Grothendieck in the Guardian

Grothendieck in the Guardian

[Paul Taylor]

An article about Alexander Grothiendieck has just appeared
in the Guardian online newspaper.  Be warned, it contains
some seriously weird stuff!  Toposes get a mention, though
"not as we know them", along with Huawei, AI and Olivia
Caramello.  Beyond that, I'm not going to comment!

Since Microsoft mangles web addresses, here is the address
with the punctuation removed:

www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence

Paul Taylor.



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Re: Grothendieck in the Guardian

[Wesley Phoa]

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Thanks - I saw this! I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical. 

Sent

> On Aug 31, 2024, at 1:13 PM, Paul Taylor <categories@paultaylor.eu> wrote:
> 
> An article about Alexander Grothiendieck has just appeared
> in the Guardian online newspaper.  Be warned, it contains
> some seriously weird stuff!  Toposes get a mention, though
> "not as we know them", along with Huawei, AI and Olivia
> Caramello.  Beyond that, I'm not going to comment!
> 
> Since Microsoft mangles web addresses, here is the address
> with the punctuation removed:
> 
> www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence
> 
> Paul Taylor.
> 
> 
> 
> ----------
> 
> You're receiving this message because you're a member of the Categories mailing list group from Macquarie University.
> 
> Leave group:
> https://url.au.m.mimecastprotect.com/s/ZQnpCXLW6DiXnxP0pS6f7CWeEe9?domain=outlook.office365.com

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Re: Grothendieck in the Guardian

[Steven Vickers]

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I actually left a comment on the Guardian:

"Even though he was immediately recognized as a genius, he was still way ahead of his time. From the early 20th century comes the mathematical theory of topology. Grothendieck’s toposes provide a vastly more general and unifying notion of what topological spaces could be, and he provocatively named them “toposes” as those things of which topology was the study. Or ought to be. Even now, topologists have generally not caught up with him.
I wonder if much his later reclusiveness came out of disappointment at the failure of others to understand his ideas."

That final paragraph is mere speculation on my part, as I never knew him. Perhaps others on the list have better insights into that.

For your amusement, I've attached a ceramic model of his head, which I made when my wife took me on a clay modelling course.

Steve.


________________________________
From: Paul Taylor <categories@PaulTaylor.EU>
Sent: Saturday, August 31, 2024 3:16 PM
To: categories@mq.edu.au <categories@mq.edu.au>
Subject: 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.


An article about Alexander Grothiendieck has just appeared
in the Guardian online newspaper.  Be warned, it contains
some seriously weird stuff!  Toposes get a mention, though
"not as we know them", along with Huawei, AI and Olivia
Caramello.  Beyond that, I'm not going to comment!

Since Microsoft mangles web addresses, here is the address
with the punctuation removed:

www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence

Paul Taylor.



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Re: Grothendieck in the Guardian

[Vaughan Pratt]

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"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

On Sun, Sep 1, 2024 at 1:16 PM Wesley Phoa <doctorwes@gmail.com> wrote:

> Thanks - I saw this! I’m relieved the journalist didn’t try to explain
> what a topos was, or indeed anything mathematical.
>
> Sent
>
> > On Aug 31, 2024, at 1:13 PM, Paul Taylor <categories@paultaylor.eu>
> wrote:
> >
> > An article about Alexander Grothiendieck has just appeared
> > in the Guardian online newspaper. Be warned, it contains
> > some seriously weird stuff! Toposes get a mention, though
> > "not as we know them", along with Huawei, AI and Olivia
> > Caramello. Beyond that, I'm not going to comment!
> >
> > Since Microsoft mangles web addresses, here is the address
> > with the punctuation removed:
> >
> > www theguardian com science article 2024 aug 31
> alexander-grothendieck-huawei-ai-artificial-intelligence
> >
> > Paul Taylor.
> >
> >
> >
> > ----------
> >
> > You're receiving this message because you're a member of the Categories
> mailing list group from Macquarie University.
> >
> > Leave group:
> >
> https://url.au.m.mimecastprotect.com/s/Q3pjCq71jxf8O2jm4cZf8CEopNi?domain=outlook.office365.com
> <https://url.au.m.mimecastprotect.com/s/Q3pjCq71jxf8O2jm4cZf8CEopNi?domain=outlook.office365.com>
>

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Re: Grothendieck in the Guardian

[Patrik Eklund]

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"Do we want to keep mathematics a dark secret, or what?"

Apparently, yes, we do.

Obviously, the definition of a topos is the same for all of us. But the 
way we use it, the way we attach it to other mathematical structures, is 
part of our own secrets. And there are apparently many such secrets.

