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Constructive algebraic integration theory without choice”, in Mathematics, Algorithms and Proofs
 Dagstuhl Seminar Proceedings, 05021, Internationales Begegnungs und Forschungszentrum (IBFI), Schloss Dagstuhl
, 2005
"... Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpret ..."
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Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpreted in MartinL type theory or in a predicative topos in the sense of Moerdijk and Palmgren. We outline how to develop most of Bishop’s theorems on integration theory that do not mention points explicitly. Coquand’s constructive version of the Stone representation theorem is an important tool in this process. It is also used to give a new proof of Bishop’s spectral theorem.
A characterization of partial metrizability: Domains are quantifiable
 Theoretical Computer Science
, 2001
"... A characterization of partial metrizability is given which provides a partial solution to an open problem stated by Kunzi in the survey paper Nonsymmetric Topology ([Kun93], problem 7 ). The characterization yields a powerful tool which establishes a correspondence between partial metrics and ..."
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Cited by 8 (3 self)
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A characterization of partial metrizability is given which provides a partial solution to an open problem stated by Kunzi in the survey paper Nonsymmetric Topology ([Kun93], problem 7 ). The characterization yields a powerful tool which establishes a correspondence between partial metrics and special types of valuations, referred to as Qvaluations (cf. also [Sch00]). The notion of a Qvaluation essentially combines the wellknown notion of a valuation with a weaker version of the notion of a quasiunimorphism, i.e. an isomorphism in the context of quasiuniform spaces. As an application, we show that #continuous dcpo's are quantifiable in the sense of [O'N97], i.e. the Scott topology and partial order are induced by a partial metric. For #algebraic dcpo's the Lawson topology is induced by the associated metric. The partial metrization of general domains improves prior approaches in two ways:  The partial metric is guaranteed to capture the Scott topology as opposed to e.g. [Smy87],[BvBR95],[FS96] and [FK97], which in general yield a coarser topology.
On the Yoneda completion of a quasimetric space
 Theoretical Computer Science
, 2002
"... Several theories aimed at reconciling the partial order and the metric space approaches to Domain Theory have been presented in the literature (e.g. [FK97], [BvBR9 8], [Smy89] and [Wag94]). We focus in this paper on two of these approaches: the Yoneda completion of generalized metric spaces of [BvBR ..."
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Cited by 8 (4 self)
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Several theories aimed at reconciling the partial order and the metric space approaches to Domain Theory have been presented in the literature (e.g. [FK97], [BvBR9 8], [Smy89] and [Wag94]). We focus in this paper on two of these approaches: the Yoneda completion of generalized metric spaces of [BvBR98], which finds its roots in work by Lawvere ([Law73], cf. also [Wag94]) and which is related to early work by Stoltenberg (e.g. [Sto67], [Sto67a] and [FG84]), and the Smyth completion ([Smy89],[Smy91],[Smy94],[Sun93] and [Sun95]). A netversion of the Yoneda completion, complementing the netversion of the Smyth completion ([Sun95]), is given and a comparison between the two types of completion is presented. The following open question is raised in [BvBR98]: &quot;An interesting question is to characterize the family of generalized metric spaces for which [the Yoneda] completion is idempotent (it contains at least all ordinary metric spaces).&quot; We show that the largest class of quasimetric spaces idempotent under the Yoneda completion is precisely the class of Smythcompletable spaces. A similar result has been obtained independently by B. Flagg and P. Sünderhauf in [FS96]
The Correspondence Between Partial Metrics and Semivaluations
"... Partial metrics, or the equivalent weightable quasimetrics, have been introduced in [Mat94] as part of the study of the denotational semantics of data flow networks (cf. also [Mat95]). The interest in valuations in connection to Domain Theory derives from e.g. [JP89], [Jon89], [Eda94] and [Hec95 ..."
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Cited by 7 (5 self)
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Partial metrics, or the equivalent weightable quasimetrics, have been introduced in [Mat94] as part of the study of the denotational semantics of data flow networks (cf. also [Mat95]). The interest in valuations in connection to Domain Theory derives from e.g. [JP89], [Jon89], [Eda94] and [Hec95]. Connections between partial metrics and valuations have been discussed in the literature, e.g. [O'N97], [BS97] and [BSh98]. In each
Localic completion of generalized metric spaces I
, 2005
"... Abstract. Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X 2 to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero selfdistance law and the triangle inequality. We describe a complet ..."
