Effective codescent morphisms, amalgamations and factorization systems (Q863915)

Let \({\mathcal C}\) be a category with pullbacks and with a factorization system \((\mathcal E,\mathcal M)\) where all morphisms in \(\mathcal M\) are monos. The main result of the paper gives a necessary and sufficient condition on a descent morphism in \({\mathcal C}\) to be effective, involving descent data of a special kind only. This theorem is then used to obtain conditions on a functor \({\mathcal C}\to{\mathcal X}\) between categories with pullbacks in order that the property of every descent morphism being effective lifts from \({\mathcal X}\) to \({\mathcal C}\). In consequence of the first result, every regular monomorphism in the category of topological spaces is an effective codescent morphism, and then the second theorem carries over this property to a number of other categories of a `topological nature'. Beside factorization systems, the notion dual to the amalgamation property also plays a key role in these investigations.