Homotopy inverse limit of a tower (${\mathrm{holim}}_n{X}_n$)

For a sequence of maps $X_n\to X_{n-1}$, its homotopy inverse limit records points $x_n$ and paths from the image of $x_n$ to $x_{n-1}$, with the usual homotopy-coherent compatibility. A fibrant replacement tower computes it by an ordinary inverse limit. For a tower whose homotopy groups are eventually constant in each degree, the limit has those stable groups: the Milnor inverse-limit exact sequence has zero derived-limit term. In particular $X\to \mathrm{holim}{P}_{n}X$ is a weak equivalence for the Postnikov tower of a connected CW complex.