### Questions on 2-covers

#### Posted by David Corfield

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?

Posted at April 30, 2008 8:56 AM UTC
## Re: Questions on 2-covers

Higher Monodromy has some relevant material.

I suppose I can pose my question for $RP(2)$, with $\pi_1 = \mathbb{Z}_2$ and $\pi_2 = \mathbb{Z}$. To kill off 1-homotopy we take the universal cover $S^2$. Then I guess to kill off 2-homotopy, we head for $S^3$.

Is it then the case that there’s a correspondence between, on the one hand, 2-groups between the trivial one and the fundamental 2-group of $RP(2)$ and, on the other, 2-covering spaces between $RP(2)$ and $S^3$.