Kerodon

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Variant 3.3.9.3. Let $X$ be the inverse limit of a tower of simplicial sets

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

If each of the morphisms $q_ n$ is a trivial Kan fibration, then the canonical map $q: X \rightarrow X(0)$ is a trivial Kan fibration.

Proof. By virtue of Proposition 1.4.4.14, the collection of trivial Kan fibrations is closed under transfinite composition in the opposite of the category of simplicial sets. $\square$