Hi all, Just to add to this, I think that a good way to think about the relationship between elementary toposes and space is: An elementary topos is a definition of a *kind* of space (rather than a specific space) — namely, the kind of space that can be found lying over it via a bounded geometric morphism. The space itself is not the topos in isolation but the specific bounded geometric morphism that identifies it as an example of a space, among many possible notions of space. By this token, one topos (in isolation) can be viewed as a space in many different ways — and sometimes these ways can be reconciled (e.g. several bounded geometric morphisms into Grothendieck toposes can be reconciled by the universal morphism into the point), but sometimes these different ways are not as easily reconciled (e.g. in the case of SET viewed as the punctual space over itself, vs. SET viewed from the perspective of the codiscrete embedding SET --> Eff; so in this case, one would think of the two incarnations of SET as two totally different spaces of a different kind) I think this reconciles the viewpoint of Grothendieck with other possible interpretations of topos theory in light of elementary toposes. Best, Jon On Tue, Sep 3, 2024, at 2:13 PM, P.T. Johnstone 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/H6F7CQnM1Wfk6Kv3VCxfpCGrdG3?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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