Choice of terminology "bound" in the context of Grothendieck toposes

Choice of terminology "bound" in the context of Grothendieck toposes

[David Roberts]

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Hi all,

This is a question for some of the people who were present at the time: a question came up on MathOverflow about the provenance of the term "bounded geometric morphism". To my mind it is clear that it relates to having a *bound* for a topos over Set being equivalent to being a Grothendieck topos, but then I could only come up with some kind of folk-etymology for that choice of word. Was there some earlier term it arose from? Or was it chosen ab initio as a suitable word to convey an idea?

Thanks,

David


--
Dr David Roberts
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Re: Choice of terminology "bound" in the context of Grothendieck toposes

[Fernando Yamauti]

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 By the way, one further annoyance is that , in higher topos theory, one also has the notion of a bounded topos (which is not difficult to generalise to arbitrary topoi). So that gives us two unequivalent definitions  with the name "bounded geometric morphism".

  Best,
   Fernando

Em ter., 24 de set. de 2024 02:01, David Roberts <david.roberts@adelaide.edu.au<mailto:david.roberts@adelaide.edu.au>> escreveu:
Hi all,

This is a question for some of the people who were present at the time: a question came up on MathOverflow about the provenance of the term "bounded geometric morphism". To my mind it is clear that it relates to having a *bound* for a topos over Set being equivalent to being a Grothendieck topos, but then I could only come up with some kind of folk-etymology for that choice of word. Was there some earlier term it arose from? Or was it chosen ab initio as a suitable word to convey an idea?

Thanks,

David


--
Dr David Roberts
http://ncatlab.org/nlab/show/David+Roberts<https://url.au.m.mimecastprotect.com/s/oTSaC2xMRkUpVLWWKCnfRC5papQ?domain=ncatlab.org>

Adjunct Associate Lecturer
School of Computer and Mathematical Sciences
University of Adelaide
SA 5005
AUSTRALIA

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Re: Choice of terminology "bound" in the context of Grothendieck toposes

[P.T. Johnstone]

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I was certainly around at the time that terminology came into use, and I may have been the first person to use it in print (in my 1977 book). In his 1975 paper "Change of base for toposes with generators", Radu Diaconescu referred to a topos F "having generators" over a base E, but that seemed a bit cumbersome;  the single word "bounded" had the right overtones, in that it referred to a single object of F (which I later took to calling simply a "bound") that controlled how complicated the objects could be relative to those of E. I don't know whether I actually suggested the term, but as far as I can recall everyone agreed that it was a sensible choice.

Incidentally, the Grothendieck school were aware of the existence of what we would call unbounded Set-toposes, only they didn't consider them to be toposes. (There is one described in SGA4 as a "faux topos".)

Peter Johnstone

________________________________
From: David Roberts <david.roberts@adelaide.edu.au>
Sent: 24 September 2024 00:59
To: Categories mailing list <categories@mq.edu.au>
Subject: Choice of terminology "bound" in the context of Grothendieck toposes

Hi all,

This is a question for some of the people who were present at the time: a question came up on MathOverflow about the provenance of the term "bounded geometric morphism". To my mind it is clear that it relates to having a *bound* for a topos over Set being equivalent to being a Grothendieck topos, but then I could only come up with some kind of folk-etymology for that choice of word. Was there some earlier term it arose from? Or was it chosen ab initio as a suitable word to convey an idea?

Thanks,

David


--
Dr David Roberts
http://ncatlab.org/nlab/show/David+Roberts<https://url.au.m.mimecastprotect.com/s/Uz1VC91W8rCkzQkPAfof9Cqqb8p?domain=ncatlab.org>

Adjunct Associate Lecturer
School of Computer and Mathematical Sciences
University of Adelaide
SA 5005
AUSTRALIA

CRICOS Provider Number 00123M
IMPORTANT: This message may contain confidential or legally privileged
information.  If you think it was sent to you by mistake,  please delete all
copies and advise the sender.  For the purposes of the SPAM Act 2003, this
email is authorised by The University of Adelaide.


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