Fundamental pro-groupoids and covering projections

We introduce a new notion of covering projection E → X of a topological spaceX which reduces to the usual notion ifX 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 1connected, 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 Sets . If (X, ∗) is a pointed connected compact metrisable space and if (X, ∗) is 1movable, then the category of covering projections of X is equivalent to the category of continuous π̌1(X, ∗)-sets, where π̌1(X, ∗) is the Čech fundamental group provided with the inverse limit topology. Introduction. It is well known that if X is a locally path-connected and semilocally 1-connected space then the category Cov projX of covering projections and transformations of X is equivalent to the category of πXsets, that is, to the functor category Sets . The aim of this work is to study the category Cov projX for any space X, without local conditions of connectedness. In 1972–73, Fox [F1, F2] introduced the notion of overlay of a metrisable space. The fundamental theorem of Fox’s overlay theory establishes the existence of a bi-unique correspondence between the d-fold overlayings of a connected metrisable space X and the representations of the fundamental trope of X in the symmetric group Σd of degree d. 1991 Mathematics Subject Classification: Primary, 55Q07, 55U40, 57M10; Secondary, 18B25, 18F20, 20L15.