I might even believe that Grothendieck's own perception, of what toposes 
really are, changed over time, and indeed in dialogue with the 
scientific community, a community which is not a closed one, but very 
much part of society.

Clearly, there may remain parts of "aus liebe zur Kunst" in topos 
theory, as for any part of mathematical theories for that matter, but 
generally speaking, there are always objectives, and requirements for 
theories to be applicable, applicability in a broader sense.

Are there real-world applications of toposes? Journalists would love to 
know, I guess.

Best,

Patrik

On 2024-09-02 08:32, Vaughan Pratt wrote:

> "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
> 
> On Sun, Sep 1, 2024 at 1:16 PM Wesley Phoa <doctorwes@gmail.com> wrote:
> 
>> Thanks - I saw this! I'm relieved the journalist didn't try to explain 
>> what a topos was, or indeed anything mathematical.
>> 
>> Sent
>> 
>>> On Aug 31, 2024, at 1:13 PM, Paul Taylor <categories@paultaylor.eu> 
>>> wrote:
>>> 
>>> An article about Alexander Grothiendieck has just appeared
>>> in the Guardian online newspaper. Be warned, it contains
>>> some seriously weird stuff! Toposes get a mention, though
>>> "not as we know them", along with Huawei, AI and Olivia
>>> Caramello. Beyond that, I'm not going to comment!
>>> 
>>> Since Microsoft mangles web addresses, here is the address
>>> with the punctuation removed:
>>> 
>>> www theguardian com science article 2024 aug 31 
>>> alexander-grothendieck-huawei-ai-artificial-intelligence
>>> 
>>> Paul Taylor.
>>> 
>>> 
>>> 
>>> ----------
>>> 
>>> You're receiving this message because you're a member of the 
>>> Categories mailing list group from Macquarie University.
>>> 
>>> Leave group:
>>> https://url.au.m.mimecastprotect.com/s/WNWDC81Vq2C6jk3jxsnfECyJMH5?domain=outlook.office365.com 
>>> [1]


Links:
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Re: Grothendieck in the Guardian

[Wesley Phoa]

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Re: Grothendieck in the Guardian

[Dusko Pavlovic]

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thank you paul!

if i squint the right way, the article very funny - and that has little to
do with grothendieck. confusing the speed of light with the speed of sound
is like something from capek or jonathan swift. and there is something like
that in every line! huawei hired a fields medalist to build AI toposes! but
it's allright because huawei is not a communist enterprise but a
self-governing corporation!...

speaking of swift, isn't it remarkable that the ways people misunderstand
each other were so vividly clear in the XVIII century that they could be
illustrated even for children, by gulliver's travels. nowadays it is sort
of the opposite. if there is any doubt about anything, just ask the
fact-checkers and they will check the facts for you. vaughan said that
topos is a generalized universe of sets. if you are not sure, the fact
checkers will confirm the truth of the matter for you. deligne is not
interested in grothendieck because mathematics is a social enterprise. if
anyone doubts that, a well-designed poll will resolve the matter. we
understand grothendieck's math, psychologists understand his behavior, the
gentle writer of this article is not an expert in any of those things but
he understands grothendieck's writing. together, we understand everything.
"just do your research" tweets elon musk. and feel the vibrations of
fibrations :)))

-- dusko

On Sat, Aug 31, 2024 at 10:13 AM Paul Taylor <categories@paultaylor.eu>
wrote:

> An article about Alexander Grothiendieck has just appeared
> in the Guardian online newspaper.  Be warned, it contains
> some seriously weird stuff!  Toposes get a mention, though
> "not as we know them", along with Huawei, AI and Olivia
> Caramello.  Beyond that, I'm not going to comment!
>
> Since Microsoft mangles web addresses, here is the address
> with the punctuation removed:
>
> www theguardian com science article 2024 aug 31
> alexander-grothendieck-huawei-ai-artificial-intelligence
>
> Paul Taylor.
>
>
>
> ----------
>
> You're receiving this message because you're a member of the Categories
> mailing list group from Macquarie University.
>
> Leave group:
>
> https://url.au.m.mimecastprotect.com/s/mnsFCmO5wZsj5MQJ9CGfxCREFKy?domain=outlook.office365.com
>

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Re: Grothendieck in the Guardian

[P.T. Johnstone]

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Dear Vaughan,

Of course  you're right. But if the journalist had explained what a topos is, it might rather have destroyed his thesis that topos theory is a "magic bullet" which is going to solve all the problems of AI. I suppose it does no harm if people at Huawei believe that; and if it causes them to throw money at people doing research in topos theory, so much the better. But I remain sceptcal. I'm reminded of the occasion when Guerino Mazzola decided that topos theory was the "magic bullet" that would solve all the problems of musical analysis, and wrote a big book to prove his point: nothing much came of that in the long term.

