A Galois approach to uniform structures
by
Jorge Picado
Mathematics Department, University of Coimbra, PORTUGAL

In this talk we discuss the role of Galois connections in uniform structures, motivated by the fact that, in the category of sets, binary relations can be either described by polarities (i.e. (contravariant) Galois connections between power sets) and axialities (i.e. covariant Galois connections, also called residuated mappings, between power sets) [M. Erné, J. Koslowski, A. Melton and G.E. Strecker (1993)].

We introduce an operation o for (contravariant) Galois connections in complete lattices L, which is, in the category of locales, an instance of the categorical notion of composition of relations defined in any cartesian category with images. Therefore, when L is a frame, the set of Galois connections in L endowed with o is a quantale, in terms of which it is then possible to characterize the uniformities of L.

On the other hand, one can also describe these uniformities in terms of residuated mappings. This way one gets the frame entourages of P. Fletcher and W. Hunsaker (1991), which leads to the conclusion that Galois connections rather than Weil uniform entourages are at the root of Fletcher-Hunsaker's approach.

Date received: May 24, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeq-13.