Uniform locales and their constructive aspects

TL;DR

This paper develops a comprehensive constructive theory of uniform locales, including entourage and covering approaches, their equivalence, and applications to metric locales and localic algebra, emphasizing internal logic and pre-uniform locales.

Contribution

It extends Johnstone's initial work by including entourage uniformities, completions, and applications to localic rings, with a focus on constructive methods and internal logic.

Findings

Established equivalence of entourage and covering approaches to uniform locales.

Described the completion as the uniform reflection of the pre-uniform locale of Cauchy filters.

Provided a method to lift algebraic operations from rationals to reals in the pointfree setting.

Abstract

Much work has been done on generalising results about uniform spaces to the pointfree context. However, this has almost exclusively been done using classical logic, whereas much of the utility of the pointfree approach lies in its constructive theory, which can be interpreted in many different toposes. Johnstone has advocated for the development of a constructive theory of uniform locales and wrote a short paper on the basic constructive theory via covering uniformities, but he never followed this up with a discussion of entourage uniformities or completions. We present a more extensive constructive development of uniform locales, including both entourage and covering approaches, their equivalence, completions and some applications to metric locales and localic algebra. Some aspects of our presentation might also be of interest even to classically minded pointfree topologists. These include the definition and manipulation of entourage uniformities using the internal logic of the geometric hyperdoctrine of open sublocales and the emphasis on pre-uniform locales. The latter leads to a description of the completion as the uniform reflection of the pre-uniform locale of Cauchy filters and a new result concerning the completion of pre-uniform localic rings, which can be used to easily lift addition and multiplication on $\mathbb{Q}$ to $\mathbb{R}$ (or $\mathbb{Q}_p$) in the pointfree setting.