Paper on slice stability in Locale Theory

Paper on slice stability in Locale Theory

[Townsend, Christopher]

I was sorry not to be at the Calais meeting the other week and had hoped
to have the paper 

 

Aspects of slice stability in Locale Theory

 

http://www.christophertownsend.org/Documents/SliceStability_v30.pdf

 

ready for it. 

 

This paper proves that a particular axiomatic approach to locale theory
is slice stable and then uses this slice stability to prove some known
locale theory/topos theory results.

All the results are known (aside from a conjecture on localic
subgroupoids) so it is really an investigation into this particular
axiomatisation rather than a presentation of new results. 

It gives what I think is a novel proof of the well known slice stability
result: Loc/Y = Loc_Sh(Y). 

 

Feedback is welcome. For example, I have always attributed the result
Loc/Y = Loc_Sh(Y) to Joyal and Tierney. Am I right? 

 

Regards, Christopher 

 

Re: Paper on slice stability in Locale Theory

[Michael Fourman]

The result was in the air - and I'm sure that Joyal in particular saw
it clearly.

For me it (or at least the logical version of it given in Fourman-
Scott) came as an answer to Dana's request for a description of an
internal cHa in a category of "Omega sets". I don't think I had
already heard it from Joyal - but I'm sure he already knew it. It was
easy to see that an object of Loc/Omega represented an internal cHa,
by generalisation of the representation of examples such as the
internal cHa O(R) in Sh(X) as O(RxX) -> O(X), or relaxation of the
representation of sheaves by local homeomorphisms.

Our paper was deliberately (and perhaps mistakenly) concrete rather
than abstract, so we didn't think to phrase the representation
explicitly as an equivalence of categories.

A related (but even more logical and obstinately pointed) result is
given in
Michael P. Fourman. T1 spaces over topological sites. J. Pure and
Applied Algebra, 27(3):223-224, March 1983.


On 6 Aug 2008, at 16:00, Prof. Peter Johnstone wrote:

> On Thu, 31 Jul 2008, Townsend, Christopher wrote:
>
>> Feedback is welcome. For example, I have always attributed the result
>> Loc/Y =3D Loc_Sh(Y) to Joyal and Tierney. Am I right?=20
>>
> I think it predates the Joyal--Tierney work by a couple of years.
> It's (more or less) present in the long Fourman--Scott paper on
> sheaves and logic in SLN 753 (the Proceedings of the 1977 Durham
> symposium), but they don't claim originality for it.
>
> Peter Johnstone
>
>
>
> _______________________________________________
> categories mailing list
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>

Professor Michael Fourman FBCS CITP
Head of Informatics
University of Edinburgh
Informatics Forum
10 Crichton Street
Edinburgh
EH8 9AB
Tel: +44 131 650 2690
michael.fourman<at>ed.ac.uk

Re: Paper on slice stability in Locale Theory

[Prof. Peter Johnstone]

On Thu, 31 Jul 2008, Townsend, Christopher wrote:

> Feedback is welcome. For example, I have always attributed the result
> Loc/Y =3D Loc_Sh(Y) to Joyal and Tierney. Am I right?=20
>
I think it predates the Joyal--Tierney work by a couple of years.
It's (more or less) present in the long Fourman--Scott paper on
sheaves and logic in SLN 753 (the Proceedings of the 1977 Durham
symposium), but they don't claim originality for it.

Peter Johnstone