Subject: categories: paper on SDT available

From: Jaap van Oosten <[email-masked]>
Date: Wed, 16 Dec 1998 12:10:31 +0100 (MET)

The following paper is available:

  Axioms and (Counter)examples in Synthetic Domain Theory

  by Jaap van Oosten and Alex K. Simpson

the paper can be found at the URL:

  http://www.math.uu.nl/publications/preprints/1080.ps.gz

ABSTRACT: Chapter 1 presents a development of basic Synthetic Domain
Theory on the basis of 4 axioms (1:\Sigma is complete; 2:\Sigma is
\neg\neg-separated; 3:\bot\in\Sigma; 4:the Phoa Principle). New results
are, that 1 and 2 imply that the \Sigma-order on \Sigma , I and F (the
initial lift algebra and the final lift coalgebra, respectively) is
(pointwise) implication, that 1 and 2 imply that complete extensional
objects (we call them complete regular \Sigma-posets) are stable under 
lifting, that under 1,2,3, axiom 4 is equivalent to \Sigma having binary
joins, and that if \Sigma is closed under N-idexed joins in \Omega, then
all complete objects are stable under lifting. We also present an analysis
of when I is an internal colimit of a diagram 0->L(0)->L^2(0)->...

Chapters 2,3,4 investigate models. We study models of the axioms in: the
Modified realizability topos Mod, the Effective topos Eff, and a particular
Grothendieck topos. In Mod, the Scott principle fails and L(2) is not
complete. In Eff, we have that the internal colimit of 0->L(0)->L^2(0)->...
is complete (whence it is not isomorphic to I), and a general theorem
characterizing I for \neg\neg-separated dominances. Finally, in a sheaf
topos we have an example where L(2) is complete but L(N) isn't.

Jaap van Oosten