Covering projections via descent theory

EDUARDO J. DUBUC

Santiago de Compostela, Espanna.

Congreso; Seminario de Categorias y aplicaciones SECA III; 2005

Universidad de Santiago

Covering projections via descent theory Topologists have dealt successfully with covering projections of non locally connected topological spaces. In their work, the descent data underneath the notion of covering projection has to be made explicit in one way or another (see Hernandez-Paricio L. J., Fundamental pro-groupoids and covering projections, Fundamenta Mathematicae 156). This means that we are in face of a situation of classical topological descent as described by Grothendieck in the introduction to Categories Fibrees et Descente, Expose VI, SGA1. The key fact in our research was that we found how to define the set of connected components of a covering projection X, even though X is not locally connected as a topological space. A covering projection is a locally constant sheaf X such that the bijections between the fibers are given by the same function for all the points in the base space B. Morphisms of covering projections are the continuous maps over B which preserve the trivialization structure. Covering projections trivialized by a cover U define a category G_U. This category is the descent category for descent data over the Cech nerve of the cover. All these constructions are functorial on covers and refinements. The category of covering projections split by all possible covers is defined by filtered colimit techniques, which take care of the morphisms between covering projections split by different covers. By its very definition it is equivalent to the classifying topos of a progroupoid determined by the groupoids associated to the Cech nerve.