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 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 ${\displaystyle Set{s}^{\pi X}}$. 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 ${\displaystyle check{\pi }_1(X,\cdot )}$-sets, where ${\displaystyle check{\pi }_1(X,\cdot )}$ is the Čech fundamental group provided with the inverse limit topology.