Some problems in categorical and asymmetric topology
by
Guillaume Brümmer
University of Cape Town
Coauthors: Hans-Peter Künzi (University of Cape Town)

We sketch a few recent developments in asymmetric topology that resulted from asking simple categorical questions. We motivate some unsolved problems and indicate a few analogous areas in which similar questions may be asked. Our terminology agrees with [1] and [5].

The following problem is simple to state but possibly hard.

1. The Tea-zero Problem. Consider the category Top₀ of T₀ topological spaces and continuous maps, a functor M : Top₀ --> Top₀ and a natural transformation m : 1 --> M. Suppose that m_X : X --> MX is a dense topological embedding and that m_MX : MX --> M² X is a homeomorphism for each T₀-space X. (The second condition says that (M, m) is idempotent.) Question : Does it follow that (M, m) is a reflection? Comment: The corresponding problem in Haus or Tych is trivial.

2. Completeness. Let QU (resp. QU₀) be the category of (T₀) quasi-uniform spaces and quasi-uniform maps. A space X in QU₀ is bicomplete iff it is injective w.r.t. the class of all epimorphic quasi-uniform embeddings. The bicompletion k_X : X --> KX has a well-known characterisation. Question : Which of the many other completeness notions in the literature (half-complete, complete, convergence-complete, D-complete, left K-complete, etc.) can be described as injectivity w.r.t. suitable classes of maps? We shall outline a simple categorical theory [3] to illustrate the implications of such questions.

3. Transitivity. Many kinds of interior-preserving open covers (e.g. locally finite, point-finite, well-monotone) induce functorial admissible quasi-uniformities on the topological spaces by a construction due to Fletcher (see [5]). Many properties of these functors are known (see e.g. [4]). Such results should also be developed in a point-free setting. Question: How can one express the Fletcher construction in quasi-uniform frames/locales?

4. Induced reflections. Let F be a functor that puts a compatible quasi-uniformity on every T₀-space, i.e. F is a section (= right inverse) of the forgetful functor T : QU₀ --> Top₀. For every X in Top₀ consider the bicompletion k_FX : FX --> KFX. Then we have a dense topological embedding Tk_FX : X --> TKFX, written briefly as r_X : X --> RX. For such (R, r) the Tea-zero Problem has a positive answer. The functor F is called K-true if KF = FR. We shall outline the main results around this concept ([2], [4]), list the main open problems, and sugggest similar questions in analogous settings.

[1] J. Adamek, H. Herrlich and G. Strecker, Abstract and concrete categories, Wiley, New York, 1990.

[2] G.C.L. Brümmer, Categorical aspects of the theory of quasi-uniform spaces, Rend. Istit. Mat. Univ. Trieste 30 Suppl. (1999), 45-74. http://mathsun1.univ.trieste.it/Rendiconti/

[3] G.C.L. Brümmer and E. Giuli, A categorical concept of completion of objects, Comment. Math. Univ. Carol. 33 (1992), 131-147.

[4] G.C.L. Brümmer and H.-P.A. Künzi, Bicompletion and Samuel bicompactification, Appl. Categ. Struct., to appear. http://www.sun.ac.za/maths/CatTop/Output/

[5] P. Fletcher and W.F. Lindgren, Quasi-uniform spaces, Marcel Dekker, New York, 1982.

http://www.sun.ac.za/maths/CatTop/Output/

Date received: July 9, 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-32.