Value semigroups, value quantales, and positivity domains

Abstract

In 1981 and 1997 Kopperman and Flagg, respectively, proved that every topological space is metrisable, provided the symmetry and separation axioms are removed from the requirements on the metric, and the metric is allowed to take values in, respectively, a value semigroup or a value quantale. Seeking to construct a value quantale from a value semigroup we focus on a small portion of the structure present in a value semigroup, comprising what we call a positivity domain, and we construct its enveloping value quantale, forming part of a detailed comparison between value semigroups and value quantales. We obtain a representation theorem for value quantales in terms of positivity domains, and we outline how products of positivity domains can be used in the theory of continuity spaces instead of (the non-existent) products of value quantales.

The topology of a quantale valued metric space

“…The slogan a metric space is a category enriched in [0, ∞] naturally raises the question as to the precise role of [0, ∞] in this connection as opposed to any other monoidal category; a question that appears not to have been explicitly investigated earlier. Indirectly, this question is lurking under the surface in recent research (Bruno (2020); Chand and Weiss (2015); Flagg (1997); Hofmann and Reis (2013); Waszkiewicz (2011, 2012); Tholen (2018); Weiss (2015Weiss ( , 2016Weiss ( , 2017Weiss ( , 2018Weiss ( , 2019; Zhang (2007)) on quantale enriched categories.…”