Peter Johnstone

________________________________
From: Vaughan Pratt <pratt@cs.stanford.edu>
Sent: 02 September 2024 06:32
To: Wesley Phoa <doctorwes@gmail.com>
Cc: categories@mq.edu.au <categories@mq.edu.au>
Subject: Re: Grothendieck in the Guardian

"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

On Sun, Sep 1, 2024 at 1:16 PM Wesley Phoa <doctorwes@gmail.com<mailto:doctorwes@gmail.com>> wrote:
Thanks - I saw this! I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical.

Sent

> On Aug 31, 2024, at 1:13 PM, Paul Taylor <categories@paultaylor.eu<mailto:categories@paultaylor.eu>> wrote:
>
> An article about Alexander Grothiendieck has just appeared
> in the Guardian online newspaper. Be warned, it contains
> some seriously weird stuff! Toposes get a mention, though
> "not as we know them", along with Huawei, AI and Olivia
> Caramello. Beyond that, I'm not going to comment!
>
> Since Microsoft mangles web addresses, here is the address
> with the punctuation removed:
>
> www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence
>
> Paul Taylor.
>
>
>
> ----------
>
> You're receiving this message because you're a member of the Categories mailing list group from Macquarie University.
>
> Leave group:
> https://url.au.m.mimecastprotect.com/s/YsayCwV1jpSGLnAnYTVf2CJYF3A?domain=outlook.office365.com<https://url.au.m.mimecastprotect.com/s/YsayCwV1jpSGLnAnYTVf2CJYF3A?domain=outlook.office365.com>

________________________________
From: Vaughan Pratt <pratt@cs.stanford.edu>
Sent: 02 September 2024 06:32
To: Wesley Phoa <doctorwes@gmail.com>
Cc: categories@mq.edu.au <categories@mq.edu.au>
Subject: Re: Grothendieck in the Guardian

"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

On Sun, Sep 1, 2024 at 1:16 PM Wesley Phoa <doctorwes@gmail.com<mailto:doctorwes@gmail.com>> wrote:
Thanks - I saw this! I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical.

Sent

> On Aug 31, 2024, at 1:13 PM, Paul Taylor <categories@paultaylor.eu<mailto:categories@paultaylor.eu>> wrote:
>
> An article about Alexander Grothiendieck has just appeared
> in the Guardian online newspaper. Be warned, it contains
> some seriously weird stuff! Toposes get a mention, though
> "not as we know them", along with Huawei, AI and Olivia
> Caramello. Beyond that, I'm not going to comment!
>
> Since Microsoft mangles web addresses, here is the address
> with the punctuation removed:
>
> www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence
>
> Paul Taylor.
>
>
>
> ----------
>
> You're receiving this message because you're a member of the Categories mailing list group from Macquarie University.
>
> Leave group:
> https://url.au.m.mimecastprotect.com/s/YsayCwV1jpSGLnAnYTVf2CJYF3A?domain=outlook.office365.com<https://url.au.m.mimecastprotect.com/s/YsayCwV1jpSGLnAnYTVf2CJYF3A?domain=outlook.office365.com>

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Re: Grothendieck in the Guardian

[Joyal, André]

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Dear Steven,

Your comments on Grothendieck in the Guardian are very good!
I like your sculpture too.

André
________________________________
De : Steven Vickers <s.j.vickers.1@bham.ac.uk>
Envoyé : 1 septembre 2024 09:06
À : categories@mq.edu.au <categories@mq.edu.au>; Paul Taylor <pt24@PaulTaylor.EU>
Objet : Re: Grothendieck in the Guardian

I actually left a comment on the Guardian:

"Even though he was immediately recognized as a genius, he was still way ahead of his time. From the early 20th century comes the mathematical theory of topology. Grothendieck’s toposes provide a vastly more general and unifying notion of what topological spaces could be, and he provocatively named them “toposes” as those things of which topology was the study. Or ought to be. Even now, topologists have generally not caught up with him.
I wonder if much his later reclusiveness came out of disappointment at the failure of others to understand his ideas."

That final paragraph is mere speculation on my part, as I never knew him. Perhaps others on the list have better insights into that.

For your amusement, I've attached a ceramic model of his head, which I made when my wife took me on a clay modelling course.

Steve.


________________________________
From: Paul Taylor <categories@PaulTaylor.EU>
Sent: Saturday, August 31, 2024 3:16 PM
To: categories@mq.edu.au <categories@mq.edu.au>
Subject: 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.


An article about Alexander Grothiendieck has just appeared
in the Guardian online newspaper.  Be warned, it contains
some seriously weird stuff!  Toposes get a mention, though
"not as we know them", along with Huawei, AI and Olivia
Caramello.  Beyond that, I'm not going to comment!

