Codenseness and openness with respect to an interior operator (Q2040883)

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 \textit{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.