Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 3.3.9.4. Let $X$ be the inverse limit of a tower of simplicial sets

\[ \cdots \xrightarrow {q_3} X(3) \xrightarrow {q_2} X(2) \xrightarrow {q_1} X(1) \xrightarrow {q_0} X(0). \]

Assume that each of the simplicial sets $X(n)$ is weakly contractible and that each of the morphisms $q_ n$ is a Kan fibration. Then the inverse limit $X = \varprojlim _{n \geq 0} X(n)$ is weakly contractible.

Proof. If each of these simplicial sets $X(n)$ is weakly contractible, then each of the morphisms $q_ n: X(n+1) \rightarrow X(n)$ is a weak homotopy equivalence. Applying Proposition 3.3.7.4, we conclude that each $q_ n$ is a trivial Kan fibration. Variant 3.3.9.3 then guarantees that the projection map $X \rightarrow X(0)$ is also a trivial Kan fibration. Since $X(0)$ is weakly contractible, it follows that $X$ is also weakly contractible. $\square$