Morphisms and Pushouts in Compact Normal Joinfit Frames

\(\mathfrak {KNJ}\) is the category of compact normal joinfit frames and frame homomorphisms and \(\mathfrak {KReg}\) is the coreflective subcategory of compact regular frames. This work investigates \(\mathfrak {KNJ}\) through its interaction with \(\mathfrak {KReg}\) via the coreflection \(\rho \). A \(\mathfrak {KNJ}\) morphism \(\phi : F \longrightarrow M\) is \(\mathcal {P}\)-essential if \(\phi \) is skeletal and the map between the frames of polars, \(\mathcal {P}(\phi ): \mathcal {P}F \longrightarrow \mathcal {P}M\) defined by \(\mathcal {P}(\phi )(p)=\phi (p)^{\perp \perp }\), is a boolean isomorphism. The \(\mathcal {P}\)-essential morphisms in \(\mathfrak {KNJ}\) are closely related to the essential embeddings in \(\mathfrak {KReg}\). We provide a characterization of the \(\mathcal {P}\)-essential morphisms in \(\mathfrak {KNJ}\) and a connection to the essential embeddings in \(\mathfrak {KReg}\). Further results about the preservation of joinfitness, the factorization of morphisms, and monomorphisms in \(\mathfrak {KNJ}\) are provided. Moreover, in the category of \(\mathfrak {KNJ}\) objects and skeletal frame homomorphisms, \(\mathfrak {KNJS}\), we construct for \(F \in \mathfrak {KNJ}\) and \(\phi :\rho F \longrightarrow H\) (an arbitrary \(\mathfrak {KReg}\) essential embedding of \(\rho F\)) the \(\mathfrak {KNJS}\) pushout of \(\rho _F: \rho F \longrightarrow F\) and \(\phi : \rho F \longrightarrow H\). Lastly, we investigate the epimorphisms and epicomplete objects in \(\mathfrak {KNJS}\).