Effective codescent morphisms, amalgamations and factorization systems

Abstract

The paper deals with the question whether it is sufficient, when investigating the problem of the effectiveness of a descent morphism, to restrict the consideration only to the descent data ( C , γ , ξ ) , where γ lies in a certain morphism class. The notion of a factorization system and the dual to the amalgamation property in the sense of Kiss, Marki, Pröhle and Tholen play the key role in our discussion. It is shown that a category C inherits from a category X the property that all descent morphisms are effective if either X is regular and C is a full coreflective, closed under pullbacks of certain epimorphisms, subcategory of X or C is regular, X has coequalizers and there exists a topological functor C → X . This implies that in the category of topological spaces, all regular monomorphisms are effective codescent morphisms (the result of Mantovani). The same is shown to be valid also for the categories of compact Hausdorff topological spaces, normal topological spaces, Banach spaces, (quasi-)uniform spaces, and (quasi-)proximity spaces. Moreover, the effectiveness of all codescent morphisms is established for the categories of Hausdorff topological spaces and (complete) metric spaces. The internal characterization of such morphisms p : B → E is given for the category of Hausdorff topological spaces, in the case of compact B and regular E .