at.algebraic topology - What is a principal refinement of a Postnikov system?

The key idea is that not all fibrations $E to B$ with fibre an Eilenberg-MacLane space $K(pi,n)$ can be constructed by pulling the principal path fibration $K(pi,n) to PK(pi,n+1) to K(pi,n+1)$ along a classifying map $B to K(pi,n+1)$. If you can construct the fibration in this way then the classifying map is the Postnikov $k$-invariant. Clearly this at least requires that the group $pi$ is abelian.

Now, it is a nice little exercise to check that existence of a principal refinement of the Postnikov tower is equivalent to $pi_1$ being nilpotent and acting nilpotently on all of the higher homotopy groups.

(Recall that a group $G$ acts nilpotently on a group $H$ if $H$ has a finite sequence of $G$-invariant subgroups $H supset H_1 supset H_2 supset cdots H_k = 1$ such that $H_i/H_{i+1}$ is abelian and the action of $G$ on it is trivial.)

The general idea of Hilton, Mislin, and Roitberg is that is it obvious how to localise abelian groups, and nilpotent groups are those which can be assembled from abelian groups one layer at a time. So we can localise nilpotent groups by working one layer at a time, and then we can localise nilpotent spaces by working up the refined Postnikov tower one stage at a time.