Generalizing $β$- and $λ$-maps

Abstract

We generalize the notions of $\beta$- and $\lambda$-maps to general selections of sublocales, obtaining different classes of localic maps. These new classes of maps are used to characterize almost normality, extremal disconnectedness, $F$-frames, $Oz$-frames, among others types of locales, in a manner akin to the characterization of normal locales via $\beta$-maps. As a byproduct we obtain a characterization of localic maps that preserve the completely below relation (that is, the right adjoints of assertive frame homomorphisms).