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Codenseness and openness with respect to an interior operator. (English) Zbl 1471.18003

This paper provides an interesting generalization of the notions of codenseness and openness with respect to an interior operator on a finitely complete category.
First of all, the authors introduce some basic properties of the notion of codenseness with respect to an interior operator \(i\) on a finitely complete category \(\mathbb C\) (with a proper \((\mathcal E,\mathcal M)\)-factorization system for morphism) with respect to \(\mathcal M\).
Secondly, they define \(i\)-codense morphisms in this category and provide some important stability properties of the class of \(i\)-codense morphisms in \(\mathbb C\).
Thirdly, using the notion of an open morphism with respect to an interior operator introduced by G. Castellini [Categorical closure operators. Boston, MA: Birkhäuser (2003; Zbl 1045.18001)], the authors give a number of new characterizations and some properties of this class of morphisms in \(\mathbb C\).
Finally, the authors present a notion of quasi-open morphisms with respect to an interior operator \(i\), and show that the quasi \(i\)-open morphisms of \(\mathbb C\) are characterized as the morphisms which reflect \(i\)-codensity.

MSC:

18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
06A15 Galois correspondences, closure operators (in relation to ordered sets)
54B30 Categorical methods in general topology

Citations:

Zbl 1045.18001
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