Quantales, Generalised Premetrics and Free Locales View Full Text


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Article Info

DATE

2017-12

AUTHORS

J. Bruno, P. Szeptycki

ABSTRACT

Premetrics and premetrisable spaces have been long studied and their topological interrelationships are well-understood. Consider the category Pre of premetric spaces and 𝜖 − δ continuous functions as morphisms. The absence of the triangle inequality implies that the faithful functor Pre→Top - where a premetric space is sent to the topological space it generates - is not full. Moreover, the sequential nature of topological spaces generated from objects in Pre indicates that this functor is not surjective on objects either. Developed from work by Flagg and Weiss, we illustrate an extension Pre↪P together with a faithful and surjective on objects left adjoint functor P→Top as an extension of Pre→Top. We show this represents an optimal scenario given that Pre→Top preserves coproducts only. The objects in P are metric-like objects valued on value distributive lattices whose limits and colimits we show to be generated by free locales on discrete sets. More... »

PAGES

1045-1058

References to SciGraph publications

  • 1997-06. Quantales and continuity spaces in ALGEBRA UNIVERSALIS
  • 2015-04. A note on the metrizability of spaces in ALGEBRA UNIVERSALIS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s10485-016-9465-8

    DOI

    http://dx.doi.org/10.1007/s10485-016-9465-8

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1024304879


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