The lattice of smooth sublocales as a Bruns–Lakser completion

Abstract

We characterise the frame morphisms f:L→M that lift to frame maps f¯:Sb(L)→Sb(M), where Sb(L) is the collection of joins of complemented sublocales of a frame L, or equivalently the Booleanization of the collection S(L) of all its sublocales. We do so by proving that Sb(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 c(a)∩o(b) for a,b∈L.