Universal categories of uniform and metric locales
type: Item
instance of: scholarly article
title: Universal categories of uniform and metric locales
zbMATH Open document ID: 0854.18005
DOI: 10.1007/BF01243595
published in: Algebra Universalis
publication date: 1997-01-22T00:00:00Z
review text: In the theory of universal categories [expounded in \textit{A. Pultr} and \textit{V. Trnková}, ``Combinatorial, algebraic and topological representations of groups, semigroups and categories'', North-Holland, Amsterdam (1980; Zbl 0418.18004)], categories like \({\mathcal T}op\) appear only after cosmetic surgery, because there are always all the constant mappings. This paper introduces formal or ``pointless'' spaces into the theory, constructing a number of classes of categories of locales -- with additional structure -- which are shown to be universal or (what is the same thing unless measurable cardinals form a proper class) alg-universal. The root example is the category of uniform locales (or dually, uniform frames). Those uniform locales which are metrizable and Boolean suffice; the morphisms can be the uniform ones, or Lipschitz with respect to given metrics, or contractive, among other possibilities. It is not known whether the category of locales (tout court) is universal.
Mathematics Subject Classification ID: 18B15
zbMATH DE Number: 920796
zbMATH Keywords: formal spaces
zbMATH Keywords: pointless spaces
zbMATH Keywords: universal categories
zbMATH Keywords: categories of locales
zbMATH Keywords: uniform locales
zbMATH Keywords: uniform frames
zbMATH Keywords: metrics
author: Bernhard Banaschewski
author: A. Pultr
reviewed by: J. R. Isbell
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01243595
OpenAlex ID: W1968320775
description: scientific article; zbMATH DE number 920796