Since Microsoft mangles web addresses, here is the address
with the punctuation removed:

www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence

Paul Taylor.



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Re: Grothendieck in the Guardian

[Steven Vickers]

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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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Re: Grothendieck in the Guardian

[David Yetter]

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I suspect Wesley expected that a journalist trying to explain toposes (particularly as Grothendieck approached them) would have made a complete hash of it and left the public with misconceptions, rather than simply no conception.

Best Thoughts,
D.Y.
________________________________
From: Vaughan Pratt <pratt@cs.stanford.edu>
Sent: Monday, September 2, 2024 12:32 AM
To: Wesley Phoa <doctorwes@gmail.com>
Cc: categories@mq.edu.au <categories@mq.edu.au>
Subject: Re: Grothendieck in the Guardian


This email originated from outside of K-State.

"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

On Sun, Sep 1, 2024 at 1:16 PM Wesley Phoa <doctorwes@gmail.com<mailto:doctorwes@gmail.com>> wrote:
Thanks - I saw this! I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical.

Sent

> On Aug 31, 2024, at 1:13 PM, Paul Taylor <categories@paultaylor.eu<mailto:categories@paultaylor.eu>> wrote:
>
> An article about Alexander Grothiendieck has just appeared
> in the Guardian online newspaper. Be warned, it contains
> some seriously weird stuff! Toposes get a mention, though
> "not as we know them", along with Huawei, AI and Olivia
> Caramello. Beyond that, I'm not going to comment!
>
> Since Microsoft mangles web addresses, here is the address
> with the punctuation removed:
>
> www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence
>
> Paul Taylor.
>
>
>
> ----------
>
> You're receiving this message because you're a member of the Categories mailing list group from Macquarie University.
>
> Leave group:
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Re: Grothendieck in the Guardian

[Vaughan Pratt]

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Steve Vickers raises the issue of when a simplification is an over-simplification.

One might think to view that as a function of the intended audience.  For an audience as comfortable with the notion of a topological space as with the notions of sets and functions, I would agree that I'd oversimplified my account.  Perhaps I was a little optimistic about mentioning graphs, but at least "graph" has no more syllables than "set", and these days half the student body is exposed to the concept thanks to Computer Science having recently passed Human Sexuality in enrollments.

I had originally intended to mention the notion of an abelian category as being related to the notion of a topos.  (Peter Freyd and I had a few exchanges on that topic in this forum a few decades ago that helped me a lot.)  However I figured that abelian groups or vector spaces as typical of objects one might find in an abelian category were above the level of sets and functions, and so decided not to mention abelian categories.

But we're all sufficiently fond of categories here to appreciate the value of being able to take the opposite of a category without feeling as ill as MSRI Director Bill Thurston claimed to be in his welcoming (and welcome) address to the Universal Algebra and Category Theory meeting at MSRI in July, 1993.  What made Bill's remark memorable for me was the sharp intake of breath heard round the room when Bill made it.  (And note how context can disambiguate a word like "welcome".)

Does "function of the intended audience" have an opposite?

Interestingly, it does.  It is called the Gunning Fog Index<https://url.au.m.mimecastprotect.com/s/XHi5Cq71jxf8OvQpmHZf8CErNF0?domain=en.wikipedia.org> for an article.  It is calculated as 0.4 times the sum of the average sentence length and the percentage of three-syllable words or longer (with certain rules about counting syllables).   The resulting number is the age group the article is optimum for.

Unfortunately this does not take into account the level and type of education needed to know what a particular word means, and "topos" can only decrease the Fog Index.  Likewise for philtrum, though acnestis will increase it, as words more likely to be encountered in Human Sexuality than Computer Science.  But at least it's a start.  A dictionary of obscure words with a score for each could be a blessing.

And so one can just write at whatever level you feel is appropriate, and calculate its fog index when done. You then know what audience your exegesis is most suitable for.

Vaughan Pratt

On Mon, Sep 2, 2024 at 6:45 AM Steven Vickers <s.j.vickers.1@bham.ac.uk<mailto:s.j.vickers.1@bham.ac.uk>> 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<mailto:pratt@cs.stanford.edu>>
Sent: Monday, September 2, 2024 6:32 AM
To: Wesley Phoa <doctorwes@gmail.com<mailto:doctorwes@gmail.com>>
Cc: categories@mq.edu.au<mailto:categories@mq.edu.au> <categories@mq.edu.au<mailto: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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Re: Grothendieck in the Guardian

[Michael Barr, Prof.]

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The problem is that, while a topos is a kind of set theory, Grothendieck thought of a topos a space and the space corresponding to Set is the one point space.  Most (educated) people's idea of space is low dimension Euclidean space and even attempting to describe the corresponding space in terms of sheaves would have left them agape.

