The asymmetric or bitopological monads in TOP revisited
by
G. C. L. Brümmer
University of Cape Town, Rondebosch 7701, South Africa

Let QU₀ denote the category of T₀ quasi-uniform spaces and uniformly continuous mappings, and TOP₀ the category of T₀ topological spaces with continuous mappings. We consider the pointed canonical completion endofunctor (K, k) in QU₀. ((K, k) is called the bicompletion in the literature.) Let F be any functorial quasi-uniformity on TOP₀, that is, a functor into QU₀ which is a section of the forgetful functor T. The pointed endofunctor T(K, k)F assigns natural extensions to the T₀ spaces. We call F lower K-true if KF is coarser than FTKF, equivalently if F is spanned by a class of bicomplete quasi-uniform spaces (see e.g. the proceedings of the Hausdorff conference in Berlin, 1992, pp. 60-71). In 1978, at the Conference on Topological Structures in Bremen, we showed in effect that if F is lower K-true, then T(K, k)F is part of a monad in TOP₀. This was done by an excursion into TYCH2TOP, the Tychonoff bitopological spaces, via Salbany's asymmetric bicoreflective embedding of TOP₀ into TYCH2TOP. The excursion was actually a composition of the obvious two adjunctions, later described by the author in Rend. Istituto Mat. Univ. Trieste 30 Suppl. (1999), 45-74 (see pp. 64-66 in particular). The contribution of the present work is to give a simple direct construction of the monad, based on the fact that KF is coarser than FTKF. Though the class of functors F under consideration is large, the monad seems to have been studied in the literature hitherto only for the case of the coarsest possible F, namely the Csaszar-Pervin quasi-uniformity.

Date received: May 31, 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-26.