INTERNATIONAL CONFERENCE IN CATEGORY THEORY: CT2009 Dedicated to the 100th Anniversary of Saunders Mac Lane University of Cape Town, 29 June - 4 July 2009

Saunders Mac Lane: personal reminiscences Ross Street Friday 3 July 2009

Introduction

Before the Conference I volunteered to speak about Saunders Mac Lane for 10 to 15 minutes. Then, on the Wednesday, we heard at Stellenbosch an excellent encapsulating quote on his stellar status and achievements.

One can learn from Wikipedia that his unused first name was Leslie but his parents early came to dislike it. Also it is claimed that the reason for the space in his surname was that his wife Dorothy found it easier to type that way. So I decided to present some personal memories in the hope this would add to the individual experiences of the audience. The format I took was that of a play in one act and four scenes. I believe Saunders would have approved of my wide colourful tie.

Scene 1: Urbana, Illinois; October or November 1968

Imagine me at the beginning of a postdoctoral fellowship at the University of Illinois, experiencing my first time away from Australia. The Mathematics Department was made up of 145 professors and 450 graduate students. Having just turned 23, I saw snow for the first time and everything was new, even the way of eating. I was excited about giving a talk at the Meeting of the Midwest Category Seminar. The night before the talk the telephone rang. It was John Gray asking me to walk over to his house where some category theorists had just had supper. Before this, apart from students, the only category theorists I had met were Max Kelly and John Gray. What sticks in my memory of that night is Myles Tierney and Jon Beck having a lively discussion with Gray about “Kan extensions” — a term I had not heard used although I knew of Kan’s paper on adjoint functors and had constructed for myself the adjoints to restriction functors. Tierney walked back to campus with me and quizzed me about what Kelly was doing on relative homological algebra. I think Fred Linton and K.T. Chen were at that dinner too: Mac Lane was not.

The time came for my talk. If there had been a talk before mine, I do not remember anything about it! I spoke on my University of Sydney thesis “Homotopy classification of filtered complexes”. I noticed Beck and Tierney in the back: then, enter Saunders Mac Lane who sat up front. After the lecture Beck and Tierney asked: “Why not have a Tor version of my result.” I explained, leaving them seemingly unconvinced. However, between talks, Mac Lane came to me and said in a patriarchal way: “Let’s take a walk.” He gave me some tips on conference lecturing. The main thing: “Don’t try to hit the audience with your best result right at the start; indeed, if at all!”

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What I did not know at the time and only learnt many years later from Sammy Eilenberg was that Eilenberg and Mac Lane were the two external examiners for my PhD thesis and they would have been writing their reports around the time of that Midwest Category Seminar. Max Kelly had managed to keep it a secret from me.

What it means is that Mac Lane knew all about my thesis. To my mind the episode shows a lot about Saunders: his formality combined with concern for the new generation.

It is 120 miles from Champaign-Urbana to Chicago. Mac Lane encouraged me to visit regularly. We talked about triangulated categories which, on his request, I followed up in seminars. He pushed me to the limits of my knowledge, never giving me the feeling he was satisfied. This extended me to think more about the usefulness and justifications of the subject. In a related vein, Saunders lovingly encouraged his wife Dorothy, whose health was slowly degenerating, to do as much as she could, including preparing meals for home-meal-hungry graduate students and visitors such as me.

Scene 2: Bowdoin College, Maine; 40 years ago, including the first Moon landing.

Most of the world’s category theorists, postdocs and graduate students attended this 1969 NSF Summer School on Categories. For six days per week over the whole summer, Mac Lane started the day with a two-hour lecture. This was the second run- through for Saunders of his book, later controversially called “Categories for the Working Mathematician”. (The first run-through was at the Australian National University in Canberra, Australia, during the previous southern hemisphere summer – January 1969 – which Bob Walters attended.) During Mac Lane’s 100 hour lecturing marathon I remember him faltering only once. It was at the end of the two hours where he wanted to prove that the left Kan extension of a finite-product-preserving functor into the category of sets was finite-product preserving. This of course concerned Lawvere theories. Overnight I wrote out a proof using the universal property of pointwise Kan extensions, carefully avoiding coends since I was unsure whether Mac Lane knew about them. Meanwhile Mac Lane’s student Eduardo Dubuc also wrote down the proof in terms of coends since he was a very impressed by the papers of Day and Kelly on completeness in enriched categories involving ends and coends. Dubuc and I submitted our pieces of paper to Mac Lane early the next morning, hoping to be helpful. I got the distinct impression that Mac Lane would use Dubuc’s version in his lectures. As it turned out, Mac Lane used neither: I think he used a technique from Lawvere’s thesis.

At the end of Mac Lane’s first two-hour lecture, he wrote up a list of topics for small sequences of “proseminars”. Then he called for all the postdocs, including Bob Paré and me, to come to the front of the room. We had to choose one of the topics and conduct the proseminar for the graduate students. I chose homological algebra. Then I met with all the graduate students interested in that topic. I wrote up a list of ten or so topics (culminating in spectral sequences) and assigned grad students to speak on them. I was very familiar with Mac Lane’s book “Homology” and we used that as text. The students were dedicated and would pass the book to the next speaker to prepare a talk. Thanks to this positive student attitude the proseminar succeeded and

2 we covered all the topics in order. As an “alien” I had been promised no NSF funding. However, soon after the Summer School, I received a cheque for $600 for my contribution. I am convinced Mac Lane was behind that.

It is interesting at this point to reflect on the different natures of Eilenberg and Mac Lane. Eilenberg lived in the penthouse apartment on top of the 16 storey dorm in which we younger people were staying. He wanted to be part of this crowd, joining us socially. Mac Lane stayed more aloof, off campus. Yet he gave all those lectures! I believe he went sailing on Sundays. Of course, Mac Lane was married and Eilenberg not.

