 Postnikov tower

A Postnikov tower of a connected CW complex $X$ consists of spaces $P_{n}X$ with ${\pi }_i(P_{n}X)={\pi }_i(X)$ for $i\le n$ and zero for $i>n$, together with compatible maps $X\to P_{n}X\to P_{n-1}X$. For simply connected $X$, each stage is a principal Eilenberg–MacLane fibration with fibre $K({\pi }_{n}X,n)$, classified by a Postnikov invariant. The canonical map $X\to {\mathrm{holim}}_n{P}_{n}X$ is a weak equivalence: in each homotopy degree the tower is eventually constant, so the derived inverse-limit obstruction vanishes.