Fundamental pro-groupoids and covering projections

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 crs(X) and an induced category pro( 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( crs(X), Sets). We also prove that the latter category is equivalent to pro(CX, Sets), where CX is the ech fundamental pro-groupoid of X. If X is locally path-connected and semilocally 1-connected, we show that crs(X) is weakly equivalent to X, the standard fundamental groupoid of X, and in this case pro( crs(X), Sets) is equivalent to the functor category SetsX. 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 (X, *)-sets, where 1(X, *) is the ech fundamental group provided with the inverse limit topology.