Topological descent via convergence structures
by
Manuela Sobral
University of Coimbra, Portugal

Descent theory was originally invented by A. Grothendieck for the purposes of algebraic geometry. The so-called Beck-Chevalley condition tells us when Grothendieck descent reduces to another categorical notion, namely to monadicity, in which case we are dealing with a certain category of algebras - the Eilenberg-Moore algebras. When the ground category is the category of topological spaces we use the expression "topological descent". For a base space B and an extension p:E --->B the descent from E to B is a general and powerful technique of solving problems on B using the extension. Being a powerful technique it uses a very special notion of extension; those extensions are called the effective descent morphisms (or the effective descent maps). They are defined categorically and their topological characterization was obtained only in 1991 by J. Reiterman and W. Tholen - more than thirty years after Grothendieck's first publication on descent theory. And it is interesting that, although descent theory seemed to have an algebraic nature, it were not algebraic but general topology methods that were successful in solving the characterization problem. Moreover, it turned out that the main tools use pre- and pseudo-topological spaces defined by certain convergence structures.

The purpose of this talk is threefold: - to show that the notion of effective descent morphism yields an important class of continuous maps of topological spaces that are appropriate extensions for the purposes of general and algebraic topology;

- to explain that, although we wanted to remove the convergence structure from the Reiterman-Tholen characterization theorem, the finite topological spaces told us not to do so...

- to exhibit an aspect of the role of convergence, namely that of making a bridge between the topological and the categorical structures; and that this "rediscovered" concept of convergence seems to be strong enough to solve essentially all problems of topological descent theory.

Date received: March 23, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafw-69.