Continuity and openness of maps on locales by way of Galois adjunctions

TL;DR

This paper explores adjoint situations in pointfree topology, characterizing various types of maps between locales using elementary concepts and providing a new proof of the Joyal-Tierney open mapping theorem.

Contribution

It introduces a unified approach to characterizations of locale maps via Galois adjunctions, simplifying reasoning in localic topology and revisiting key theorems.

Findings

Characterizations of meet-preserving, localic, and open maps using adjunctions.

A new proof of the Joyal-Tierney open mapping theorem.

Analysis of interchange laws between operators in locale theory.

Abstract

We study four adjoint situations in pointfree topology that interchange images and preimages with closure and interior operators and establish with them a number of characterisations for meet-preserving maps, localic maps, open maps (in a broad sense) and open localic maps between locales. The principal and most attractive feature of these adjunctions is that they are all concerned with elementary ideas and basic concepts of localic topology: the use of the concrete language of sublocales and its technique simplifies the reasoning. We then revisit open localic maps in detail and present a new proof of Joyal-Tierney open mapping theorem. We end with a study of the interchange laws between preimages/images and closure/interior operators, making clear the similarities and differences with the classical realm.