A convenient category of locales

TL;DR

This paper extends the framework of Kan extendable subcategories to define and prove the cartesian closure of the category of compactly generated strongly Hausdorff locales, generalizing previous work on fibrewise topological spaces.

Contribution

It introduces a new application of Kan extendable subcategories to locales, establishing the cartesian closure of compactly generated strongly Hausdorff locales.

Findings

The category of compactly generated strongly Hausdorff locales is cartesian closed.

The framework for fibrewise topological spaces applies to locales.

The approach unifies topological and locale-theoretic concepts.

Abstract

The notion of Kan extendable subcategories was initially introduced to define the category of compactly generated fibrewise topological spaces over a T1 base space and to establish its cartesian closure. In this paper, we show that the same framework can likewise be applied to define the category of compactly generated strongly Hausdorff locales and to prove that it, too, is cartesian closed