Michael
________________________________
From: David Yetter <dyetter@ksu.edu>
Sent: Monday, September 2, 2024 12:52 PM
To: Vaughan Pratt <pratt@cs.stanford.edu>; Wesley Phoa <doctorwes@gmail.com>
Cc: categories@mq.edu.au <categories@mq.edu.au>
Subject: Re: Grothendieck in the Guardian

I suspect Wesley expected that a journalist trying to explain toposes (particularly as Grothendieck approached them) would have made a complete hash of it and left the public with misconceptions, rather than simply no conception.

Best Thoughts,
D.Y.
________________________________
From: Vaughan Pratt <pratt@cs.stanford.edu>
Sent: Monday, September 2, 2024 12:32 AM
To: Wesley Phoa <doctorwes@gmail.com>
Cc: categories@mq.edu.au <categories@mq.edu.au>
Subject: Re: Grothendieck in the Guardian


This email originated from outside of K-State.

"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

On Sun, Sep 1, 2024 at 1:16 PM Wesley Phoa <doctorwes@gmail.com<mailto:doctorwes@gmail.com>> wrote:
Thanks - I saw this! I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical.

Sent

> On Aug 31, 2024, at 1:13 PM, Paul Taylor <categories@paultaylor.eu<mailto:categories@paultaylor.eu>> wrote:
>
> An article about Alexander Grothiendieck has just appeared
> in the Guardian online newspaper. Be warned, it contains
> some seriously weird stuff! Toposes get a mention, though
> "not as we know them", along with Huawei, AI and Olivia
> Caramello. Beyond that, I'm not going to comment!
>
> Since Microsoft mangles web addresses, here is the address
> with the punctuation removed:
>
> www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence
>
> Paul Taylor.
>
>
>
> ----------
>
> You're receiving this message because you're a member of the Categories mailing list group from Macquarie University.
>
> Leave group:
> https://url.au.m.mimecastprotect.com/s/CeJ2Cp81gYCnzZB21IPf7CGchfM?domain=outlook.office365.com<https://url.au.m.mimecastprotect.com/s/CeJ2Cp81gYCnzZB21IPf7CGchfM?domain=outlook.office365.com>

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Re: Grothendieck in the Guardian

[Eduardo J. Dubuc]

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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
> 
> 
> You're receiving this message because you're a member of the Categories 
> mailing list group from Macquarie University. To take part in this 
> conversation, reply all to this message.
> View group files 
> <https://url.au.m.mimecastprotect.com/s/MrKAC91W8rCkm43zxSof9CqjHf1?domain=outlook.office365.com>   | Leave group <https://url.au.m.mimecastprotect.com/s/dUdWC0YKgRsG2Nok4iDhYC9sG45?domain=outlook.office365.com>   | Learn more about Microsoft 365 Groups <https://url.au.m.mimecastprotect.com/s/wxbeCgZ05JfAlE0qNH2iEC41Vuj?domain=aka.ms>
> 

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Re: Grothendieck in the Guardian

[John Baez]

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On Mon, Sep 2, 2024 at 2:57 PM Steven Vickers <s.j.vickers.1@bham.ac.uk<mailto:s.j.vickers.1@bham.ac.uk>> wrote:

> Dear Vaughan,

> It's easy to make the summary "A topos is simply one of many possible generalizations of sets and their functions ...", but that's definitely an over-simplification.

I thought Vaughan was trying to say how the Guardian could have explained toposes.  Such an explanation had darn well better be simplified, even "over-simplified".

But I will attack Vaughan from the other side (hi Vaughan!).  I don't think the Guardian editors would allow a journalist to give such a technical and mysterious explanation.

But maybe I'm underestimating math education in the UK.  Do typical Guardian readers know what "functions between sets" are?   I think in the US most people, if they remember their math classes at all, have only heard of functions from the real numbers to the real numbers, like polynomials and the dreaded "trig functions".

But I really don't have a good sense of what most people know about math, or what the article could have usefully done to explain toposes.

Best,
jb




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Re: Grothendieck in the Guardian

[Bas Spitters]

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> Are there real-world applications of toposes? Journalists would love to know, I guess.
Toposes are fx used in program verification: e.g. https://url.au.m.mimecastprotect.com/s/WqzrC1WLjwsMpAlP9ILf9CVbcnO?domain=iris-project.org
This ranges from the topos of trees to more advantaged topos theory
used in modal type theories or homotopy type theory.

