The Galois approach to uniform structures

Abstract

Entourages in uniform spaces, being binary relations, may be easily described by polarities (i.e. Galois connections between power sets) and axialities (i.e. residuated pairs between power sets). Here we consider this in the pointfree context of the category of frames, establishing characterizations of uniformities in terms of either the quantale of Galois connections or the quantale of residuated pairs. In virtue of this we conclude that frame uniformities may be thought of as either a family of polarities or as a family of axialities. The latter is precisely Fletcher-Hunsaker's approach to frame uniformities, showing that Galois connections rather than entourages are at the root of this approach. Then we prove that the equivalence between the somewhat different approaches to uniform frames of Fletcher-Hunsaker and Picado is a consequence of a very general phenomenon on Galois connections. Besides, we present a categorical description for the composition of Galois connections that justifies its definition.

Quaestiones Mathematicae 28(2005), 355–373.