Categorical Topology, by Walter Tholen

Topology Atlas Document # zaaa-45.htm | Production Editor: R. Flagg

Topology Atlas Invited Contributions, vol. 1, issue 1 (1996), 1-2.

Categorical Topology
by

Walter Tholen

(York University, North York, Ontario, Canada)

The general aim of the field is to better understand topological phenomena using Category Theory. There are three partly overlapping directions of "Categorical Topology" if one uses the term in the narrow sense adopted by people with combined interests in General Topology and Category Theory, which we describe below. (Under any broader interpretation of the term, this summary would have to start with a description of the development leading to the very creation of the notion of category by Eilenberg and Mac Lane in the forties, as a result of a "clash" between Algebra and Topology, and continue with the many modern developments in Algebraic Topology and Geometry involving Category Theory.)

1. Topological Categories.

The notion emerged soon after Horst Herrlich's pioneering monograph "Topologische Reflexionen und Coreflexionen" (Springer LNM 78, 1968). It facilitates a better understanding of fundamental constructions (like product or quotient space) in categories which behave similarly to the category of topological spaces and proves to be useful in the study of its sub- and supercategories. The question of cartesian closedness (the possibility of forming function spaces) figures prominently in this research. Good survey articles with an extensive list of open problems include: H. Herrlich and M. Husek, Categorical Topology, in: "Recent Progress in General Topology (eds. M. Husek and J. van Mill), Elsevier, Amsterdam 1992, pp. 369-403; H. Herrlich and M. Husek, Some Open Categorical Problems in Top, Appl. Categ. Structures 1 (1993) 1-19.

2. Locales.

Looking at the lattice of open sets of a topological space, which satisfies an infinte distributive law, J. Isbell in his famous paper "Atomless parts of spaces" (Math. Scand. 31 (1972) 5-32) used the notion of frame or locale to pursue topological concepts by lattice-theoretic methods. "The point of pointless topolgy" (P.T. Johnstone, Bull. AMS 8 (1983) 41-53) is based on a number of beautiful results, like the choice-free proof of Tychonoff's Theorem. The book "Stone Spaces" by P.T. Johnstone (Cambridge University Press, 1982) still gives a good overview of the subject. For a more recent survey, see his article "The Art of Pointless Thinking: A Student's Guide to the Category of Locales", in: "Category at Work" (eds. H. Herrlich and H.-E. Porst), Heldermann Verlag, Berlin 1991, pp. 85-107.

3. Closure Operators.

In a category equipped with a notion of subobject and closure as introduced by D. Dikranjan and E. Giuli (Topology Appl. 27 (1987) 129-143), one may pursue topological concepts in a context no longer confined to Top-like categories. For instance, Tychonoff's Theorem holds in this context (M.M. Clementino and W. Tholen, Proc. AMS, to appear). Research in this area overlaps with both of the two areas described above, since the approach is Set-free as in 2, and since many (but not all) of the applications are aimed at categories of interest in 1. "The Categorical Structure of Closure Operators" is described in a recent book by D. Dikranjan and W. Tholen (Kluwer Academic Publishers, Dordrecht 1995).

Received by the editors: December 6, 1995; Revised: December 7, 1995.