Towards the end of the summer, one of the graduate students presented Saunders with a multicoloured lairy wide tie she had made with many little mirrors sewn into it. He wore it with pride.

Scene 3: New Orleans, Louisiana; academic year 1969–1970.

Tulane University was having whole academic years on different fields of mathematics. September 1969 began its “Year on Category Theory”. Perhaps the semigroupists formed the strongest permanent group in the Mathematics Department with Alf Clifford, Paul Mostert and Karl Hofmann, but there were quite a few other good people such as Frank Birtel, John Dauns and the abelian groups guru László Fuchs. A huge number of category theorists visited during the year. Jack Duskin and I were appointed for the year as assistant professors to teach at Tulane’s Newcombe College for young southern ladies; we each had an office in both the College and Tulane’s Math building Gibson Hall.

Mac Lane visited for the first semester and gave his third run for his book “CFTWM”. His student Dubuc and I became greater friends and, encouraged by Saunders, wrote a joint paper while Dubuc finished his thesis very much built on Day-Kelly. Mac Lane’s lectures were always dynamic and I enjoyed the reinforcement of the material on hearing it again in a smaller group.

I believe John Gray visited twice that year. He lectured on a closed structure on the category of 2-categories. Someone asked why he called the internal hom for the closed structure “Fun”. I interjected that it was obviously something to do with functors, to which John jokingly snapped back: “No Ross! It’s because it really is fun!” More seriously, Gray speculated that this might be a genuine case of a closed category in the sense of Eilenberg-Kelly which was not monoidal. Jack Duskin protested immediately. Jack had been in Paris and knew many things from the Grothendieck and Ehresmann Schools; in particular he had learnt things from Bénabou about morphisms of bicategories. Saunders picked up on the protest and agreed with Jack, insisting that the three of them have a night out on New Orleans cuisine to settle the matter. I regret that I was not forceful enough to suggest that I should tag along that night. The next day Gray was convinced that the tensor product did exist. He later wrote the construction in his Springer Lecture Notes in Math volume 391, although that book does not prove the Stasheff-Mac Lane pentagon which had to await Gray’s later beautiful paper in which the braid groups may have first shown their face in category theory. It is a bit ironic that what is now called the (lax) Gray tensor product had this history.

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Mac Lane’s lecturing was replaced in the second semester by that of Zdeněk Hedrlín and Bernhard Banaschewski.

Soon after Mac Lane left Tulane he invited Brian Day to Chicago to teach and research for two years. Brian told me he is very grateful he had this opportunity to travel abroad to a fine University.

Scene 4: Category Theory Conferences: a few examples.

In the 1970s, by some quirk, Max attended these mid-year meetings in odd years and I in even: typically it was Oberwolfach plus somewhere else (like the Isle of Thorns or Louvain-La-Neuve). It was a privilege and pleasure to be with both Eilenberg and Mac Lane in those settings.

One evening at the piano in Oberwolfach, Saunders Mac Lane and Peter Freyd performed an (unrehearsed I am sure) operatic duet to the script of Cartan-Eilenberg’s “Homological Algebra”. Saunders’ booming musical voice began: “Let lambda denote a commutative ring . . .” during which Peter would equally boomingly cut in with another dramatic quote. Great fun indeed.

On an equally cultural note, at the end of these conferences, Saunders would recite a poem he had written mentioning something on each talk at the meeting.

Let me fast forward to the Isle of Thorns in 1985 where I first presented my work with André Joyal on braided monoidal categories. Mac Lane’s coherence theorem for symmetric monoidal categories had used the disjoint union of the symmetric groups as a monoidal category. We had modified this for the braid groups, introducing some geometry in place of combinatorics. I was disappointed that Mac Lane seemed quite tired during my talk. Perhaps he did not think it too different from things he had done. Eilenberg had already collaborated with me on higher rewriting systems and after my lecture became very excited, telling me about his pointed braid monoids in rewriting theory. Certainly by 1987 at Louvain-La-Neuve, Mac Lane too was enthusiastic about braidings. Before my lecture I remember listing the people who had contributed to the material I was speaking about. Sammy protested to me that he did not want to be part of any “team”. To the best of my memory that conference was the last time I saw Eilenberg and Mac Lane together. While we are celebrating Mac Lane’s 100th, observe that Eilenberg would now be 95.

At the 1991 Montréal CT, Mac Lane supported Joyal’s and my efforts to get our paper “An introduction to Tannaka duality and quantum groups” into the Como volume SLNM 1488. As can be the case when working with André, we were well behind the deadline: for example, despite many drafts our “first paper” on braided monoidal categories was still to be published. Mac Lane was still a strong and assertive figure at this conference, asking questions and making comments at the ends of talks. In particular, he made it clear he did not like lots of overhead transparencies.

In 1995 at Halifax, I recall that Saunders went on the Conference outing involving a hearty walk. He was still pushing to understand the connection between quantum

4 groups and physics: “I see lots of little q’s around, but where is the quantum mechanics?”

By 1999 in Coimbra, aged nearly 90, he had to opt out of the walk half way through. Yet he was still strong enough to amaze the Fado musicians by thundering a song on the zeros of the Riemann zeta function!

Peter, the son of Hungarian Australian mathematicians George and Esther Szekeres, told them as they turned 90 that, according to statistics, their life expectancy was 2 years. But not only that it remained two years for that whole decade! Saunders died on 14 April 2005.

Symbols that come to mind for Saunders Mac Lane are

∃ ! in connection with his contribution to our understanding of limits and because we are so grateful that he existed and was unique.

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