The following result seems to be frequently quoted:


Consider the fibration $K(\pi,n)=\Omega K(\pi,n+1)\to PK(\pi,n+1)\to K(\pi,n+1)$. Let $B$ be any topological space (which is not too pathologic). Then the fiberations on $B$ with fiber $K(\pi,n)$ are classified by the space $K(\pi, n+1)$. In other words, the pull-back map
$$ [B,K(\pi,n+1)]\to \{\textrm{fibrations on } B \textrm{ with }K(\pi,n)\textrm{ as fibers, up to fiber homotopy}\}$$
is a bijection.


Does anyone know a proof of this result, or a reference? Thanks a lot.



Classifying space for fibrations with Eilenberg-MacLane space as fibers
There is a very careful analysis of this question in Lemma 3.4.2, page 57, of More Concise Algebraic Topology, by Kate Ponto and myself. Assuming that $E$ and $B$ are connected, a fibration $E\longrightarrow B$ with fiber $K(A,n)$ for an abelian group $A$ is a pullback of of the path space fibration over $K(A,n+1)$ if and only if $\pi\_1(B)$ acts trivially on $A$. 


