Postnikov towers for $\Omega X$

Let \[ \cdots\to X_n\to X_{n-1}\to\cdots \] be a Postnikov tower for \(X\). Looping preserves fibrations, so \[ \cdots\to\Omega X_n\to\Omega X_{n-1}\to\cdots \] is a tower of fibrations. Since \[ \pi_i(\Omega X_n)=\pi_{i+1}(X_n), \] the space \(\Omega X_n\) has the correct truncated homotopy groups to be the corresponding Postnikov stage of \(\Omega X\).

The successive Postnikov fibration is classified by a \(k\)-invariant. Looping its classifying map expresses \[ \Omega X_{n+1}\to\Omega X_n \] as a pullback of a path-space fibration. Equivalently, the three-term sequence \[ \Omega X_{n+1}\longrightarrow\Omega X_n\longrightarrow\Omega X_{n-1} \] is a principal fibration, with fiber an Eilenberg–MacLane loop space. Thus \[ \boxed{\text{looping a Postnikov tower for }X\text{ gives a Postnikov tower of principal fibrations for }\Omega X.} \]