Fundamental pro-groupoids and covering projections

Luis Javier Hernández-Paricio¹

¹

Fundamenta Mathematicae, Tome 156 (1998) no. 1, pp. 1-31.

Résumé

We introduce a new notion of covering projection E → X of a topological space X which reduces to the usual notion if X is locally connected. We use locally constant presheaves and covering reduced sieves to find a pro-groupoid π crs (X) and an induced category pro (π crs (X), Sets) such that for any topological space X the category of covering projections and transformations of X is equivalent to the category pro (π crs (X), Sets). We also prove that the latter category is equivalent to pro (π CX, Sets), where π CX is the Čech fundamental pro-groupoid of X. If X is locally path-connected and semilocally 1-connected, we show that π crs (X) is weakly equivalent to π X, the standard fundamental groupoid of X, and in this case pro (π crs (X), Sets) is equivalent to the functor category $Sets^{π X}$. If (X,*) is a pointed connected compact metrisable space and if (X,*) is 1-movable, then the category of covering projections of X is equivalent to the category of continuous $\check π_1 (X,*)$-sets, where $\check π_1 (X,*)$ is the Čech fundamental group provided with the inverse limit topology.

Détail

DOI : 10.4064/fm-156-1-1-31

Keywords: covering projection, covering transformation, pro-groupoid, Čech fundamental pro-groupoid, covering reduced sieve, locally constant presheaf, category of fractions, subdivision, fundamental groupoid, Čech fundamental group, G-sets, continuous G-sets