"Sober Spaces and Continuations" (draft paper)
This is to invite your comments on Sober Spaces and Continuations Paul Taylor http://www.dcs.qmul.ac.uk/~pt/ASD/ (Please note that a lot of work has been done on this since the version that was put on the web without advertisement on 16 August.) A topological space is sober if it has exactly the points that are dictated by its open sets. We explain the analogy with the way in which computational values are determined by the observations that can be made of them. We propose a new definition of sobriety formulated in terms of lambda calculus and elementary category theory, with no reference to lattice structure, but show that, for topological spaces, this coincides with the standard lattice-theoretic definition. We show how to extend the primitive symbolic or categorical structure to make its types sober. For the natural numbers, the additional structure provides definition by description and general recursion. This is NOT ``denotational semantics of continuations using sober spaces'', though that could easily be derived. On the contrary, this paper provides the underlying lambda-calculus on the basis of which abstract Stone duality will re-axiomatise general topology. I would like to hear your comments on any of the following points: * intuitions about what it means to determine a computational value by making observations of it; * the categorical construction that is in Hayo Thielecke's thesis; * the "sober lambda calculus" that I derive from it; * re-axiomatisation of general topology using this, and in particular my notions of "prime" and "sober" expressed in the lambda calculus; * Russell's theory of descriptions, and general recursion; * computational effects that arise from Thielecke's "force". This paper serves as an introduction to my "Abstract Stone Duality" programme. However, it does NOT do the monadic construction either categorically or in terms of the "axiom of comprehension" that I presented at the APPSEM meeting in Darmstadt in March. I am currently writing these up in another paper, provisionally called "Comprehension in ASD". (I started to "get the upper hand" with the details of the proofs of the normalisation theorem about two weeks ago, and hope to be able to release a version containing all the maths, but not the narrative, before the teaching year starts again. Paul
Paul Taylor wrote:
This is to invite your comments on
Sober Spaces and Continuations
Paul Taylor
Dear Paul, thanks for your interesting article. I would like to point out that three relevant papers are missing from the references: [1] Fuehrmann, C. Direct-style and comtinuation-passing style models of control. See http://www.cs.bham.ac.uk/~cxf/research.htm. (1999). [2] Fuehrmann, C. Direct models of the computational lambda-calculus. In Proceedings of MFPS 15, ENTCS 20 (1999). [3] Selinger, P. Control categories and duality. MSCS 11:207-260 (2001). Many or all of your general categorical results (sections 3-4 and 6-7) appear in these papers. For instance, your notion of an object being "sober" (Def. 4.6) coincides with what Fuehrmann calls "satisfying Moggi's equalizing requirement", with associated theorems. I like the word "sober", because it is elegant and suggestive in this context, but Fuehrmann still deserves credit for the concept and the theorem. I liked the sections in which you apply these categorical methods to the category of locally compact topological spaces; it is interesting to see how the categorical concepts can be characterized in the concrete case. -- Peter
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