Paper on slice stability in Locale Theory
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
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
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
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participants (3)
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Michael Fourman -
Prof. Peter Johnstone -
Townsend, Christopher