Some aspects of topological descent

The paper deals with (effective) descent morphisms for subfibrations X of the basic fibration Top/X, for topological spaces X and classes of continuous functions stable under pullback. For a category with pullbacks, we prove the stability under pullback of effective -descent morphisms for a class satisfying some suitable conditions. This plays a rôle in relating effective -descent to effective global-descent and enables us to obtain a criterion for effective étale-descent. We also show that the inclusion of the class of effective global-descent maps in the class surjective effective étale-descent is strict.Partial financial support by Centro de Matemática da Universidade de Coimbra is gratefully acknowledged.