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Abstract. Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X 2 to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero selfdistance law and the triangle inequality. We describe a completion of gms’s by Cauchy filters of formal balls. In terms of Lawvere’s approach using categories enriched over [0, ∞], the Cauchy filters are equivalent to flat left modules. The completion generalizes the usual one for metric spaces. For quasimetrics it is equivalent to the Yoneda completion in its netwise form due to Künzi and Schellekens and thereby gives a new and explicit characterization of the points of the Yoneda completion. Nonexpansive functions between gms’s lift to continuous maps between the completions. Various examples and constructions are given, including finite products. The completion is easily adapted to produce a locale, and that part of the work is constructively valid. The exposition illustrates the use of geometric logic to enable pointbased reasoning for locales. 1.
Relators and Metric Bisimulations (Extended Abstract)
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE 11 (1998), PAGES 17
, 1998
"... Coalgebras of set functors preserving weak pullbacks are particularly wellbehaved. Invoking a result by Carboni, Kelly, and Wood (1990), we show that this can be explained by the fact that such functors can be uniquely extended to a relator. This insight next suggests a de nition of metric bisimula ..."
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Cited by 3 (0 self)
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Coalgebras of set functors preserving weak pullbacks are particularly wellbehaved. Invoking a result by Carboni, Kelly, and Wood (1990), we show that this can be explained by the fact that such functors can be uniquely extended to a relator. This insight next suggests a de nition of metric bisimulation.
Quantitative Semantics, Topology, and Possibility Measures
, 1997
"... Given a topological space X and a complete lattice L, we study the space of L predicates F L (X) = [X ! L op ] op , continuous maps from X to L op in its Scotttopology. It yields a functor F L (\Delta) from TOPL, a full subcategory of TOP subsuming continuous domains, to SUP, the category ..."
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Cited by 3 (2 self)
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Given a topological space X and a complete lattice L, we study the space of L predicates F L (X) = [X ! L op ] op , continuous maps from X to L op in its Scotttopology. It yields a functor F L (\Delta) from TOPL, a full subcategory of TOP subsuming continuous domains, to SUP, the category of complete suplattices and maps preserving suprema. Elements of F 2 (X) are continuous predicates (= closed sets), and elements of F [0;1] (X) may be viewed as probabilistic predicates. Alternatively, one may consider the complete suplattice P L (X) = O(X)\Gamma ffi L of maps ¯: O(X) ! L preserving suprema (= possibility measures), which results in another functor P L (\Delta) from TOP to SUP. We show that these functors are equivalent for two restrictions. First, we leave SUP unchanged and restrict TOPL to CONT, the category of continuous domains in their Scotttopology; second, we fix TOP but restrict L to cocontinuous lattices. Conversely, if F L (X) and P L (X) are isomorphic f...
On the categorical meaning of Hausdorff and Gromov distances
 I. Topology Appl
"... Abstract. Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V. The Hausdorff functor which, for every Vcategory X, provides the powerset of X with a suitable Vcategory structure, is part of a monad on VCat whose Eile ..."
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Abstract. Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V. The Hausdorff functor which, for every Vcategory X, provides the powerset of X with a suitable Vcategory structure, is part of a monad on VCat whose EilenbergMoore algebras are ordercomplete. The Gromov construction may be pursued for any endofunctor K of VCat. In order to define the Gromov “distance ” between Vcategories X and Y we use Vmodules between X and Y, rather than Vcategory structures on the disjoint union of X and Y. Hence, we first provide a general extension theorem which, for any K, yields a lax
A Powerdomain of Possibility Measures
, 1997
"... We provide a domaintheoretic framework for possibility theory by studying possibility measures on the lattice of opens of a topological space. The powerspaces Poss(X) and Poss [0;1] (X) of all such maps extend to functors in the natural way. We may think of possibility measures as continuous valu ..."
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We provide a domaintheoretic framework for possibility theory by studying possibility measures on the lattice of opens of a topological space. The powerspaces Poss(X) and Poss [0;1] (X) of all such maps extend to functors in the natural way. We may think of possibility measures as continuous valuations by replacing `+' with `' in their modular law. The functors above send continuous maps to supmaps and continuous domains to completely distributive lattices; in the latter case they are locally continuous. Finite suprema of scalar multiples of point valuations form a basis of the powerdomains above if O(X) is the Scotttopology of a continuous domain. The notion of [0; 1] and [0; 1]modules corresponds to that of continuous cones if addition on the reals and on the module is replaced by suprema. The powerdomain Poss(D) is the free [0; 1]module and Poss [0;1] (D) the free [0; 1]module over a continuous domain D. 1 Possibility Measures This extended abstract attempts to recast som...