On the Yoneda completion of a quasi-metric space (2002)
| Venue: | Theoretical Computer Science |
| Citations: | 6 - 4 self |
BibTeX
@MISC{Künzi02onthe,
author = {H. P. Künzi and M. P. Schellekens},
title = {On the Yoneda completion of a quasi-metric space},
year = {2002}
}
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Abstract
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 net-version of the Yoneda completion, complementing the net-version 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]: "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)." We show that the largest class of quasi-metric spaces idempotent under the Yoneda completion is precisely the class of Smyth-completable spaces. A similar result has been obtained independently by B. Flagg and P. Sünderhauf in [FS96]
