Kerodon

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

Corollary 3.3.9.5. 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 a contractible Kan complex and that each of the morphisms $q_ n$ is a Kan fibration. Then the inverse limit $X = \varprojlim _{n \geq 0} X(n)$ is a contractible Kan complex.