categories: Bill Lawvere

[Anders Kock <kock@math.au.dk>]

Bill Lawvere

As Bill’s first Ph.D. student, I feel that I should try to sketch the scientific atmosphere created by Bill  and others, and in which I grew up. I came to Chicago in 1963 as a Ph.D. student in algebraic topology, and on that occasion attendened lectures by Bill on  functorial semantics, and quantifiers as adjoints. I likewise recall lectures by Mac Lane on sheaf theory, where again adjointness was a pivot. Mac Lane had also brought Benabou to Chicago (or maybe that was not until 1967?). In my world picture, I felt that I had the privilege of seeing Category Theory, as a subject in its own right, come into being.

I left Chicago in 1964, without yet having my degree.  In 1966, Beno Eckmann had managed to organize a special category year at ETH in Zürich, where  also Bill was invited. So I went to Zürich to resume my Ph.D. studies, but now with Bill as formal advisor. I shared office with Bill in the house at Zehnderweg 13. This house was a lively place this year. Besides Bill, many other American category theorists  (Linton, Beck, Tierney, John Gray,...) were there , and also Lambek, and the local Fritz Ulmer, Michel André,...; Marta Bunge attended the seminars, she was at that time staying in the nearby Freiburg i.Br. (and her later thesis had Bill as a (co-)advisor).

The key scientific advance brought about by  the Zürich year was the idea  of triple, or monad (which also became crucial in much of my own later work). The Springer “Lecture Notes in Mathematics” vol 80 (1969)  (“Seminar on Triples and Categorical Homology Theory Theory”) records well the scientific significance of the “Zürich year”. Bill’s contribution to the volume has title “Ordinal sums and equational doctrines”.

I do not intend to comment further on Bill’s scientific work at that time, but rather describe how our lives, from 1966 and for a many years thereafter, intertwined.

As mentioned, Bill and I shared office in ETH in 1966 (perhaps better: Bill  shared his office with me). I would work there in the morning and until late in the afternoon, where Bill would come, and then for hours discuss mathematics with me, while smoking many Benson and Hedges cigarettes. He had a need to elaborate his ideas for an audience, like me.

  I think it was during this year that he met his later wife Fatima. Shortly  before I went to Zürich in 1966, I had met what was to become my wife, Hanne, in Aarhus, and therefore, while in Zürich, I travelled (by train) back to Aarhus several times.  Bill and Fatima had  left Zürich in December to go to Chicago. I also left Zürich (back to Aarhus) in the middle of December, got married on December 17, and a few weeks later, went to Chicago with Hanne to resume my studies with Bill.

Bill and Fatima had invited us to stay with them in a big house in Chicago which they had rented, so that is where Hanne and I spent our honeymoon. Our two families have been close friends since.  We met all four of us again in Halifax a few years later (we also brought our two small children along).

This brings the story back to mathematics. Dalhousie University in Halifax had organized a category year 1969-70, to be led by Bill; I had the privilege to participate.  Just as Zürich was the year of monads, the Halifax year was the year of elementary toposes, as initiated by Bill in 1969, and which was chrystallized and expanded in 1969-70 by him and Myles Tierney.

After returning to Aarhus in 1970,  I,   together with Gavin Wraith, used the notes that I had taken from Tierney’s talks in Halifax, as basis for a  seminar, resulting in our book Elementary Toposes (Aarhus Lecture Notes Series No. 30, September 1971). This book had some role in early dissemination of this elementary notion of topos, as did the collection edited by Bill , “Toposes, Algebraic Geometry and Logic” (Springer Lecture Notes in Mathematics 274, 1972), which was based on a conference in Halifax in early 1971.

I had by then already returned to Aarhus. Bill came as a visiting professor  to our math department in the academic year 1971-72.

Bill and I met later at conferences at various places. In particular, I should mention three “Open-House” events I was organizing in Aarhus, in May 1973, May 1978, and June 1983,  respectively. Because of the presence of Bill,  these events attracted several good category theorists. The meetings  were quite loosely organized, and each lasting a couple of weeks; at the first of these Open House events, Bill gave nine talks, between May 8 and May 29, mainly on topos theory. For the 1978 and 1983 Open Houses, published proceeding exist, entitled “Topos Theoretic Methods in Geometry”, and “Category Theoretic Methods in Geometry” ,  as Aarhus Various Publication Series no. 30 (1979) and no. 35 (1983), respectively. The 1978 event  included some accounts, by Bill and others, of some of the early  synthetic differential geometry, and in the proceedings is included his seminal 1967 Chicago  talk on categorical dynamics.  In the 1983 event, on Bill’s initiative, Alfred Frölicher participated, to present some of the early stages  in the development of the idea of the Cartesian closed category of Convenient Vector Spaces.

I shall not elaborate further on our later scientific and warm personal relationship, just mention that for several years, we made (transatlantic) phone calls  to each other on February 9, the birthday of Bill as well as of Hanne.



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