Authors’ summary: 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 \(\mathcal RL\), the family of continuous real valued functions on a locale, or pointfree space.
Thus 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. This is an appropriate context for this investigation because every \(\mathbf W\)-object can be canonically represented as a subobject of some \(\mathcal RL\).
We show that the suprema which are pointwise in the Madden representation can be characterized purely algebraically. They are precisely the suprema which are context-free, in the sense of being preserved by every \(\mathbf W\)-homomorphism out of \(G\). We show that closure under such suprema characterizes the \(\mathbf W\)-kernels among the convex \(\ell\)-subgroups. Finally, we prove that all existing joins in a \(\mathbf W\)-object \(G\) are pointwise iff its Madden frame \(L\) is Boolean, and that all existing countable joins in \(G\) are pointwise if \(L\) is a \(P\)-frame, but not conversely.
This leads up to the appropriate analog of the Nakano-Stone Theorem: a (completely regular) locale \(L\) has the feature that \(\mathcal RL\) is conditionally pointwise complete (\(\sigma\)-complete), i.e., every bounded (countable) family from \(\mathcal RL\) has a pointwise supremum in \(\mathcal RL\), 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 \(\mathcal RL\) for \(L\) a Boolean locale (\(P\)-locale). Finally, we show that a \(\mathbf W\)-object \(G\) is pointwise \(\sigma\)-complete iff it is epicomplete.