Separation versus Connectedness
by
Maria Manuel Clementino
Coauthors: Walter Tholen

We discuss and compare two distinct categorical approaches to connectedness and to disconnectedness/separatedness: one based on the correspondence between left- and right-constant subcategories, as explored for topological spaces by Herrlich, Preuß, and by Arhangel'ski and Wiegandt, and another one based on closure operators in the sense of Giuli and Dikranjan, which distinguishes between those objects with dense diagonals and those with closed diagonals. In the presence of ``enough points" (as in set-based topological categories), the latter approach exceeds the former in terms of generality by far, and we are able to provide characterizations for the subcategories produced under either approach in categorical generality, which entail the known topological theorems.

The advantage of this procedure is that it produces the ``right" corresponding fibrewise notions for morphisms as well; one just has to switch from the ambient category to its comma categories. Hence a connected (separated) map f: X --> Y is nothing but a connected (separated) object in the category of objects over Y. In this work we concentrate on factorization theorems, of which Collins' concordant-dissonant factorizations are the topological prototypes. We explore these in both contexts, that of the left-right correspondence and that of closure operators. However, at the morphism level it is no longer true that closure operators provide the more general context, because of the lack of ``points" in comma categories. This observation helps us to settle a long-standing problem in conjunction with the Giuli-Husek Diagonal Theorem, by showing that in the category of topological spaces over the 1-sphere, it is no longer true that a quotient-reflective subcategory is presentable as the subcategory of c-separated objects for any closure operator c.

Date received: June 24, 1996

Copyright © 1996 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 Mathematical Conference Abstracts. Document # caah-14.