Grothendieck

Grothendieck

[Michael Barr, Prof.]

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Thinking about the question of Lawver-Tierney's elementary topos, I thought it might be interesting to recount my one personal experience with Grothendieck.

It was the summer of 1971.  There was a logic conference in, IIRC, England, sponsored by NATO.  Some people got together and decided to organize a logic counter-conference not sponsored by NATO.  It took place at a residential school in a town called Uldum about 50 km or so south of Aarhus on the Jutland peninsula and they invited Grothendieck.  Although he had at that time given up mathematics (he was present at the ICU Nice a year earlier, but when Bill Lawvere tried to talk to him—presumably about elementary toposes—G. said he wasn't interested), he accepted the invitation because of the circumstance.  And gave a talk.  Since I was spending that summer in Aarhus and had a car, I drove down to hear G.

He first described the Verdier axioms in some detail.  What logicians made of that I do not know.  Then he turned to the audience and said, doesn't this remind you of set theory?  Someone should study should study set theory from this point of view.  I might add that for me, the Verdier axioms didn't look like set theory at all, but I am not a logician.  At any rate, during the question period, I told him that there was a set of axioms for a topos that really did make it look like set theory.  So G. Asked me to come to the blackboard and describe them.  So I gave the L-T axioms except I added complete with generators to make them fully equivalent to Verdier's.  He said that that was interesting and I sat down.  That seems to have been the first time he had heard about the L-T axioms.

Michael ​


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Re: Grothendieck

[Vaughan Pratt]

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I'm glad that my response to Wes Phoa's "I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical." has prompted this discussion of the differences between Grothendieck's topological understanding of the topos concept and Lawvere and Tierney's elementary formulation.

One thing I like about Chu(Set,K) for the case K = {0,1} is that both those understandings nicely embed in it.  As with Lafont and Streicher (LICS'91), who showed how to embed Vct_K in Chu(Set,K) for the underlying set of any field k (which I found hard to believe on a first reading and needed Yves to reassure me about it), I find applications for larger K than {0,1} very useful, e.g. for both true concurrency and branching time (Math. Struct. in Comp. Science, 13:4, 485-529, August 2003, also http://boole.stanford.edu/pub/seqconc.pdf<https://url.au.m.mimecastprotect.com/s/AducCP7L1NfK4R2kXsru6Cx_OUr?domain=boole.stanford.edu>).  Beyond that I also find useful my "Communes via Yoneda, from an elementary perspective" (Fundamenta Informaticae XXI (2001) 1001–1017, also http://boole.stanford.edu/pub/yon.pdf<https://url.au.m.mimecastprotect.com/s/LB9LCQnM1Wfk6pKyPu9CpCGrtMG?domain=boole.stanford.edu>).  Each of these goes well beyond K = {0,1}.

In 1999, at the Coimbra summer school on category theory where John Baez, Cristina Pedichio, and I each gave five lectures in the first week of that summer school, Myles Tierney told me during a picnic that he didn't like my lecture notes on Chu(Set,K).  In retrospect I wish I'd pressed him for his objections.  Today I would guess that he might have felt better with Chu(V,K) where V was an arbitrary elementary topos, as a middle ground between V = Set and the even more general V in Mike Barr's beauiful little *-autonomous categories book, but at the time I was pretty naive about toposes.

Despite the points made in this thread such as "a Grothendieck topos makes Set a one-object topos", I still feel that the reporter would have been on much safer ground with Set as an elementary topos than trying to convey Grothendieck's original thinking that makes Set an "over-simplification", if that's what's at issue here.

Vaughan


On Thu, Sep 5, 2024 at 9:07 PM Michael Barr, Prof. <barr.michael@mcgill.ca<mailto:barr.michael@mcgill.ca>> wrote:
Thinking about the question of Lawver-Tierney's elementary topos, I thought it might be interesting to recount my one personal experience with Grothendieck.

It was the summer of 1971.  There was a logic conference in, IIRC, England, sponsored by NATO.  Some people got together and decided to organize a logic counter-conference not sponsored by NATO.  It took place at a residential school in a town called Uldum about 50 km or so south of Aarhus on the Jutland peninsula and they invited Grothendieck.  Although he had at that time given up mathematics (he was present at the ICU Nice a year earlier, but when Bill Lawvere tried to talk to him—presumably about elementary toposes—G. said he wasn't interested), he accepted the invitation because of the circumstance.  And gave a talk.  Since I was spending that summer in Aarhus and had a car, I drove down to hear G.

He first described the Verdier axioms in some detail.  What logicians made of that I do not know.  Then he turned to the audience and said, doesn't this remind you of set theory?  Someone should study should study set theory from this point of view.  I might add that for me, the Verdier axioms didn't look like set theory at all, but I am not a logician.  At any rate, during the question period, I told him that there was a set of axioms for a topos that really did make it look like set theory.  So G. Asked me to come to the blackboard and describe them.  So I gave the L-T axioms except I added complete with generators to make them fully equivalent to Verdier's.  He said that that was interesting and I sat down.  That seems to have been the first time he had heard about the L-T axioms.

Michael ​


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Grothendieck

[Moh'd Alsani]

Very sad news.

Alexander Grothendieck 28 March 1928 – 13 November 2014

http://www.liberation.fr/sciences/2014/11/13/alexandre-grothendieck-ou-la-mort-d-un-genie-qui-voulait-se-faire-oublier_1142614



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