Some aspects of topological descent (Q1364913)

For \({\mathfrak E}\) a class of morphisms in Top, the author studies the class \(\text{Eff} ({\mathfrak E})\) of maps which are effective for \({\mathfrak E}\)-descent in the sense of \textit{G. Janelidze} and \textit{W. Tholen} [ibid. 2, No. 3, 245-281 (1994; Zbl 0805.18005)]. Specifically, generalizing a result obtained previously by the author and the reviewer [CMS Conf. Proc. 13, 421-433 (1992; Zbl 0792.18002)], she gives sufficient conditions on \({\mathfrak E}\) for \(\text{Eff} ({\mathfrak E})\) to be stable under pullback. For \({\mathfrak E}\) the class of local homeomorphisms, she characterizes the surjective maps in \(\text{Eff} ({\mathfrak E})\) and gives a finite example of a surjective map in \(\text{Eff} ({\mathfrak E})\) which fails to belong to \(\text{Eff(Mor }{\mathfrak {Top}})\), thus solving a problem left open in the Janelidze-Tholen paper.