Name
Abstract | ||
|---|---|---|
Using on the one hand closure operators in the sense of Dikranjan and Giuli and on the other hand left- and right-constant 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 right-constant subcategories lead to a straight generalization of Collins' concordant and dissonant maps in the category
$$\mathcal{T}op$$
of topological spaces. By contrast, closure operators are neither able to describe these types of maps in
$$\mathcal{T}op$$
, 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 by-product we obtain a negative solution to the ten-year-old problem whether the Giuli–Hušek Diagonal Theorem holds true in every decent category, and exhibit a counter-example in the category of topological spaces over the 1-sphere. |
Year | DOI | Venue |
|---|---|---|
1998 | 10.1023/A:1008636615842 | Applied Categorical Structures |
Keywords | Field | DocType |
closure operator,left- and right-constant subcategory,separated and connected morphism,dissonant and concordant morphism | Diagonal,Discrete mathematics,Topology,Social connectedness,Category of topological spaces,Closure operator,Topological space,Factorization,Morphism,Monotone polygon,Mathematics | Journal |
Volume | Issue | ISSN |
6 | 3 | 1572-9095 |
Citations | PageRank | References |
4 | 1.34 | 1 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Maria Manuel Clementino | 1 | 61 | 25.61 |
| Walter Tholen | 2 | 77 | 39.38 |