Quotient maps of locales
type: Item
instance of: scholarly article
title: Quotient maps of locales
zbMATH Open document ID: 0967.18002
DOI: 10.1023/A:1008662331958
author: Till Plewe
published in: Applied Categorical Structures
publication date: 2001-08-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 54B15
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 18A20
zbMATH DE Number: 1563335
zbMATH Keywords: category of locales
zbMATH Keywords: regular epimorphisms
zbMATH Keywords: quotient map of locales
reviewed by: Peter Johnstone
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1563335