In algebraic topology, there is the well-known notion of coverings of a space. It is very nice, it has many properties, but I find it frustrating that:

1) some hypotheses are needed for them to work perfectly, like semi-locally simply-connectedness for the existence of coverings,

2) those covering maps don't seem to correspond to anything interesting from a categorical point-of-view.

On the other hand, there is this notion of coverings of a groupoid (see for example Brown's book). In my opinion, those are much nicer, since to any property of a covering of a space corresponds an analogue property in groupoids by applying the fundamental groupoid functor, and since the two objections disappear. For example, a covering of a groupoid can be defined as a functor that as right lifting properties with respect to some simple functors, so this notion makes sense categorically.

So my (naive) questions are the following. Are there particular reasons to stick to this historical notion of coverings of a space? What would remain of the theory if we replace this notion by defining, for example, a covering as a continuous function which induces a covering between the fundamental groupoids? Do you know if there is some work on that direction?

a posteriorijustification of the semi-locally simply-connectedness : for a continuous $f : X\to Y$, it gives a left adjoint to $f^\ast : \mathsf{Sh}_{\rm loc}(Y) \to \mathsf{Sh}_{\rm loc}(X)$. (The "loc" subscript means locally constant.) $\endgroup$ – Pierre Cagne Jan 18 '18 at 10:41