Localic entourage uniformities revisited
by
Jorge Picado
University of Coimbra, Portugal

Classically the description of uniform structures via entourages is based on the calculus of relations on sets, that is, the 2-category of relations on sets. The fact that this 2-category is bi-closed w.r.t. relational composition yields, in particular, two other different descriptions in terms of, respectively, Galois polarities and axialities.

In this talk we discuss the extension of these ideas to locales. Since the category of locales is not regular (contrarily to the category of sets), the calculus of relations is not so well behaved (it is not an allegory in the words of Freyd and Scedrov [3]); one has to restrict to compact Hausdorff locales or to discrete locales in order to get a 2-categorical structure as in sets. In spite of this, it is still possible to describe uniformities in terms of this calculus [4]. We present a categorical framework that, in particular, explains why this approach is equivalent to the approaches of [1] and [2].

[1] M. J. Ferreira and J. Picado, The Galois approach to uniform structures, preprint, 2000.

[2] P. Fletcher and W. Hunsaker, Entourage uniformities for frames, Monatsh. Math. 112 (1991) 271-279.

[3] P. J. Freyd and A. Scedrov, Categories, Allegories, North-Holland, 1990.

[4] J. Picado, Weil uniformities for frames, Comment. Math. Univ. Carolin. 36 (1995) 357-370.

Date received: July 19, 2001

Copyright © 2001 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 # cagx-47.