Grothendieck often said sites are a "provisional" definition. In the Buffalo lectures he says the Giraud axioms are closer to the intuition -- so much so that Grothendieck says “I have a tendency to forget which properties Giraud uses” and just think of his characterization by exactness properties of sets, so far as concern finite limits and arbitrary colimits. But he also says several times “so called sites” are needed for some proofs. (I am not sure why he thought that. From my point of view, anything you can say with a site you can say in nearly the same way about the generators in the Giraud axioms.) Here is one way he could have benefitted from the Lawvere-Tierney axioms. Simply by taking those and requiring set-sized colimits (as the Giraud axioms also do, explicitly) he would get his original (Grothendieck) toposes in a more concise way. On Wed, Sep 4, 2024 at 9:40 AM Steven Vickers <s.j.vickers.1@bham.ac.uk> wrote:

Thanks for the link, Colin - I enjoyed reading it. Can I ask a couple of questions? [some cut]

1. Was he aware of the possibility, or desirability, of using point-free topology for the internal notion of topological space? The Joyal-Tierney paper didn't appear until 1984. You mention "topological sheaf" a couple of times, which suggests to me a fibrewise point-set approach in that sheaves (local homeomorphisms) have discrete fibres.

I don't know. He does make one remark in the opposite sense. He remarks that the topos of sheaves on a topological space characterizes the space uniquely--if the space is sober, and then he says sober spaces are the good ones anyway. If you have a space where [some] irreducible closed sets have no generic points, he says, you will sooner or later add generic points. This makes evident sense in the context of advancing from classical varieties to schemes. And really Grothendieck's main concrete interest in toposes was scheme sites related to the gros and petit etale sites. Conceivably Grothendieck's reservations about set-theoretic axioms (p. 203)

might be resolved if we go to point-free spaces, in the same kind of way as they remove the need for choice from Heine-Borel and Tychonoff, though I don't suppose we're anywhere close to knowing how to do it for the Weil conjectures.

Maybe, but I think his only concern about set theory was set-theoretic size. And I think there is an important reason for that. I mean important at least towards understanding Grothendieck. Contrary to many other mathematicians, he absolutely did care to have a rigorous logical foundation. But like most mathematicians he did not care much *what* foundation, as long as there was some known rigorous one. He saw that ZFC was accepted, and (as he does say in the lectures) once he had experts assure him that universes are considered consistent with ZFC he felt his work on set theory was done. He did not consider this a final answer, and he advised looking into smaller sets that might do the job. But done well enough for the SGA. And he was not going to pursue it further. I have a chapter on this point forthcoming ``Grothendieck did not believe in universes, he believed in topos and schemes,'' for a book M. Panza, D. Struppa and J-J. Szczecinarz eds. *Grothendieck's Mathematical and Philosophical Legacy*, in page proofs,Springer Nature. The paper title is a quote of Pierre Cartier, with grateful memories of beautiful conversations and many things he taught me. 2. For arbitrary colimits, we have to know what "arbitrary" means. Is there

any evidence that Grothendieck thought about this, or did he just accept that classical set theory would supply all the indexations? More generally it depends on a chosen base topos, and the Elephant has an elaborate application of indexed categories to explain how that works.

This I think is a serious question. So far as I know Grothendieck never plumbed the depths of indexed category theory. But he ran up against the subject. Probably much more of that is implicit in his work than explicit, and that would need hard exploration. best, Colin +++++++++++++++++

There is a way round this in that if an elementary topos has an nno then some infinite internal colimits exist independently of any choice of base topos. It's very much an open question how far this takes you, though there are definite inroads into real analysis.

All the best,

Steve. ------------------------------ *From:* Colin McLarty <colin.mclarty@case.edu> *Sent:* Wednesday, September 4, 2024 2:04 AM *To:* P.T. Johnstone <ptj1000@cam.ac.uk> *Cc:* Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk>; Eduardo J. Dubuc <edubuc@dm.uba.ar>; categories@mq.edu.au < categories@mq.edu.au> *Subject:* Re: Grothendieck in the Guardian

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Grothendieck said repeatedly, in the 33 hours of tape recorded 1973 Buffalo lectures on topos theory, that there are "two intuitions of topos." Those lectures were in English. One intuition of topos is a category with "the exactness properties of the category of sets -- at least so far as finite limits and arbitrary colimits are concerned." He gives this whole formula repeatedly. (And he says this means you can do mathematics in any topos.) The second is that a topos is a generalized topological space, though of course a topos is very large set theoretically and the spaces can be quite small -- as the category of sets is a one point space.

