Universal categories of uniform and metric locales
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.
scientific article; zbMATH DE number 920796