Pointfree pointwise suprema in unital archimedean $\ell$-groups

TL;DR

This paper extends the concept of pointwise supremum to a pointfree setting within unital archimedean $ ext{l}$-groups, characterizing when these structures are pointwise complete in relation to boolean locales.

Contribution

It introduces a broad definition of pointwise completeness for $ ext{W}$-objects and characterizes them as those arising from boolean locales, generalizing classical theorems.

Findings

$ ext{R}L$ is conditionally pointwise complete iff $L$ is boolean.

Pointwise complete $ ext{W}$-objects are of the form $ ext{R}L$ for boolean $L$.

Pointwise $\sigma$-completeness is equivalent to being epicomplete.

Abstract

We generalize the concept of the pointwise supremum of real-valued functions to the pointfree setting. The concept itself admits a direct and intuitive formulation which makes no mention of points. But our aim here is to investigate pointwise suprema of subsets of $\mathcal{R}L$, the family of continuous real valued functions on a locale, or pointfree space. Our setting is the category $\mathbf{W}$ of archimedean lattice-ordered groups ($\ell$-groups) with designated weak order unit, with morphisms which preserve the group and lattice operations and take units to units. A main result is the appropriate analog of the Nakano-Stone Theorem: a (completely regular) locale $L$ has the feature that $\mathcal{R}L$ is conditionally pointwise complete ($\sigma $-complete), i.e., every bounded (countable) family from $\mathcal{R}L$ has a pointwise supremum in $\mathcal{R}L$, iff $L $ is boolean (a $P$-locale). We adopt a maximally broad definition of unconditional pointwise completeness ($\sigma$-completeness): a divisible $\mathbf{W}$-object $G$ is pointwise complete ($\sigma$-complete) if it contains a pointwise supremum for every subset which has a supremum in any extension. We show that the pointwise complete ($\sigma$-complete) $\mathbf{W}$-objects are those of the form $\mathcal{R}L$ for $L$ a boolean locale ($P$-locale). Finally, we show that a $\mathbf{W}$-object $G$ is pointwise $\sigma$-complete iff it is epicomplete.