Extremal Prime Filters and Universality of Some Categories

Abstract

Abstract Due to the existence of constants, classical topological categories cannot be universal in the sense of containing each concrete category as a full subcategory. In the point-free case, this obstruction vanishes and the question of universality makes sense again. The main problem, namely that as to whether the category of locales and localic morphisms is universal is still open; we prove, however, the universality of the following categories: - pairs (locale, sublocale) with the localic morphisms preserving the distinguished sublocales, - frames with frame homomorphisms reflecting the maximal prime ideals, - Priestley spaces with f-maps preserving the maximal elements.