Fundamental pro-groupoids and covering projections

Resumen

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(πCX, Sets), where πCX 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 πX, the standard fundamental groupoid of X, and in this case pro(π crs(X), Sets) is equivalent to the functor category SetsπX. 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 π̌(X, *)-sets, where π̌1(X, *) is the Čech fundamental group provided with the inverse limit topology.