Universal categories of uniform and metric locales

Abstract

Recall that a category is called universal if it contains an isomorphic copy of any concrete category as a full subcategory. In particular, if$C$ is universal then every monoid can be represented as the endomorphism monoid of an object in$C$. A major obstacle to universality in categories of topological nature are the constant maps (which prevent, for instance, representing nontrivial groups as endomorphism monoids). Thus, to obtain, say, a universal category of uniform spaces, the constants have to be prohibited by artificial additional conditions (for instance, conditions of an openness type). Since in generalized spaces (locales) we do not necessarily have points, the question naturally arises as to whether we can get rid of surplus conditions in search of universality there. In this paper we prove that the category of uniform locales with all uniform morphisms is universal. Indeed we establish the universality already for the subcategory of very special uniform locales, namely Boolean metric ones. Moreover, universality is also obtained for more general morphisms, such as Cauchy morphisms, as well as for special metric choices of morphisms (contractive, Lipschitz). The question whether one can avoid uniformities remains in general open: we do not know whether the category of all locales with all localic morphisms is universal. However, the answer is final for the Boolean case: by a result of McKenzie and Monk ([10], see Section 4) one cannot represent groups by endomorphisms of Boolean algebras without restriction by an additional structure.

We use only basic categorical terminology, say, that from the introductory chapters of [9]. All the necesasary facts concerning generalized spaces (frames, locales) and universality are explicitly stated. More detail on frames (locales) can be found in [8] and on universality and embeddings of categories in [11].