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* categories: Bill Lawvere
@ 2023-02-08 15:39 Anders Kock
  2023-02-09  4:58 ` categories: " Ross Street
  0 siblings, 1 reply; 6+ messages in thread
From: Anders Kock @ 2023-02-08 15:39 UTC (permalink / raw)
  To: categories

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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* categories: Re: Bill Lawvere
  2023-02-08 15:39 categories: Bill Lawvere Anders Kock
@ 2023-02-09  4:58 ` Ross Street
  0 siblings, 0 replies; 6+ messages in thread
From: Ross Street @ 2023-02-09  4:58 UTC (permalink / raw)
  To: Anders Kock; +Cc: list

Dear Anders

Thank you for that tribute and interesting history particularly poignant on  Bill's birthday.

On 9 Feb 2023, at 2:39 am, Anders Kock <> wrote:

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.

That reminds me, and perhaps I have mentioned this before?
Sammy Eilenberg told me once that each category theorist must have a special category on which they are expert.
I told that to John Gray who said: Ross, the time has come for that category to be Cat.
John's graduate course on category theory was based on that philosophy, heavily influenced by Bill's LaJolla paper.


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* categories: Bill Lawvere
@ 2023-02-20  1:56 Michael Healy
  0 siblings, 0 replies; 6+ messages in thread
From: Michael Healy @ 2023-02-20  1:56 UTC (permalink / raw)
  To: categories

Dear colleagues,

I am deeply affected by Bill Lawvere's departure. I met him rather late, at  the Category Theory conference at UBC in 1997. I knew very little, not having been schooled in that area of mathematics. So I introduced myself to two new people whose works I had begun reading---Bill, and also Saunders Mac Lane. They were very welcoming to a newcomer. In fact, it felt like Bill was taking a new enthusiast under his wing, so to speak. He spent time with me during breaks and I will forever benefit from his time there and at later  conferences and in correspondence, and from the contributions he made to mathematics.

Very sincerely,
Mike Healy

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* categories: Bill Lawvere
@ 2023-02-02  9:12 Ross Street
  0 siblings, 0 replies; 6+ messages in thread
From: Ross Street @ 2023-02-02  9:12 UTC (permalink / raw)
  To: categories


Yesterday our Seminar started again after our summer break. Local people are attending in person although zoom is still used. I arrived early since there were some things I wanted to find in my office: non-computer-file material. I moved rooms recently so some papers are in mysterious places. Not so  my remaining books. These are all in nice alphabetical order according to author.

Recently I started talking mathematics with a last-year-of-high-school student and so far it is ongoing. I wanted to consult my copy of the book  ``Conceptual Mathematics'' to see how some things were treated there. The authors Bill Lawvere and Steve Schanuel inscribed my copy on the inside cover. Those words, confirming what people have been writing about Bill, spurred me  to write these personal memories.

Bill wrote: ``For Ross Street, In appreciation of the many inspiring mathematical discussions in Sydney and around the World, and in recognition of co-participation in the great struggle for improved teaching of science. And with warm affection, Bill Lawvere.'' (After that came: ``Me too, Steve Schanuel.'')

As a PhD student at the University of Sydney, the papers in the LaJolla 1965 Conference on Categorical Algebra had a big influence on me. In particular, I found the one by Bill (first in the volume) quite remarkable. I had started worrying about foundations of mathematics a few years earlier. Bill's  ideas were not what I had read in Russell!

I first met Bill in 1968-69 while I was a postdoc at the University of Illinois, 120 miles south of Chicago. In mid-1969 (when a USA citizen was taking one small step), we met for longer at the Summer School on Category Theory, Bowdoin College, Maine. The lecture of Bill that I remember was about metric spaces as enriched categories. Anyone familiar with my publications and those of my colleague Bob Walters will know what an influence that work had on us.

I think it was at Oberwolfach in 1972 when I nervously gave my talk on Yoneda structures that Bill suggested maybe my 2-category should have a tensor product. Around that time, Brian Day took some notes of Bill's lectures in Aarhus on quite a few topics including monoidal bicategories.

Soon after we saw John Gray's summary of talks in a conference at the ETH, Z\"urich. The talk by Myles Tierney announced one huge step for mathematics: the concept of Elementary Topos. This was joint work with Bill, of course. The application connecting forcing to sheaf theory was absolutely brilliant. However, it is the simplicity of the topos axioms that is truly beautiful. With further suggestions from Bill and input from Anders Kock, Chris Mikkelsen and Bob Par\'e, the axioms reduced even further. The elementary topos fan club just grows and grows (to elephant proportions). I was greatly honoured to hear that Bill was giving my Survey of Topos Theory notes to his  graduate students to read.