On Mon, Sep 2, 2024 at 11:26 PM Patrik Eklund <peklund@cs.umu.se> wrote:
>
> "Do we want to keep mathematics a dark secret, or what?"
>
> Apparently, yes, we do.
>
> Obviously, the definition of a topos is the same for all of us. But the way we use it, the way we attach it to other mathematical structures, is part of our own secrets. And there are apparently many such secrets.
>
> I might even believe that Grothendieck's own perception, of what toposes really are, changed over time, and indeed in dialogue with the scientific community, a community which is not a closed one, but very much part of society.
>
> Clearly, there may remain parts of "aus liebe zur Kunst" in topos theory, as for any part of mathematical theories for that matter, but generally speaking, there are always objectives, and requirements for theories to be applicable, applicability in a broader sense.
>
> Are there real-world applications of toposes? Journalists would love to know, I guess.
>
> Best,
>
> Patrik
>
>
>
> On 2024-09-02 08:32, Vaughan Pratt wrote:
>
> "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
>
> On Sun, Sep 1, 2024 at 1:16 PM Wesley Phoa <doctorwes@gmail.com> wrote:
>
> Thanks - I saw this! I'm relieved the journalist didn't try to explain what a topos was, or indeed anything mathematical.
>
> Sent
>
> > On Aug 31, 2024, at 1:13 PM, Paul Taylor <categories@paultaylor.eu> wrote:
> >
> > An article about Alexander Grothiendieck has just appeared
> > in the Guardian online newspaper. Be warned, it contains
> > some seriously weird stuff! Toposes get a mention, though
> > "not as we know them", along with Huawei, AI and Olivia
> > Caramello. Beyond that, I'm not going to comment!
> >
> > Since Microsoft mangles web addresses, here is the address
> > with the punctuation removed:
> >
> > www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence
> >
> > Paul Taylor.
> >
> >
> >
> > ----------
> >
> > You're receiving this message because you're a member of the Categories mailing list group from Macquarie University.
> >
> > Leave group:
> > https://url.au.m.mimecastprotect.com/s/lr4lC2xMRkUpk2gB9C1hRC5Br8Y?domain=outlook.office365.com

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Re: Grothendieck in the Guardian

[Clemens Berger]

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Dear Paul and Wesley, 

 thanks for having pointed to the Guardian's article ! Last August
"France Culture" hosted a radio broadcast in five parts about
Grothendieck's life: 


https://url.au.m.mimecastprotect.com/s/iV8hCNLJxki0NqWWwfmfpCybJDB?domain=radiofrance.fr


 I wamly recommend this broadcast to everyone understanding French. The
common point with the Guardian's article are interviews with
Grothendieck's children, yet there are many more including one with
Pierre Deligne and one with an ingeneer of Huawei. 

 While I was upset after having read the Guardian's article, feeing that
its global conclusion was an illustration of the relationship that might
exist between mathematical genius and mental illness, the French
broadcast delivers a much brighter picture of Grothendieck's life
showing his deep emotional involvment in everything he did, including
mathematics. 

 All the best, 

 Clemens. 

Le 2024-08-31 22:24, Wesley Phoa a écrit : 

> Thanks - I saw this! I'm relieved the journalist didn't try to explain what a topos was, or indeed anything mathematical. 
> 
> Sent
> 
>> On Aug 31, 2024, at 1:13 PM, Paul Taylor <categories@paultaylor.eu> wrote:
>> 
>> An article about Alexander Grothiendieck has just appeared
>> in the Guardian online newspaper. Be warned, it contains
>> some seriously weird stuff! Toposes get a mention, though
>> "not as we know them", along with Huawei, AI and Olivia
>> Caramello. Beyond that, I'm not going to comment!
>> 
>> Since Microsoft mangles web addresses, here is the address
>> with the punctuation removed:
>> 
>> www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence
>> 
>> Paul Taylor.
>> 
>> 
>> 
>> ----------
>> 
>> You're receiving this message because you're a member of the Categories mailing list group from Macquarie University.
>> 
>> Leave group:
>> https://url.au.m.mimecastprotect.com/s/i1VkCOMK7YcpAL88EIvh8CGVUiU?domain=outlook.office365.com [1]
 

Links:
------
[1]
https://url.au.m.mimecastprotect.com/s/i1VkCOMK7YcpAL88EIvh8CGVUiU?domain=outlook.office365.com

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Re: Grothendieck in the Guardian

[Ross Street]

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Yes Clemens. That **is** a major concern.
==Ross//

On 3 Sep 2024, at 7:50 PM, Clemens Berger <Clemens.BERGER@univ-cotedazur.fr> wrote:

an illustration of the relationship that might exist between mathematical genius and mental illness



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Re: Grothendieck in the Guardian

[Steven Vickers]

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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/qRDBCBNqgBC7VXLrZSzfkC2VWXi?domain=arxiv.org.

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

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.


