The lattice of smooth sublocales as a Bruns-Lakser completion

TL;DR

This paper characterizes frame morphisms that lift to maps between collections of complemented sublocales, showing that these collections form a Bruns-Lakser completion of certain locally closed sublocales.

Contribution

It establishes an isomorphism between the Booleanization of sublocales and the Bruns-Lakser completion of locally closed sublocales in frames.

Findings

Characterization of frame morphisms lifting to sublocale collections

Identification of $ extsf{S}_b(L)$ with Bruns-Lakser completion

Isomorphism between Booleanized sublocales and locally closed sublocales

Abstract

We characterise the frame morphisms $f:L\to M$ that lift to frame maps $\overline{f}:\mathsf{S}_b(L)\to \mathsf{S}_b(M)$, where $\mathsf{S}_b(L)$ is the collection of joins of complemented sublocales of a frame $L$, or equivalently the Booleanization of the collection $\mathsf{S}(L)$ of all its sublocales. We do so by proving that $\mathsf{S}_b(L)$ is isomorphic to the Bruns--Lakser completion of the meet-semilattice formed by the locally closed sublocales, i.e. the sublocales of the form $\mathfrak{c}(a)\cap \mathfrak{o}(b)$ for $a,b\in L$.