Subject: categories: re: An Abstract Stone Duality

From: Jaap van Oosten <[email-masked]>
Date: Wed, 6 Jan 1999 12:35:26 +0100 (MET)

This is a reaction to Paul Taylor's advertisement for his paper
on Abstract Stone Duality.

Paul has considered appropriate to mention the paper "Axioms and
(Counter)examples in Synthetic Domain Theory" by Alex Simpson and
myself, and to polemicize against it in the following way:
  
> You might like to consider this paper alongside the recent work of Alex
> Simpson and Jaap van Oosten on Synthetic Domain Theory,
> 	http://www.math.uu.nl/publications/preprints/1080.ps.gz
> which is very much concerned with the infinitary axiom.  In particular
> Alex is keen to emphasise the additional completeness of the ambient
> category that is needed to construct infinitary co/limits and solutions
> of domain equations, by comparison with directed joins and fixed points
> of functions.  He considers that some attention to the axiom of replacement
> is needed here, and I agree with him, though our treatments are different.
> 
> My paper is written for a general mathematical audience, and argues its
> point of view from a position of extreme Cartesisn doubt.  I feel that
> Alex and Jaap's work is very arcane by comparison, as it depends heavily
> on Hyland's effective topos.

Thanks, Paul, for mentioning our paper, but since the above text is a
blatant misrepresentation of its content, a correction is in order.
Our paper develops the theory on the basis of 4 axioms, of which one
(the \neg\neg-separatedness of \Sigma) is rather special, as we explicitly
acknowledge. Otherwise the treatment is completely general, and "completeness
of the ambient category" nowhere enters the picture (we do have, however,
some treatment of whether the lift functor preserves internal colimits of
chains). The remark about Replacement refers to other work by Alex. 

The axiomatic treatment raises independence questions, some of which we solve
by considering models. One of these models is the effective topos. Nowhere
do we hint that the effective topos should have a privileged place among
models.

Anyway, it is funny that our work, which builds on the tradition of turning
SDT into an axiomatic theory (a process which I think is still unfinished),
tradition which was started by Pino Rosolini, Wesley Phoa, Martin Hyland and
Paul himself, is now found to be "arcane" (mind you, the whole subject is
less than twenty years old) by Paul.

Jaap van Oosten