References

[Bar81]

Henk Barendregt. The Lambda Calculus: its Syntax and Semantics. Number 103 in Studies in Logic and the Foundations of Mathematics. North-Holland, 1981. Second edition, 1984.

[Bou66]

Nicolas Bourbaki. Topologie Générale. Hermann, 1966. English translation, “General Topology”, distrubuted by Springer-Verlag, 1989.

[CS88]

John Horton Conway and Neil Sloane. Sphere Packings, Lattices and Groups. Number 290 in Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, 1988.

[GHK++80]

Gerhard Gierz, Karl Heinrich Hoffmann, Klaus Keimel, Jimmie Lawson, Michael Mislove, and Dana Scott. A Compendium of Continuous Lattices. Springer-Verlag, 1980. Second edition, Continuous Lattices and Domains, published by Cambridge University Press, 2003.

[Har40]

G. H. Hardy. A Mathematician’s Apology. Cambridge University Press, 1940. Frequently reprinted.

[HL78]

Karl Hofmann and Michael Lawson. Spectral theory of continuous lattices. Transactions of the American Mathematical Society, 246:285–310, 1978.

[HM81]

Karl Hofmann and Michael Mislove. Local compactness and continuous lattices. In Bernhard Banaschewski and Rudolf-Eberhard Hoffmann, editors, Continuous Lattices, volume 871 of Springer Lecture Notes in Mathematics, pages 209–248, 1981.

[JKM99]

Achim Jung, Mathias Kegelmann, and Andrew Moshier. Multi lingual sequent calculus and coherent spaces. Fundamenta Informaticae, 37(4):369–412, 1999.

[JKM01]

Achim Jung, Mathias Kegelmann, and Andrew Moshier. Stably compact spaces and closed relations. Electronic Notes in Theoretical Computer Science, 45, 2001. MFPS 17.

[Joh82]

Peter Johnstone. Stone Spaces. Number 3 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1982.

[Joh84]

Peter Johnstone. Open locales and exponentiation. Contemporary Mathematics, 30:84–116, 1984.

[Joh85]

Peter Johnstone. Vietoris locales and localic semi-lattices. In R.-E. Hoffmann and K. H. Hoffmann, editors, Continuous Lattices and their Applications, number 101 in Pure and Applied Mathematics, pages 155–180. Marcel Dekker, 1985.

[JS96]

Achim Jung and Philipp Sünderhauf. On the duality of compact vs. open. In S. Andima, R.C. Flagg, G. Itzkowitz, P. Misra, Y. Kong, and R.D. Kopperman, editors, Papers on general topology and applications: Eleventh Summer Conference at University of Southern Maine, volume 806 of Annals of the New York Academy of Sciences, pages 214–230, 1996.

[Jun90]

Achim Jung. The classification of continuous domains. In Logic in Computer Science, pages 35–40. IEEE Computer Society Press, 1990.

[Kel55]

John Kelley. General Topology. Van Nostrand, 1955. Reprinted by Springer-Verlag, Graduate Texts in Mathematics, 27, 1975.

[Plo77]

Gordon Plotkin. LCF considered as a programming language. Theoretical Computer Science, 5:223–255, 1977.

[Sco70]

Dana Scott. Outline of a mathematical theory of computation. In 4th Annual Princeton Conference on Information Sciences and Systems, pages 169–176, 1970. Superseded by Oxford technical report PRG-2.

[Sco82]

Dana Scott. Domains for denotational semantics. In M. Nielson and E. M. Schmidt, editors, Automata, Languages and Programming: Proceedings 1982. Springer-Verlag, 1982. Lecture Notes in Computer Science 140.

[Smy77]

Michael Smyth. Effectively given domains. Theoretical Computer Science, 5:257–274, 1977.

[Tay91]

Paul Taylor. The fixed point property in synthetic domain theory. In Gilles Kahn, editor, Logic in Computer Science 6, pages 152–160. IEEE, 1991.

[Tay99]

Paul Taylor. Practical Foundations of Mathematics. Number 59 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1999.

[Thi97]

Hayo Thielecke. Categorical Structure of Continuation Passing Style. PhD thesis, University of Edinburgh, 1997. Also available as technical report ECS-LFCS-97-376.

