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From: Anders Kock
To: "categories@mta.ca"
Subject: categories: Bill Lawvere
Date: Wed, 8 Feb 2023 15:39:25 +0000
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Bill Lawvere
As Bill=92s first Ph.D. student, I feel that I should try to sketch the sci=
entific atmosphere created by Bill and others, and in which I grew up. I c=
ame to Chicago in 1963 as a Ph.D. student in algebraic topology, and on tha=
t occasion attendened lectures by Bill on functorial semantics, and quanti=
fiers 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 C=
hicago (or maybe that was not until 1967?). In my world picture, I felt tha=
t I had the privilege of seeing Category Theory, as a subject in its own ri=
ght, come into being.
I left Chicago in 1964, without yet having my degree. In 1966, Beno Eckman=
n had managed to organize a special category year at ETH in Z=FCrich, where=
also Bill was invited. So I went to Z=FCrich to resume my Ph.D. studies, b=
ut 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, ma=
ny other American category theorists (Linton, Beck, Tierney, John Gray,...=
) were there , and also Lambek, and the local Fritz Ulmer, Michel Andr=E9,.=
..; 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 Z=FCrich year was the idea=
of triple, or monad (which also became crucial in much of my own later wor=
k). The Springer =93Lecture Notes in Mathematics=94 vol 80 (1969) (=93Semi=
nar on Triples and Categorical Homology Theory Theory=94) records well the =
scientific significance of the =93Z=FCrich year=94. Bill=92s contribution t=
o the volume has title =93Ordinal sums and equational doctrines=94.
I do not intend to comment further on Bill=92s scientific work at that time=
, but rather describe how our lives, from 1966 and for a many years thereaf=
ter, 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 la=
te in the afternoon, where Bill would come, and then for hours discuss math=
ematics 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 Z=FCrich in 1966, I had met what was to become my wife, H=
anne, in Aarhus, and therefore, while in Z=FCrich, I travelled (by train) b=
ack to Aarhus several times. Bill and Fatima had left Z=FCrich in Decembe=
r to go to Chicago. I also left Z=FCrich (back to Aarhus) in the middle of =
December, got married on December 17, and a few weeks later, went to Chicag=
o 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. Ou=
r 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 privile=
ge to participate. Just as Z=FCrich was the year of monads, the Halifax ye=
ar was the year of elementary toposes, as initiated by Bill in 1969, and wh=
ich was chrystallized and expanded in 1969-70 by him and Myles Tierney.
After returning to Aarhus in 1970, I, together with Gavin Wraith, used t=
he notes that I had taken from Tierney=92s talks in Halifax, as basis for a=
seminar, resulting in our book Elementary Toposes (Aarhus Lecture Notes Se=
ries No. 30, September 1971). This book had some role in early disseminatio=
n of this elementary notion of topos, as did the collection edited by Bill =
, =93Toposes, Algebraic Geometry and Logic=94 (Springer Lecture Notes in Ma=
thematics 274, 1972), which was based on a conference in Halifax in early 1=
971.
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 sho=
uld mention three =93Open-House=94 events I was organizing in Aarhus, in Ma=
y 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 f=
irst of these Open House events, Bill gave nine talks, between May 8 and Ma=
y 29, mainly on topos theory. For the 1978 and 1983 Open Houses, published =
proceeding exist, entitled =93Topos Theoretic Methods in Geometry=94, and =
=93Category Theoretic Methods in Geometry=94 , as Aarhus Various Publicati=
on Series no. 30 (1979) and no. 35 (1983), respectively. The 1978 event in=
cluded some accounts, by Bill and others, of some of the early synthetic d=
ifferential geometry, and in the proceedings is included his seminal 1967 C=
hicago talk on categorical dynamics. In the 1983 event, on Bill=92s initi=
ative, Alfred Fr=F6licher participated, to present some of the early stages=
in the development of the idea of the Cartesian closed category of Conveni=
ent Vector Spaces.
I shall not elaborate further on our later scientific and warm personal rel=
ationship, just mention that for several years, we made (transatlantic) pho=
ne calls to each other on February 9, the birthday of Bill as well as of H=
anne.
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