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charset="utf-8" Content-Transfer-Encoding: quoted-printable I'm glad that my response to Wes Phoa's "I=E2=80=99m relieved the journalis= t didn=E2=80=99t try to explain what a topos was, or indeed anything mathem= atical." has prompted this discussion of the differences between Grothendie= ck'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 =3D {0,1} is that both tho= se understandings nicely embed in it. As with Lafont and Streicher (LICS'9= 1), who showed how to embed Vct_K in Chu(Set,K) for the underlying set of a= ny field k (which I found hard to believe on a first reading and needed Yve= s to reassure me about it), I find applications for larger K than {0,1} ver= y 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.ed= u/pub/seqconc.pdf). Beyond that I also find useful my= "Communes via Yoneda, from an elementary perspective" (Fundamenta Informat= icae XXI (2001) 1001=E2=80=931017, also http://boole.stanford.edu/pub/yon.p= df). Each of these goes well beyond K =3D {0,1}. In 1999, at the Coimbra summer school on category theory where John Baez, C= ristina Pedichio, and I each gave five lectures in the first week of that s= ummer 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 = =3D Set and the even more general V in Mike Barr's beauiful little *-autono= mous 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 Gro= thendieck's original thinking that makes Set an "over-simplification", if t= hat's what's at issue here. Vaughan On Thu, Sep 5, 2024 at 9:07=E2=80=AFPM Michael Barr, Prof. > wrote: Thinking about the question of Lawver-Tierney's elementary topos, I thought= it might be interesting to recount my one personal experience with Grothen= dieck. 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 log= ic counter-conference not sponsored by NATO. It took place at a residentia= l school in a town called Uldum about 50 km or so south of Aarhus on the Ju= tland peninsula and they invited Grothendieck. Although he had at that tim= e given up mathematics (he was present at the ICU Nice a year earlier, but = when Bill Lawvere tried to talk to him=E2=80=94presumably about elementary = toposes=E2=80=94G. said he wasn't interested), he accepted the invitation b= ecause of the circumstance. And gave a talk. Since I was spending that su= mmer 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 th= is 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, durin= g the question period, I told him that there was a set of axioms for a topo= s 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 sa= id that that was interesting and I sat down. That seems to have been the f= irst time he had heard about the L-T axioms. Michael =E2=80=8B You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | Learn more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --000000000000272d2306216ce433 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
I'm glad that my res= ponse to Wes Phoa's "I=E2=80=99m relieved the journalist didn=E2=80=99= t try to explain what a topos was, or indeed anything mathematical." h= as prompted this discussion of the differences between Grothendieck's topological understanding of the topos concept and Lawvere and Tierney's e= lementary formulation.

One thing I like abo= ut Chu(Set,K) for the case K =3D {0,1} is that both those understandings ni= cely 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 n= eeded 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 (Ma= th. Struct. in Comp. Science, 13:4, 485-529, August 2003, also http://boole.stanford.edu/pub/seqconc.pdf).  Beyond that I also fi= nd useful my "Communes via Yoneda, from an elementary perspective"= ; (Fundamenta Informaticae XXI (2001) 1001=E2=80=931017, also http://boole.stanford.edu/pub/yon.pdf).  Each of these goes well b= eyond K =3D {0,1}.

In 1999, at the Coim= bra summer school on category theory where John Baez, Cristina Pedichio, an= d 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 =3D Set and the even more general V in Mike Barr's beauiful little *-autonomous categor= ies book, but at the time I was pretty naive about toposes.

Despite the points m= ade in this thread such as "a Grothendieck topos makes Set a one-objec= t 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-si= mplification", if that's what's at issue here.

Vaughan


On Thu, Sep 5, 2024 at 9:07=E2=80=AFP= M Michael Barr, Prof. <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 Grothen= dieck.

It was the summer of 1971.  There was a logic conference in, IIRC, Eng= land, sponsored by NATO.  Some people got together and decided to orga= nize 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 invite= d 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=E2=80=94presumably about elementary toposes=E2=80=94= G. said he wasn't interested), he accepted the invitation because of the ci= rcumstance.  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 st= udy set theory from this point of view.  I might add that for me, the Verdier axioms didn't look like set theory at all, bu= t I am not a logician.  At any rate, during the question period, I tol= d him that there was a set of axioms for a topos that really did make it lo= ok like set theory.  So G. Asked me to come to the blackboard and describe them.  So I gave the L-T axioms e= xcept I added complete with generators to make them fully equivalent to Ver= dier'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 =E2=80=8B
 
 
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