[Vic88]

Steven Vickers. Topology Via Logic, volume 5 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1988.

[Wil70]

Peter Wilker. Adjoint product and hom functors in general topology. Pacific Journal of Mathematics, 34:269–283, 1970.

The papers on abstract Stone duality may be obtained from

[O]

Paul Taylor, Foundations for Computable Topology. in Giovanni Sommaruga (ed.), Foundational Theories of Mathematics, Kluwer 2011.

[A]

Paul Taylor, Sober spaces and continuations. Theory and Applications of Categories, 10(12):248–299, 2002.

[B]

Paul Taylor, Subspaces in abstract Stone duality. Theory and Applications of Categories, 10(13):300–366, 2002.

[C]

Paul Taylor, Geometric and higher order logic using abstract Stone duality. Theory and Applications of Categories, 7(15):284–338, 2000.

[D]

Paul Taylor, Non-Artin gluing in recursion theory and lifting in abstract Stone duality. 2000.

[E]

Paul Taylor, Inside every model of Abstract Stone Duality lies an Arithmetic Universe. Electronic Notes in Theoretical Computer Science 122 (2005) 247-296.

[F]

Paul Taylor, Scott domains in abstract Stone duality. March 2002.

[G–]

Paul Taylor, Local compactness and the Baire category theorem in abstract Stone duality. Electronic Notes in Theoretical Computer Science 69, Elsevier, 2003.

[G]

Paul Taylor, Computably based locally compact spaces. Logical Methods in Computer Science, 2 (2006) 1–70.

[H–]

Paul Taylor, An elementary theory of the category of locally compact locales. APPSEM Workshop, Nottingham, March 2003.

[H]

Paul Taylor, An elementary theory of various categories of spaces and locales. November 2004.

[I]

Andrej Bauer and Paul Taylor, The Dedekind reals in abstract Stone duality. Mathematical Structures in Computer Science, 19 (2009) 757–838.

[J]

Paul Taylor, A λ-calculus for real analysis. Journal of Logic and Analysis, 2(5), 1–115 (2010)

[K]

Paul Taylor, Interval analysis without intervals. February 2006.

[L]

Paul Taylor, Tychonov’s theorem in abstract Stone duality. September 2004.

[N]

Paul Taylor, Computability in locally compact spaces. 2010.

[AA]

Paul Taylor, Equideductive categories and their logic. 2010.

[BB]

Paul Taylor, An existential quantifier for topology. 2010.

[CC]

Paul Taylor, Cartesian closed categories with subspaces. 2009.

[DD]

Paul Taylor, The Phoa principle in equideductive topology. 2010.

[EE]

Paul Taylor, Discrete mathematics in equideductive topology. 2010.

[FF]

Paul Taylor, Equideductive topology. 2010.

[GG]

Paul Taylor, Underlying sets in equideductive topology. 2010.

Acknowledgements

This paper evolved from [G–], which included roughly Sections 5–8 and 11 of the present version. It was presented at Category Theory and Computer Science 9, in Ottawa on 17 August 2002, and at Domains Workshop 6 in Birmingham a month later. The characterisation of locally compact objects using effective bases (Section 6) had been announced on categories on 30 January 2002.

The earlier version also showed that any object has a filter basis, and went on to prove Baire’s category theorem, that the intersection of any sequence of dense open subobjects (of any locally compact overt object) is dense. These arguments were adapted from the corresponding ones in the theory of continuous lattices [GHK++80, Sections I 3.3 and 3.43].

I would like to thank Andrej Bauer, Martƒn Escardó, Peter Johnstone, Achim Jung, Jimmie Lawson, Graham White and the CTCS and LMCS referees for their comments. Graham White has given continuing encouragement throughout the abstract Stone duality project, besides being an inexhaustible source of mathematical ideas.

This research is now supported by UK EPSRC project GR/S58522, but this funding was obtained in part on the basis of the work in this paper. Apart from 5.19, 12.13, 12.14, 15.3, 15.5, 16.15 and 17.6, the work here was carried out during a period of unemployment, supported entirely from my own savings. However, I would have been unable to do this without the companionship and emotional support of my partner, Richard Symes.

© 2002–5 Paul Taylor.