Effective descent maps of topological spaces

Abstract

The basic technique in A. Joyal's and M. Tierney's work on “An extension of the Galois theory of Grothendieck” is descent theory for morphisms of locales (in a topos). They showed that open surjections are effective descent morphisms in the category of locales. I. Moerdijk gave an axiomatic proof of this result which shows that the same result holds true also in the category Top of topological spaces. G. Janelidze and W. Tholen proved that every locally sectionable map in Top is an effective descent morphism, and that effective descent morphisms are universal quotient maps in Top. In this paper, we give • • bu a complete characterization of effective descent maps in bdTop, • bu an example of a universal quotient map in bdTop which is not an effective descent morphism. This is done by first transfering the problem into a friendlier environment than Top, namely into the topological quasitopos hull of Top, the category of pseudotopological spaces. Here effective descent morphisms are simply quotient maps. Although the notion of effective descent morphism depends on the category, it is possible to reinterpret the pseudotopological characterization in purely topological terms, under extensive use of filter theory.