Are chain-complete posets co-wellpowered? (Q2140998)

The category of complete partial orders, \textbf{CPO}, has as objects those posets where every up-directed set has a join. \textbf{CPO} has as morphisms all functions between objects which preserve directed joins. Empty directed joins are included, and the empty join is a least element, so every object of \textbf{CPO} has a least element \(0\) and every morphism of \textbf{CPO} will preserve \(0\). The authors show that \textbf{CPO} is not co-wellpowered. This statement means that some object of \textbf{CPO} has a proper class of quotients. To establish this statement, the authors construct complete partial orders \(K_{\alpha}\), \(\alpha\) an ordinal, where \(|K_{\alpha}|\geq \aleph_{\alpha}\) holds for all \(\alpha\). They produce injective epimorphisms \(f_{\alpha,\beta}\colon K_{\alpha}\to K_{\beta}\). for every \(\alpha<\beta\). This suffices to show that \(K_0\) is the domain of a proper class of inequivalent epimorphisms, namely a proper class of the \(f_{0,\beta}\)'s. (In this paper, \(K_0\) is taken to be a countable coproduct of \(2\)-element chains, so this particular poset witnesses that \textbf{CPO} is not co-wellpowered.) It is then shown that \textbf{CPO} is \emph{weakly} co-wellpowered. This statement means that every object of \textbf{CPO} has only a set of \emph{strong} quotients. A strong quotient is an equivalence class of strong epimorphisms. Since \textbf{CPO} has pullbacks, strong epimorphisms coincide with extremal epimorphisms, so one may read this result as saying that every object of the category is the domain of only a set of equivalence classes of extremal epimorphisms.