On connectedness via closure operators.(English)Zbl 0993.18004

The author enriches the investigation of categorical connectedness properties (i.e., of left- and right-constant subcategories) induced by categorical closure operators, by defining and investigating for each category $${\mathcal X}$$, equipped with a closure operator $$c$$, the subcategories of $$c$$-coarse and $$c$$-fine objects. Under mild conditions on $${\mathcal X}$$, left- resp. right-constant subcategories turn out to be particular instances of subcategories consisting of coarse resp. fine objects. Moreover, at the morphism level, i.e., for sliced categories this new approach allows the simultaneous study of concordant and dissonant maps, and also of monotone and light maps.

MSC:

 18B30 Categories of topological spaces and continuous mappings (MSC2010) 54B30 Categorical methods in general topology 54D05 Connected and locally connected spaces (general aspects) 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 18E40 Torsion theories, radicals 06A15 Galois correspondences, closure operators (in relation to ordered sets)
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