Codenseness and Openness with Respect to an Interior Operator

Abstract

Working in an arbitrary category endowed with a fixed $({\mathcal {E}}, {\mathcal {M}})$-factorization system such that ${\mathcal {M}}$ is a fixed class of monomorphisms, we first define and study a concept of codense morphisms with respect to a given categorical interior operator i. Some basic properties of these morphisms are discussed. In particular, it is shown that i-codenseness is preserved under both images and dual images under morphisms in ${\mathcal {M}}$ and ${\mathcal {E}}$, respectively. We then introduce and investigate a notion of quasi-open morphisms with respect to i. Notably, we obtain a characterization of quasi i-open morphisms in terms of i-codense subobjects. Furthermore, we prove that these morphisms are a generalization of the i-open morphisms that are introduced by Castellini. We show that every morphism which is both i-codense and quasi i-open is actually i-open. Examples in topology and algebra are also provided.