From mboxrd@z Thu Jan 1 00:00:00 1970 X-Spam-Checker-Version: SpamAssassin 3.4.4 (2020-01-24) on inbox.vuxu.org X-Spam-Level: * X-Spam-Status: No, score=1.8 required=5.0 tests=DATE_IN_PAST_03_06, LOCALPART_IN_SUBJECT,RCVD_IN_MSPIKE_H2 autolearn=no autolearn_force=no version=3.4.4 Received: (qmail 2183 invoked from network); 8 Feb 2023 19:53:49 -0000 Received: from smtp2.mta.ca (198.164.44.75) by inbox.vuxu.org with ESMTPUTF8; 8 Feb 2023 19:53:49 -0000 Received: from rr.mta.ca ([198.164.44.159]:41728) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1pPqV0-0008Ip-7v; Wed, 08 Feb 2023 15:52:58 -0400 Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1pPqU4-0001AS-LD for categories-list@rr.mta.ca; Wed, 08 Feb 2023 15:52:00 -0400 From: Anders Kock To: "categories@mta.ca" Subject: categories: Bill Lawvere Date: Wed, 8 Feb 2023 15:39:25 +0000 Accept-Language: da-DK, en-US Content-Language: da-DK Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Precedence: bulk Reply-To: Anders Kock Message-Id: 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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]