Generalizing $\beta$- and $\lambda$-maps

TL;DR

This paper extends the concepts of $eta$- and $\lambda$-maps to broader classes of localic maps, providing new characterizations of various locale properties and relations.

Contribution

It introduces generalized $eta$- and $\lambda$-maps for sublocale selections, enabling new characterizations of locale properties like almost normality and extremal disconnectedness.

Findings

Characterization of almost normal locales

New classes of localic maps for various locale properties

Identification of maps preserving the completely below relation

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).