I wrote this up, published as ``Grothendieck's 1973 topos lectures in Buffalo NY,'' in English, in F. Jaeck ed. *Lectures grothendieckiennes [2017 - 2018]*, Soci\'et\'e Math\'ematique de France. published January 2022, pp. 189--204. The whole book can be read free online at https://url.au.m.mimecastprotect.com/s/CcQWCYW86EsL32O18F0fVCxQPZq?domain=sp...

Everything he says there coheres well with what he wrote in 1958 up to SGA1, though of course the detailed treatment in SGA4 is focused on technical proofs rather than intuition. And on the other side, it all coheres well with *Recoltes et Semailles *though again the focus is different.

Colin

On Tue, Sep 3, 2024 at 4:14 PM P.T. Johnstone <ptj1000@cam.ac.uk> wrote:

Dear Steve,

Yes, I agree that a topos in isolation (whether elementary or not) is not a generalized space. It acquires spatial qualities through its interaction with other toposes through the medium of geometric morphisms – that is what I meant when I said it has geometric (and logical) potentialities.

Peter

------------------------------ *From:* Steven Vickers <s.j.vickers.1@bham.ac.uk> *Sent:* 03 September 2024 18:40 *To:* P.T. Johnstone <ptj1000@cam.ac.uk>; Eduardo J. Dubuc < edubuc@dm.uba.ar> *Cc:* categories@mq.edu.au <categories@mq.edu.au> *Subject:* Re: Grothendieck in the Guardian

Dear Peter,

I accept that changing "elementary topos" is probably not going to happen.

Do you agree, though, that an elementary topos, in itself, is not a generalized space? It becomes one only when equipped with a bounded geometric morphism to a fixed base S, hence the importance of BTop/S.

That muddies the motto that a topos is a generalized space.

Steve. ------------------------------ *From:* P.T. Johnstone <ptj1000@cam.ac.uk> *Sent:* Tuesday, September 3, 2024 5:48 PM *To:* Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk>; Eduardo J. Dubuc <edubuc@dm.uba.ar> *Cc:* categories@mq.edu.au <categories@mq.edu.au> *Subject:* Re: Grothendieck in the Guardian

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Dear Steve, dear Eduardo,

I can't take issue with Eduardo's anecdote, since I never met Grothendieck myself. But I have heard from other sources that his main unhappiness about elementary toposes was annoyance that he had failed to spot the notion of subobject classifier (which he referred to as "the Lawvere element") that made the elementary development possible. If he had, SGA4 might have looked very different!

And I don't think it's helpful to try to find another name for "elementary topos". The whole point of the story about the blind men and the elephant is that "wherever you touch it, it's still the same animal"; every topos, wherever it comes from (even ones like realizability toposes, whose origin is entirely logical) contains within itself both geometric and logical potentialities, and it's the interplay between these that gives the subject its power.

Incidentally, whilst logical functors are important, it's also important to remember that geometric morphisms (or at least their inverse image parts) are also "structure-preserving functors" in at least two senses (see A4.1.18 and B2.2.7 in the Elephant). So they would be natural morphisms to study even if one came to toposes from an entirely "logical" background.

Peter Johnstone

------------------------------ *From:* Steven Vickers <s.j.vickers.1@bham.ac.uk> *Sent:* 03 September 2024 11:59 *To:* Eduardo J. Dubuc <edubuc@dm.uba.ar> *Cc:* categories@mq.edu.au <categories@mq.edu.au> *Subject:* Re: Grothendieck in the Guardian

Dear Eduardo,

I'm gratified to see your anecdote about Grothendieck being unhappy with "elementary topos". It confirms what I saw as a big problem with the current usage of "topos" when I wrote my article "Point-free generalized spaces, pointwise", https://url.au.m.mimecastprotect.com/s/xNE7CZY146s5M924GHjhjCBtbLN?domain=ar...

