Is there a natural example of a $K(\hat{\mathbf Z}, 1)$?

Question

Does there exist a nice classifying space for $\hat{\mathbf Z}$, the profinite completion of $\hat{\mathbf Z}$?

Answer 1

I might be completely out of my depth here, but since it's Math.StackExchange, I'll leave a few remarks.

In the case of $\hat{\mathbb{Z}}$ considered as a topological group, a slightly more general question was asked on MathOverflow. There are not that many answers, but it seems that the subject is already quite difficult when you consider $\mathbb{Z}_{p}$, the $p$-adic integers.

In the case of $\hat{\mathbb{Z}}$ with discrete topology, you cannot really hope for a vastly "geometric" model (say, with skeleta compact manifolds), as $\hat{\mathbb{Z}}$ is not finitely generated. However, the usual construction of $B \hat{\mathbb{Z}}$ (as the geometric realization of the nerve of the group) might be a workable thing. In general, there is a huge literature (example) on the subject of completion of spaces and probably this is the thing you should look at. My guess is that you would then obtain $K(\hat{\mathbb{Z}}, 1)$ by completing the circle at all primes, but you might need an expert to know for sure.