Corollary 11.9.5.2. Suppose we are given a tower of simplicial sets
\[ \cdots \xrightarrow {q_3} X(3) \xrightarrow {q_2} X(2) \xrightarrow {q_1} X(1) \xrightarrow {q_0} X(0). \]
If $X(0)$ is a Kan complex and each $q_ n$ is a Kan fibration, then the inverse limit $X = \varprojlim _{n \geq 0} X(n)$ is a Kan complex.