Categories with strong monomorphic strong coimages (Q656154)

Let \(\mathcal{C}\) be an arbitrary category and \(SE(\mathcal{C})\) (respectively, \(SM(\mathcal{C})\) be the subcategory of \(\mathcal{C}\) with the same objects and whose morphisms are strong epimorphisms (respectively, strong monomorphisms) of \(\mathcal{C}\). Also, let the pro-category pro-\(\mathcal{C}\) of \(\mathcal{C}\) be the universal category with directed inverse limits containing \(\mathcal{C}\) as a full subcategory. In this paper, the author investigates the question: when an object \(X\) of pro-\(\mathcal{C}\) is isomorphic to an object of pro-\(SE(\mathcal{C})\) or pro-\(SM(\mathcal{C})\). Moreover, he presents conditions which give stability of objects of pro-\(\mathcal{C}\). In this sense, the author considers \(\mathcal{C}\) a category with strong monomorphic strong coimages and obtains the following results: - if \(f:X\rightarrow Y\) is a strong epimorphism of pro-\(\mathcal{C}\) and X is isomorphic to an object of pro-\(SE(\mathcal{C})\), then \(Y\) is isomorphic to an object of pro-\(SE(\mathcal{C})\); - for an object \(X\) of pro-\(\mathcal{C}\), \(X\) is isomorphic to an object of pro-\(SM(\mathcal{C})\) if and only if there is a strong monomorphism \(f:X\rightarrow P\), where \(P\) is an object of \(\mathcal{C}\); - for \(P,Q\) objects of \(\mathcal{C}\), \(X\) object of pro-\(\mathcal{C}\), if \(f:P\rightarrow X\) is a strong epimorphism of pro-\(\mathcal{C}\) and \(g:X\rightarrow Q\) is a strong monomorphism of pro-\(\mathcal{C}\), then \(X\) is stable.