Another approach to topological descent theory

Abstract

In the category Top of topological spaces and continuous functions, we prove that descent morphisms with respect to the class IE of continuous bijections are exactly the descent morphisms, providing a new characterization of the latter in terms of subfibrations IE(X) of the basic fibration given by Top/X which are, essentially, complete lattices. Also effective descent morphisms are characterized in terms of effective morphisms with respect to continuous bijections. For classes IE satisfying suitable conditions, we show that the class of effective descent morphisms coincides with the one of effective IE-descent morphisms.