Miscellaneous papers on Abstract Stone Duality

Paul Taylor

30 June 2008

Local Compactness and the Baire Category Theorem in ASD: a feasibility study

PDF (655 kb)
DVI (316 kb)
PostScript (compressed) (218 kb)
A5 PS booklet (compressed) (206 kb)
What are these?
[12 Feb 2009]
Presented at Category Theory and Computer Science 9, Ottawa, August 2002. Published in Electronic Notes in Theoretical Computer Science 69, Elsevier, 2003.
This was the preliminary version of Computably based locally compact spaces. However, it also shows that every object has a filter basis, and proves a form of Baire's category theorem, that, for any countable family of open dense subsets, the intersection is also dense.

Scott Domains in ASD

PDF (456 kb)
DVI (207 kb)
PostScript (compressed) (167 kb)
A5 PS booklet (compressed) (155 kb)
What are these?
[12 Feb 2009]
The bulk of this paper re-works Scott's information systems within ASD, although this is largely made redundant by the treatment of domain theory in § G 12.
It begins by showing that Scott domains must be overt in order to form a cartesian closed category. The necessary and sufficient condition for overtness is that the consistency predicate in the information system be decidable.
It also constructs a model of the untyped lambda-calculus, Church's thesis (in the form of a saturated surjection N→ΣN) and the halting set (an open subspace of N that is not closed.)
An appendix characterises discardable and copyable maps wrt the monad, in the sense of Thielecke.
To the above development of Scott's domain theory within ASD should be added a treatment of Plotkin's PCF and its parallel extensions. The composite translation from PCF to domain theory to ASD to continuations would be the same as the known continuation-passing translations of PCF.
The point of this is that, since the complications of ASD dissolve when we apply it to known simple computational calculi, we would also expect ASD to give simple computational accounts of more difficult mathematical phenomena.

Tychonov's Theorem in ASD

PDF (298 kb)
DVI (144 kb)
PostScript (compressed) (220 kb)
A5 PS booklet (compressed) (207 kb)
What are these?
[12 Feb 2009]
Presented at Domain Theory Workshop 7, at the Technical University of Darmstadt, 29 August - 2 September 2004, using these 10MB scanned slides.
These notes construct Cantor space (2N) in ASD and show that it is compact. This is discussed in the context of the failure of this result in Recursive Analysis (because of the existence of Kleene Trees and the failure of König's Lemma there) and objections that were raised by Martín Escardó.
It is also shown that Baire space, i.e. the exponential NN, is not definable in ASD, and also that any overt compact subspace of Cantor space is either empty or a retract.
Finally, the notion of "canopy" is introduced - the lattice dual of bases in § G 6.

This is www.PaulTaylor.EU/ASD/misclc.php and it was derived from ASD/misclc.tex which was last modified on 20 February 2011.