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
>
>
> You're receiving this message because you're a member of the Categories
> mailing list group from Macquarie University. To take part in this
> conversation, reply all to this message.
> View group files
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Re: Grothendieck in the Guardian

[P.T. Johnstone]

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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/jm31C2xMRkUpkLyO6TnfRC5UBQ9?domain=arxiv.org<https://url.au.m.mimecastprotect.com/s/jm31C2xMRkUpkLyO6TnfRC5UBQ9?domain=arxiv.org>.

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

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.


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
>
>
> You're receiving this message because you're a member of the Categories
> mailing list group from Macquarie University. To take part in this
> conversation, reply all to this message.
> View group files
> <https://url.au.m.mimecastprotect.com/s/dvNMC3QNl1SpmjN0xTqhoCQYwul?domain=outlook.office365.com<https://url.au.m.mimecastprotect.com/s/dvNMC3QNl1SpmjN0xTqhoCQYwul?domain=outlook.office365.com>>   | Leave group <https://url.au.m.mimecastprotect.com/s/kXmoC4QO8xSBJZk36sBiYC4oOtL?domain=outlook.office365.com<https://url.au.m.mimecastprotect.com/s/kXmoC4QO8xSBJZk36sBiYC4oOtL?domain=outlook.office365.com>>   | Learn more about Microsoft 365 Groups <https://url.au.m.mimecastprotect.com/s/E72RC5QP8ySZ0yN4RH2soCkhX5U?domain=aka.ms<https://url.au.m.mimecastprotect.com/s/E72RC5QP8ySZ0yN4RH2soCkhX5U?domain=aka.ms>>
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Re: Grothendieck in the Guardian

[Eduardo J. Dubuc]

[-- Attachment #1: Type: text/plain, Size: 9825 bytes --]


Dear Peter, I am aware of the recognition by Grothendieck of Lawvere 
contributions, specially concerning the subobject classifier, that he 
refereed to as "the Lawvere element" or rather "The Lawvere object", a 
term also used by Cartier.

Carboni told me that he and Lawvere visited Grothendiek when he was 
already a hermit, he would not speak a word, but when he saw Lawvere he 
did!, and said "oh!, Bill !", and seemed very happy of Bill's visit. But 
only wrote things in pieces of paper as a way to communicate.

Grothendieck's annoyance, in his Chicago visit, in an informal 
interchange at a table in Jimmy's bar in Hyde Park, was about to call or 
name "Topos" the Lawvere concept, not about that he had failed to spot 
the notion of subobject classifier.

Grothendieck reflected a lot about which name to give to his main and 
fundamental concept, and came up with "Topos". He felt that when people 
say "Topos", they should be referring to his notion, and not to any 
other one.

Eduardo Dubuc

On 03/09/2024 1:48 PM, P.T. Johnstone wrote:
> 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/hwUnCk815RCOnDj5LH2fOCGpvs0?domain=arxiv.org 
> <https://url.au.m.mimecastprotect.com/s/hwUnCk815RCOnDj5LH2fOCGpvs0?domain=arxiv.org>.
> 
> 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
> 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.
> 
> 
> 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
>  >
>  >
>  > You're receiving this message because you're a member of the Categories
>  > mailing list group from Macquarie University. To take part in this
>  > conversation, reply all to this message.
>  > View group files
>  > 
> <https://url.au.m.mimecastprotect.com/s/GwZIClx1OYU2oW8X9u9hoCzbTKm?domain=outlook.office365.com <https://url.au.m.mimecastprotect.com/s/GwZIClx1OYU2oW8X9u9hoCzbTKm?domain=outlook.office365.com>>   | Leave group <https://url.au.m.mimecastprotect.com/s/8bJNCmO5wZsj5V3W0TBixCR7qbA?domain=outlook.office365.com <https://url.au.m.mimecastprotect.com/s/8bJNCmO5wZsj5V3W0TBixCR7qbA?domain=outlook.office365.com>>   | Learn more about Microsoft 365 Groups <https://url.au.m.mimecastprotect.com/s/mHWYCnx1Z5U7G42myUZsYCJICKV?domain=aka.ms <https://url.au.m.mimecastprotect.com/s/mHWYCnx1Z5U7G42myUZsYCJICKV?domain=aka.ms>>
>  >

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Re: Grothendieck in the Guardian

[P.T. Johnstone]

[-- Attachment #1: Type: text/plain, Size: 9611 bytes --]

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

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.

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/_ub2C4QO8xSBJZPzziOfYC4PGdV?domain=arxiv.org<https://url.au.m.mimecastprotect.com/s/_ub2C4QO8xSBJZPzziOfYC4PGdV?domain=arxiv.org>.

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

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.


