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Email Gateway (Oct 28 2023 at 20:36):From: JS Lemay <js.lemay@mq.edu.au>
From: Martin Escardo <escardo.martin@gmail.com>
Sent: Sunday, October 29, 2023 7:03:55 AM (UTC+10:00) Canberra, Melbourne, Sydney
To: Categories mailing list
Subject: Higgs object
Dear topos theorists,
Recently I found myself coming across Higgs' object.
I only know two references that mention it, the Elephant, and Freyd's
paper "Choice and well-ordering".
Is there an original paper by Denis Higgs, or at least a manuscript by
him that you may have seen or have available?
When was the first time the Higgs' object showed up in topos theory?
References or recollections are welcome.
Thanks,
Martin
Email Gateway (Oct 29 2023 at 21:09):From: JS Lemay <js.lemay@mq.edu.au>
From: Martin Escardo <escardo.martin@gmail.com>
Sent: Monday, October 30, 2023 8:09:01 AM (UTC+10:00) Canberra, Melbourne, Sydney
To: Categories mailing list <categories@mq.edu.au>
Subject: Re: Higgs object
I got lots of private replies - thanks.
The most informative one was that Hugo Volger published in 1975 a paper that includes "Higgs involution theorem", namely that any embedding of \Omega into itself is an involution (and hence an isomorphism).
(1) Hugo Volger. "Ultrafilters, ultrapowers and finiteness in a topos". Journal of Pure and Applied Algebra 6 (1975) 345-356
(2) Then there is Johnstone's paper "Automorphisms of \Omega". Algebra Universalis, 9 (1979) 1-7.
And then the references I mentioned below already when I asked the question, the Elephant and a paper by Freyd.
It seems to me that the Higgs object was introduced by Johnstone in the above paper (and repeated in the Elephant). In the internal language, it is the object {q : \Omega | for all p : Omega, p or (p implies q)}.
So the answers didn't uncover anything new, in terms of mathematics, but they include the new historical fact, to me, that it was Volger who first published Higgs' Involution Theorem.
There doesn't seem to be anything published by Higgs himself about this.
Best,
Martin
On 28/10/2023 21:03, Martin Escardo wrote:
Dear topos theorists,
Recently I found myself coming across Higgs' object.
I only know two references that mention it, the Elephant, and Freyd's paper "Choice and well-ordering".
Is there an original paper by Denis Higgs, or at least a manuscript by him that you may have seen or have available?
When was the first time the Higgs' object showed up in topos theory?
References or recollections are welcome.
Thanks,
Martin
Email Gateway (Oct 30 2023 at 10:49):From: JS Lemay <js.lemay@mq.edu.au>
From: ptj@maths.cam.ac.uk <ptj@maths.cam.ac.uk>
Sent: Monday, October 30, 2023 9:31:40 PM (UTC+10:00) Canberra, Melbourne, Sydney
To: escardo.martin@gmail.com <escardo.martin@gmail.com>
Cc: Categories mailing list <categories@mq.edu.au>
Subject: Re: Higgs object
Dear Martin,
Apologies for not responding sooner; I was busy over the weekend, and
I wanted to consult my own copy of `Automorphisms of Omega' before
replying.
I think I learned of Higgs's result, that every monic endomorphism
of Omega is an involution, from Gavin Wraith in about 1973. As far
as I know, Denis never wrote it down. When I saw the result mentioned
in Volger's 1975 paper, it occurred to me that someone ought to
work out an explicit description of the automorphism group; this may
well have been known to Higgs, but I didn't get it from either Wraith
or Volger. I also described the internal object of automorphisms, so
I could be credited with having invented the Higgs object. But it was
Peter Freyd who gave it that name, when he realized it played an
important role in his work on choice and well-ordering; it's possible
that Peter had evidence that Denis had in fact defined it, but I
never saw it.
Best regards,
Peter Johnstone
On Oct 29 2023, Martin Escardo wrote:
I got lots of private replies - thanks.
The most informative one was that Hugo Volger published in 1975 a paper
that includes "Higgs involution theorem", namely that any embedding of
\Omega into itself is an involution (and hence an isomorphism).(1) Hugo Volger. "Ultrafilters, ultrapowers and finiteness in a topos".
Journal of Pure and Applied Algebra 6 (1975) 345-356(2) Then there is Johnstone's paper "Automorphisms of \Omega". Algebra
Universalis, 9 (1979) 1-7.And then the references I mentioned below already when I asked the
question, the Elephant and a paper by Freyd.It seems to me that the Higgs object was introduced by Johnstone in the
above paper (and repeated in the Elephant). In the internal language, it
is the object {q : \Omega | for all p : Omega, p or (p implies q)}.So the answers didn't uncover anything new, in terms of mathematics, but
they include the new historical fact, to me, that it was Volger who first
published Higgs' Involution Theorem.There doesn't seem to be anything published by Higgs himself about this.
Best,
MartinOn 28/10/2023 21:03, Martin Escardo wrote:
Dear topos theorists,Recently I found myself coming across Higgs' object.
I only know two references that mention it, the Elephant, and Freyd's
paper "Choice and well-ordering".Is there an original paper by Denis Higgs, or at least a manuscript by
him that you may have seen or have available?When was the first time the Higgs' object showed up in topos theory?
References or recollections are welcome.
Thanks,
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