
details
Author(s)
Maria Manuel Clementino; Walter Tholen;
Title Separated and connected maps
Abstract Using on the one hand closure operators in the sense of Dikranjan and Giuli and on the other hand left and rightconstant subcategories in the sense of Herrlich, Preuss, Arhangel'skii and Wiegandt, we apply two categorical concepts of connectedness and separation/disconnectedness to comma categories in order to introduce these notions for morphisms of a category and to study their factorization behaviour. While at the object level in categories with enough points the first approach exceeds the second considerably, as far as generality is concerned, the two approaches become quite distinct at the morphism level. In fact, left and rightconstant subcategories lead to a straight generalization of Collins' concordant and dissonant maps in the category Top of topological spaces. By contrast, closure operators are neither able to describe these types of maps in Top, nor the more classical monotone and light maps of Eilenberg and Whyburn, although they give all sorts of interesting and closely related types of maps. As a byproduct we obtain a negative solution to the tenyears old problem whether the GiuliHusek Diagonal Theorem holds true in every decent category, and exhibit a counterexample in the category of topological spaces over the 1sphere.
Journal Appl. Categ. Structures
Volume 6
Year 1998
Issue 3
Page(s) 373401