.

Regarding "Lawvere sets", Anel and Joyal have proposed the word "logos" for the categories of sheaves, the algebraic counterparts of the spatial notion of topos. Perhaps "Lawvere logos" is another candidate for "elementary topos"?

Steve.

------------------------------ *From:* Eduardo J. Dubuc <edubuc@dm.uba.ar> *Sent:* Tuesday, September 3, 2024 2:06 AM *To:* Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk>; Vaughan Pratt <pratt@cs.stanford.edu> *Cc:* categories@mq.edu.au <categories@mq.edu.au> *Subject:* Re: Grothendieck in the Guardian

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Dear Steven, I very much agree with all in your posting, and I would like to add some comments:

"A topos is simply one of many possible generalization of sets and their functions ... "

is very misleading

in fact as a topos Sets is a point. The underlying category of a topos shares the exactness properties of the category of sets, and only secondarily it is an elementary topos

Morphisms of elementary topoi are the logical ones, and they sit in the other side of the duality, morphism of topoi (or their inverse images which go in the same direction than morphisms of elementary topoi) do no agree with the elementary topos essential structure of the underlying category.

As Steven say,

"The generalized topological spaces are at the heart of Grothendieck's motivation"

I remember I was a student at Chicago when Grothendieck visited, and after his talk, around a table at a bar (he would drink only water) he said that he was very upset that Lawvere had called his concept a topos.

I imagine he could have called "generalized set" for example.

Much confusion would have being avoided, and today elementary topi would be called "Lawvere Sets", an even strong recognition of the importance of Lawvere and his creation.

Eduardo.

On 02/09/2024 10:45 AM, Steven Vickers wrote:

...

Dear Vaughan,

It's easy to make the summary "A topos is simply one of many possible generalization of sets and their functions ...", but that's definitely an over-simplification. As Mac Lane and Moerdijk say, toposes have two facets: as generalized universes of sets (which is what you said), and as generalized topological spaces (which is what I was alluding to in my own Guardian comment). In fact Johnstone's Elephant explicitly tries to bring out even more facets.

The generalized topological spaces are at the heart of Grothendieck's motivation. On the one hand, that is in the sense of algebraic topology, in that topological invariants such as cohomologies can be calculated for them - and can be exploited in algebraic geometry. On the other hand, it is also in the sense of general topology, in that a topos can be fruitfully be viewed as a space whose points are the models of a geometric theory that the topos classifies.

The trouble is, the generalized topological space is easy to lose sight of, even easier if you move to elementary toposes (which are not classifying toposes in their own right, but only relative to other toposes), so many mathematicians just see the generalized universes of sets.

Actually, the "essential properties that make sets so valuable in mathematics" can be an obstruction to seeing the generalized topological spaces. The issue is that some of the "essential properties", such as cartesian closedness and subobject classifiers, do not interact successfully with geometric morphisms, the generalization of continuous maps. (They are not preserved by inverse image functors.) For an unobstructed view of the generalized topological spaces, at least in the sense of general topology, it is best to reject those non-geometric constructions.

To summarize: if you view toposes as "simply" the generalized universes of sets, then you risk overlooking the generalized topological spaces.

Steve.

------------------------------------------------------------------------ *From:* Vaughan Pratt <pratt@cs.stanford.edu> *Sent:* Monday, September 2, 2024 6:32 AM *To:* Wesley Phoa <doctorwes@gmail.com> *Cc:* categories@mq.edu.au <categories@mq.edu.au> *Subject:* Re: Grothendieck in the Guardian *CAUTION:* This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe.

"I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical."

Why would anyone object to journalists doing exactly those things? Do we want to keep mathematics a dark secret, or what?

A topos is simply one of many possible generalization of sets and their functions that allows many other mathematical objects besides sets to be imbued with some of the essential properties that make sets so valuable in mathematics.

For example graphs and their maps form a topos with very similar properties to sets and their functions, such as having the notion of a power set. But not all properties, for example the law of the excluded middle, which holds for sets but not graphs.

Vaughan Pratt

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