reference request: Postnikov towers for non-simply-connected spaces I've read that for a space $X$ which is connected but not necessarily simply-connected, we can no longer obtain the $n^{\rm th}$ layer $P_nX$ of the Postnikov tower for $X$ as the pullback of a path-loops fibration (with contractible total space).  Instead, there is a pullback diagram
$$ \begin{array}{ccc} P_nX & \rightarrow & B\pi_1 X \\ \downarrow & & \downarrow \\ P_{n-1}X & \rightarrow & K(\pi_nX,n+1) \times_{\pi_1 X} E \pi_1X \end{array} $$
(where the fiber is of course still $K(\pi_nX,n)$).  Does anyone know of a reference for this?  I checked the two books I have on hand -- Hatcher and Switzer -- but neither of them covers this.
 A: The first Postnikov invariant  for a pointed, connected space $X$ is $k^3\in H^3(\pi_1(X), \pi_2(X))$ and is also represented by a "crossed sequence" 
$$0 \to \pi_2(X) \to C_2 \to C_1 \to \pi_1(X) \to 1$$
where $C_2 \to C_1$ is a crossed module. There is quite a lot of information on such invariants, including some specific calculations of these, in the book partially titled Nonabelian Algebraic Topology  (EMS Tracts in Mathematics vol 15, 2011). 
More examples are given in Graham Ellis and Luyen Van Le,  "Homotopy 2-Types of Low Order" J. Experimental Math, 2014. 
Loday's 1982 JPAA paper   "Spaces   with  finitely   many   non trivial        homotopy     groups" has a lot of relations with higher Postnikov invariants. 
A: I think your $B\pi_1X$ should be something else... What is the right-hand vertical map supposed to be?
In any case, this should do:
Link
C. Robinson "Moore-Postnikov Systems for non-simply connected spaces"
In particular the beginning of section 4 (his $\hat{K}$ is the same as the Borel construction that you have, I think.)
