Overview of topological descent theory
by
Manuela Sobral
Departamento de Matematica Universidade de Coimbra 3001-454 Coimbra - Portugal

Let \Phi:C^op --> Cat be a pseudofunctor, p: E --> B a morphism in C and Eq(p) the kernel pair of p, regarded as an internal category in C. The category Des_\Phi(p) of descent data is defined as the category of internal actions of Eq(p) on Cat and the morphism p is said to be an effective \Phi-descent morphism if the canonical comparison functor \Phi(B) --> Des_\Phi(p) is an equivalence of categories. This definition is well known to be equivalent to A. Grothendieck's original definition, and to several others under various additional conditions.

Topological descent theory studies the situation where C = Top is the category of topological spaces and \Phi is determined by a pullback stable class bkE of continuous maps as follows:

• \Phi(B) is the full subcategory of (Top \downarrow B) with objects all pairs (A, \alpha) with \alpha: A --> B in bkE,
• \Phi(p): \Phi(B) --> \Phi(E) is induced by the pullback functor p^*: (Top \downarrow B) --> (Top \downarrow E),

Given an open covering family (U_\lambda) of B, we could take p to be the canonical map from the coproduct of the U_\lambda's to B. The fact that this map is an effective \Phi-descent morphism for ``almost" all \Phi's (and in particular almost all bkE's) is a classical observation. It explains how the Grothendieck's idea of descent contains various aspects of the idea of localization. It thus also shows how important is the problem of finding wide classes of effective descent morphisms, with expected many powerful applications in geometry and topology. Three such classes had been described for the global descent: locally sectionable (G. Janelidze and W. Tholen), open (M. Sobral; it was then observed by I. Moerdijk that this result easily follows from his axiomatic approach designed for locales and toposes) and perfect (J. Vermeulen) maps. After that, J. Reiterman and W. Tholen gave a complete characterization in terms of ultrafilter convergence, using the notion of pseudo-topological space (=Choquet space) defined by certain convergence structures. In particular the J. Reiterman and W. Tholen's paper contained the first example of a non-effective global descent morphism in Top (i.e., a map p for which the comparison functor Top \downarrow B --> Des(p) is full and faithful but not an equivalence). Some years later I found such an example for finite topological spaces, and then G. Janelidze and I decided to study the finite case in detail. This brought amazing simple motivations for all previous results and showed that they are closely related to old ideas of A. Grothendieck and J. Giraud. Moreover, it suggested a new approach to some open problems (e.g. of étale-descent) using M. Barr's presentation of the convergence structures as lax algebras over the ultrafilter monad. The finite/lax algebra approach is also supported now by several interesting results beyond descent theory of M. M. Clementino and D. Hofmann.

The purpose of this talk is to describe briefly the abovementioned and related results-including the most recent reformulations.

Date received: May 16, 2002