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 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 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 by-product we obtain a negative solution to the
ten-years old problem whether the Giuli-Husek Diagonal
Theorem holds true in every decent category, and exhibit a
counter-example in the category of
topological spaces over the 1-sphere.
