Jean-Louis Loday. Indeed such views did not much change till in 1984 I was able to report on the work with Loday on the nonabelian tensor product.

Since I had planned to go to a conference in Marseilles-Luminy in June, 1982, I wrote to AG with a load of offprints and preprints explaining that I had heard of this interest and wondered if I could visit him to discuss it.

He wrote back to say that he was very out of things, so a meeting may not be useful, but that he did write three letters to Larry Breen in 1975 and would I be interested in seeing them. Of course I replied in essence “Yes please”. The letters were in French, two of them typed, and the third 40 pages of handwritten A5 paper not so easy to decipher. So I translated and typed it as best as I could, and checked it out with others at the conference and with Larry Breen.

AG became enthusiastic about the idea of \(\infty\)-groupoids, and also about potential applications of a 2-dimensional van Kampen Theorem to compactifications of modular toposes of complex curves. I have not been able to pursue the latter ideas.

In October I sent him a comment explaining that our strict \(\infty\)-groupoids modelled only very restricted homotopy types, which I now call “linear homotopy types”, since there is no quadratic or higher information, such as Whitehead products. This led him to start thinking hard about various weak versions of \(\infty\)-groupoids, and their value in modelling homotopy types.

Gradually, our correspondence developed and became very friendly, and he eventually signed himself as “Yours affectionately, Alexander”. Tim Porter also joined in the correspondence, which continued in a friendly even chatty tone.

I early asked him if I could circulate any of the correspondence, and he was happy with that. This circulation led to others getting in contact with him.

In the end, my correspondence with him amounted to about 78 letters.

In early 1983, I and Larry Breen received the first part of the manuscript called “Pursuing Stacks”, “written in English in response to a correspondence in English”, and prefaced by a “letter to Quillen”, which was in fact never answered. I duly circulated this work to a few people. By the end of June, the manuscript had amounted to over 600 pages! It was written in the form of a diary, typed but with handwritten corrections. AG insisted that if it was to be published, it should be “as is”, since he felt that young people should be aware that even well known people make mistakes. I am glad to say that Pursuing Stacks is being edited by Georges Maltsiniotis and is to be published by the Société Mathématique, together with his correspondence with various people in these years.

Alexander had clearly hoped I would work on some of this. But in April 1982 Jean-Louis Loday and I came up with the notion of nonabelian tensor product of groups as arising from some pushouts of crossed squares, and realised that this put the importance of a complete proof of our conjectured van Kampen type theorem on a higher urgency.

In 1985 I has arranged a visit to Toulouse, and asked Alexander if I could visit him on the way. He picked me up at Orange rail station and took me back to his home, Les Aumettes, out in the country, and a long way from Montpellier where he worked. He showed me to the room where I would sleep, and hoped I would not be disturbed by a Buddhist shrine in the room. I said that was no problem. He had kindly bought some sausages for my meals, as he was a strict vegetarian. Meals were preceded by a Buddhist chant, which sounded very good to me. While it was light, he went to the garden and chopped up some vine roots for the stove.

The following day he drove me out into the mountains for a picnic lunch, and then back to his house for some supper. I regret that my photographs of this were jinxed by being overlapped with those of my following visit!

We discussed many things over the two evenings. He showed me a beginning draft of “Recollte et Semaille”. I sometimes wonder if I could have used more nous to dissuade him from some of what seemed to be some attacks on people. But I had little knowledge of the people and of the background, and did not want to argue with my kind host.

He was not so keen to discuss mathematics, but I did try to steer the conversation round to that. One comment stuck in my mind when he claimed he could compute the “Teichmüller Groupoid” seemingly by induction on the handles, using clutching functions. I have never understood this, but it led to thinking about what might be higher groupoid structures for symmetries of graphs, and so for the notion of symmetry for the topos of graphs and morphisms. This idea was helpful to a joint supervision with Chris Wensley of the thesis of John Shrimpton on symmetries of directed graphs.

The drive

The next day we drove to Monpellier for my lecture. Alexander had a somewhat battered estate car, avoided motorways and drove fast. We picked up two hitchhikers who must have been amazed at the conversation in the front, as I was going on about double groupoids. Finally we arrived outside the Department, a little late, and had the following conversation: “Do you mean to say that \(n\)-fold groupoids model homotopy \(n\)-types?” “Yes.” “I suppose you have said that at some time in your letters?” “Yes!” “But that is absolutely beautiful!”
Alexander dressed very informally, with an open crocheted sweater, and liked to talk informally to the students at Montpellier. At some time he remarked: “There is something enervating about the atmosphere of a great research institute.” Perhaps this is some background to his departure from IHES in 1970, and his then apparent rejection of mathematics. This rejection was taken quite seriously by the mathematical community, and yet it is clear that in his time at Montpellier he was writing many thousands of pages of mathematics. He even sent 2000 pages of notes on his approach to Galois theory to two USA mathematicians, who in the end left them in storage!

For a few years 1982–1991 he was in contact with many mathematicians.

In letters I had explained that Tim Porter and I had grant applications turned down. In 1984 Alexander submitted a grant application under the name “Esquisse d’un programme”, and circulated it to a number of people.

In 2006 I asked Mikhail Kapranov about the influence of these writings in the Soviet Union. He responded: “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.”

In 1986 AG wrote a reference for me which included the following:

“This programme (which I have started pushing through in the volume 1 of
“Pursuing Stacks”) has some substantial overlap with R. Brown’s. Getting
aware of this was the starting point, in 1982, of a very stimulating correspondence between R. Brown and myself, which has been continuing till now. It is
this correspondence mainly, and the friendly and competent interest of Ronnie
Brown in mathematical ramblings, which was the decisive impetus to take up
again and push ahead some of the old ponderings of mine, materializing in the
writing up of “The Modelizing Story” (the volume alluded to above).”

The later story

Gradually, Alexander’s interests turned to dreams and religion, and away from mathematics, so that much of the programme described in “Esquisse” was not pursued by him, though it has been a great stimulus to many, as has Pursuing Stacks. In April 1991 I sent him a postcard from Iona, and he responded in a friendly way, even saying he had taken up mathematics for five months after a gap of four full years; there was also a hint that he expected some end of all things. To my surprise, in mid 1991 he took himself away from the world.

In 2004, reading of comments of many on AG, and on Pursuing Stacks, I realised that many did not know its origins, that I had a kind of treasure in this correspondence, and so sent some of it to Georges Maltsiniotis. He found it very interesting, so in the end I sent him the complete set of originals.

Conclusion

Others have written comprehensively on Alexander’s mathematical achievements. From his letters he also comes across as great writer, with a feel for the rhythm of the English language.
He thus seems to come under Shakespeare’s words: “…as imagination bodies forth the forms of things unknown/ the poet’s pen turns them to shapes, and gives to airy nothing/ a local habitation and a name.” He also made comments on methodology, for example on snobism, speculation, and how specific computations came out of  “understanding”.

It was a thrill to correspond in this way, and I hope that the eventual availability of the correspondence will enhance the picture of Alexander Grothendieck.

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