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