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Erschienen in:
26.10.2020
Codenseness and Openness with Respect to an Interior Operator
Erschienen in: Applied Categorical Structures | Ausgabe 2/2021
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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.