The DeMorganization of a locale

TL;DR

This paper explores the DeMorganization of locales, showing it is a fitted sublocale and providing a concrete description, with a key result linking DeMorganization to Booleanization in metrizable locales.

Contribution

It introduces the concept of DeMorganization in the localic context, demonstrating its geometric nature and establishing its equivalence to Booleanization for certain metrizable locales.

Findings

DeMorganization is always a fitted sublocale.

For metrizable locales without isolated points, DeMorganization equals Booleanization.

Extremally disconnected metric locales without isolated points are Boolean.

Abstract

In 2009, Caramello proved that each topos has a largest dense subtopos whose internal logic satisfies De Morgan law (also known as the law of the weak excluded middle). This finding implies that every locale has a largest dense extremally disconnected sublocale, referred to as its DeMorganization. In this paper, we take the first steps in exploring the DeMorganization in the localic context, shedding light on its geometric nature by showing that it is always a fitted sublocale and by providing a concrete description. Explicit examples of DeMorganizations for toposes that do not satisfy De Morgan law are rather difficult to find. We present a contribution in that direction, with the main result of the paper showing that for any metrizable locale (without isolated points), its DeMorganization coincides with its Booleanization. This, in particular, implies that any extremally disconnected metric locale (without isolated points) must be Boolean, generalizing a well-known result for topological spaces to the localic setting.