Postnikov section and Postnikov tower

Let $(X,x_0)$ be a connected based space and let $n\ge 1$. An $n$th Postnikov section is a based map

$p_n:X⟶P_{n}X$

to a connected based space such that $(p_n{)}_{\ast }:{\pi }_i(X,x_0)\to {\pi }_i(P_{n}X,p_n{x}_0)$ is an isomorphism for $1\le i\le n$ and ${\pi }_i(P_{n}X)=0$ for $i>n$. The isomorphism on ${\pi }_1$ transports the usual ${\pi }_1$-actions on every retained higher group.

A Postnikov tower is a choice of sections $p_n$, the convention $P_{0}X=\ast$, and maps

$q_n:P_{n}X⟶P_{n-1}X(n\ge 1)$

and specified based homotopies $q_n{p}_n\simeq p_{n-1}$. Unless a strict model has been chosen, the tower is therefore a diagram in the based homotopy category rather than a literally commuting inverse sequence.

This definition asserts neither that $X\to {\mathrm{holim}}_n{P}_{n}X$ is an equivalence nor that an ordinary inverse limit recovers $X$. Those are separate convergence claims.