= Homotopy inverse limit of a tower
{title2=$\operatorname{holim}_nX_n$}
= Homotopy inverse limit
{synonym}
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 $X\to\operatorname{holim}P_nX$ is a weak equivalence for the <Postnikov tower> of a connected <CW complex>.
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