## 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

### Keywords:

interior operator; codenseness; openness; quasi-openness

Zbl 1045.18001
Full Text:

### References:

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