Embedding locales and formal topologies into positive topologies

Archive for Mathematical Logic 57 (7-8):755-768 (2018)
Abstract
A positive topology is a set equipped with two particular relations between elements and subsets of that set: a convergent cover relation and a positivity relation. A set equipped with a convergent cover relation is a predicative counterpart of a locale, where the given set plays the role of a set of generators, typically a base, and the cover encodes the relations between generators. A positivity relation enriches the structure of a locale; among other things, it is a tool to study some particular subobjects, namely the overt weakly closed sublocales. We relate the category of locales to that of positive topologies and we show that the former is a reflective subcategory of the latter. We then generalize such a result to the category of suplattices, which we present by means of cover relations. Finally, we show that the category of positive topologies also generalizes that of formal topologies, that is, overt locales.
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DOI 10.1007/s00153-017-0605-0
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References found in this work BETA

Inductively Generated Formal Topologies.Thierry Coquand, Giovanni Sambin, Jan Smith & Silvio Valentini - 2003 - Annals of Pure and Applied Logic 124 (1-3):71-106.
Aspects of General Topology in Constructive Set Theory.Peter Aczel - 2006 - Annals of Pure and Applied Logic 137 (1):3-29.
Compactness in Locales and in Formal Topology.Steven Vickers - 2006 - Annals of Pure and Applied Logic 137 (1):413-438.
Positivity Relations on a Locale.Francesco Ciraulo & Steven Vickers - 2016 - Annals of Pure and Applied Logic 167 (9):806-819.

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