Questions on 2-covers

The following may well have been talked about by John in his lectures but I didn’t see it explicitly there, so I’ll ask.

Since at level 1, we have

A Galois connection between subgroups of the fundamental group ${\pi }_1(X)$ and path-connected covering spaces of X for path-connected $X$. A universal covering space is simply connected.

should we not expect a level 0 analogue:

A ‘connection’ between subsets of the set of connected components ${\pi }_0(X)$ and covering spaces of $X$ whose locally constant fibres are either empty or $\{*\}$?

This puts these latter ‘covering spaces’ into correspondence with the poset of subsets of ${\pi }_0(X)$, so that the universal covering space with truth valued fibres is the empty set.

So do we have then:

A Galois 2-connection between sub-2-groups of the fundamental 2-group ${\pi }_2(X)$ and path-connected 2-covering spaces (with groupoid fibres) of X for path-connected $X$, a universal 2-covering space being 2-connected?

So, if we think of a nice space with nontrivial first and second homotopy, say the loop space of the 2-sphere, do we have a correspondence between 2-covering spaces and sub-2-groups of its fundamental 2-group? And is there a universal 2-connected 2-cover with 1- and 2-homotopy killed off?