When Bill visited the University of Sydney, Steve Schanuel visited Macquarie. These were wonderful times for our mathematics. Both men reached out beyond the category theorists, inspiring the two departments. Other people overlapped with these visits: for example, Dold, Heller at Macquarie (with Schanuel keen on trace, Euler characteristic and homotopy theories) and Carboni (with Lawvere keen on enrichment in monoidal bicategories) at Sydney.

I was invited to SUNY Buffalo several times. With Duskin, Isbell, Lawvere, Schanuel, Schack, and others, there was plenty to discuss in the Department  common room (``Steve's office''). I was honoured to deliver the John Myhill Memorial Lectures for 1993. Bill showed great enthusiasm for the topics.

As category theory is embraced by the wider community, Bill's contributions  will live on and be admired.


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* categories: Bill Lawvere
@ 2023-01-30  9:43 Jirí Adámek
  0 siblings, 0 replies; 6+ messages in thread
From: Jirí Adámek @ 2023-01-30  9:43 UTC (permalink / raw)
  To: categories net

I am very sad that Bill Lawevere, a great mathematician and a great man,
has passed away. Jiri Rosicky and I had the extraordinary good luck of
working on several joint articles with Bill, and we have very happy memories
of that collaboration. It started after the publication of our book on
locally presentable categories in 1994: Bill approached us shortly after
that with the suggestion to study what properties make locally presentable
'tick'. We had difficuties understanding his suggestion, and during the
discussions about that we have realized that nobody had studied the
algebraic duality that would be the analogy of the Gabriel-Umer duality to
finitary varieties. This led to our first short joint paper.

Later Bill came with the suggestion to have a look at 'how algebraic
is algebra': is there a higher-order monad whose algebras are finitary
varieties? In one of his early mails on that topic he wrote that the
answer would probably depend on Ulam cardinals. (A concept we two had
not heard of before.) We spent months of intensive work on this topic
during which Bill's impulses were the much needed support in times when
there seemed to be no light at the end of the tunnel. But eventually
light came, and we published two joint papers, about varieties and about
locally finitely presentable categories. One of our basic results needed
the assumption of non-existence of measurable cardinals. Only then did
we two learned that they are also called Ulam cardinals- our admiration
for Bill's insight could not have been greater...

One day Bill came with the grand idea of a initiating a collection of
monographs on topics he found important. At first a group of enthusiastic
colleagues agreed to start working on the project, but as time went on,
one after another decided to quit. In the end, just one monograph was
realized, the joint book with Jiri Rosicky and Enrico Vitale on algebraic
theories. Bill  followed our work on it closely, supporting us and
criticizing what he did not find right. We were proud when he agreed to
write a preface to it.

During all the time of our collaboration with Bill he was extremely
patient and friendly. It is indeed a very  sad news that this great and
kind man has passed away.

Jiri Adamek

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* categories: Bill Lawvere
@ 2023-01-26 13:58 Steve Awodey
  0 siblings, 0 replies; 6+ messages in thread
From: Steve Awodey @ 2023-01-26 13:58 UTC (permalink / raw)
  To: categories net

Dear Colleagues,

It’s very sad to hear about Bill Lawvere’s passing. I haven’t seen him since a meeting at Colgate several years ago, 
where he came with his son, who I believe is a math teacher.

Bill was of course a great mathematician who changed the way we think about logic and the foundations of mathematics, but he was also a kind and generous  person. 
When I was a student of Saunders at Chicago, Bill often visited for the weekend and lectured, discussed ideas with me, and encouraged me.  I suppose he was there to talk to Ieke, who visited often then, 
because he and Saunders were working on the book.  But Bill was always happy  to patiently explain things to me and to suggest things for me to work on.

When he later came to CMU for a seminar, I told him he would have a bit more  time than a usual lecture, up to 2 hours.
The room had large blackboards on 3 walls, and he used all 3 of them, moving  desks out of the way and telling the audience to turn their chairs.  After 3  hours I had to cut him off - he had only heard the part about having more time!

I am very grateful for the patience and generosity that Bill showed me as a student, 
and for his mathematical ideas which have fascinated and motivated me ever since then.

I also recall that Saunders had the very highest opinion of Bill.

Best wishes,


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2023-02-08 15:39 categories: Bill Lawvere Anders Kock
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2023-02-20  1:56 categories: " Michael Healy
2023-02-02  9:12 Ross Street
2023-01-30  9:43 Jirí Adámek
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