Quotient maps of locales
One disadvantage of the category of locales, as compared with that of topological spaces, is that there is no simple characterization of the regular epimorphisms in it, comparable to the characterization of regular epimorphisms of spaces as quotient maps. Various substitute notions of `quotient map of locales' have been proposed by several authors; one purpose of the present paper is to show that two of these, the `weak quotient maps' of \textit{Li Yongming} and the `equationally closed subframes' of \textit{Aleš Pultr} and \textit{Anna Tozzi}, are in fact equivalent. But the author does much more than this: he also shows that localic regular epimorphisms between spatial locales need not be regular epimorphisms of spaces, that closed surjections of locales need not be regular epimorphisms (though they are so if their domains are subfit), and (answering a question of the reviewer) that composites of regular epimorphisms of locales need not be regular epimorphisms. In addition to all these negative results, the author also gives a necessary and sufficient condition for a locale map to be a regular epimorphism, though it seems likely to be too complicated for practical use in many cases.
scientific article; zbMATH DE number 1563335