categories: Bill Lawvere

[Ross Street <ross.street@mq.edu.au>]

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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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