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
>
>
> You're receiving this message because you're a member of the Categories
> mailing list group from Macquarie University. To take part in this
> conversation, reply all to this message.
> View group files
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Re: Grothendieck in the Guardian

[Colin McLarty]

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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/ltpSCr810kC8APjP0T7foC4yXHp?domain=spartacus-idh.com

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
>
> *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.
>
> 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/IbNQCvl1g2S7WpgpjcXh8CQ2ZTV?domain=arxiv.org
> <https://url.au.m.mimecastprotect.com/s/IbNQCvl1g2S7WpgpjcXh8CQ2ZTV?domain=arxiv.org>
> .
>
> 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
>
> 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.
>
>
> 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
> >
> >
> > You're receiving this message because you're a member of the Categories
> > mailing list group from Macquarie University. To take part in this
> > conversation, reply all to this message.
> > View group files
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Re: Grothendieck in the Guardian

[Ross Street]

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On 4 Sep 2024, at 2:48 AM, P.T. Johnstone <ptj1000@cam.ac.uk> wrote:

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!

Memory tells me that a subobject classifier does occur in SGA, perhaps in an example, with the \Omega notation which
Lawvere and Tierney adopted. So Bill Lawvere was surprised when he heard Grothendieck's "Lawvere element"
suggestion. Of course, the power that the cartesian closedness and the subobject classifier unleash was developed
by Bill and Myles. Their's also was the ability to express a Grothendieck topology structure concisely as a monad on
 \Omega and to see it in Paul Cohen's forcing construction.

Ross


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Re: Grothendieck in the Guardian

[Colin McLarty]

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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
>
> *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.
>
> 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=spartacus-idh.com
>
> 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
>
> *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.
>
> 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=arxiv.org
> <https://url.au.m.mimecastprotect.com/s/xNE7CZY146s5M924GHjhjCBtbLN?domain=arxiv.org>
> .
>
> 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
>
> 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.
>
>
> 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
> >
> >
> > You're receiving this message because you're a member of the Categories
> > mailing list group from Macquarie University. To take part in this
> > conversation, reply all to this message.
> > View group files
> > <
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Re: Grothendieck in the Guardian

[Jon Sterling]

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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 
> 
> *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.
>
> 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=arxiv.org 
> <https://url.au.m.mimecastprotect.com/s/H6F7CQnM1Wfk6Kv3VCxfpCGrdG3?domain=arxiv.org>.
>
> 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 
> 
> 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.
>
>
> 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
>>
>>
>> You're receiving this message because you're a member of the Categories
>> mailing list group from Macquarie University. To take part in this
>> conversation, reply all to this message.
>> View group files
>> <https://url.au.m.mimecastprotect.com/s/KV2tCRONg6svrp4RYUNh4C1u_QH?domain=outlook.office365.com <https://url.au.m.mimecastprotect.com/s/KV2tCRONg6svrp4RYUNh4C1u_QH?domain=outlook.office365.com>>   | Leave group <https://url.au.m.mimecastprotect.com/s/Gfb4CVARmOHxlEq5BTyiGCELZA9?domain=outlook.office365.com <https://url.au.m.mimecastprotect.com/s/Gfb4CVARmOHxlEq5BTyiGCELZA9?domain=outlook.office365.com>>   | Learn more about Microsoft 365 Groups <https://url.au.m.mimecastprotect.com/s/660MCWLVn6i5jQrP3CnsPCo8muw?domain=aka.ms <https://url.au.m.mimecastprotect.com/s/660MCWLVn6i5jQrP3CnsPCo8muw?domain=aka.ms>>
>>

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Re: Grothendieck in the Guardian

[Ross Street]

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Camell Kachour has just let me know that 10 years ago he wrote to Myles Tierney.

Myles kindly responded (21 Nov 2014) with:

==================================
Dear Camell,

Attached is the remark concerning Omega. It is from the first (ever)
edition of "Cohomologie étale des schémas" by Artin and Grothendiek,
1963-1964.

Best

Myles
===================================

I have put Myles' attachment at:
http://science.mq.edu.au/~street/SGA.pdf

Ross

On 4 Sep 2024, at 11:51 AM, Ross Street <ross.street@mq.edu.au> wrote:


On 4 Sep 2024, at 2:48 AM, P.T. Johnstone <ptj1000@cam.ac.uk> wrote:

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!

Memory tells me that a subobject classifier does occur in SGA, perhaps in an example, with the \Omega notation which
Lawvere and Tierney adopted. So Bill Lawvere was surprised when he heard Grothendieck's "Lawvere element"
suggestion. Of course, the power that the cartesian closedness and the subobject classifier unleash was developed
by Bill and Myles. Their's also was the ability to express a Grothendieck topology structure concisely as a monad on
 \Omega and to see it in Paul Cohen's forcing construction.

Ross



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