Kerodon

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Example 4.5.6.8 (Isofibrant Towers). Let $\mathscr {F}: \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ be a diagram, which we identify with a tower of simplicial sets

\[ \cdots \rightarrow \mathscr {F}(3) \rightarrow \mathscr {F}(2) \rightarrow \mathscr {F}(1) \rightarrow \mathscr {F}(0). \]

Then $\mathscr {F}$ is isofibrant (in the sense of Definition 4.5.6.3) if and only if each of the simplicial sets $\mathscr {F}(n)$ is an $\infty $-category and each of the transition functors $\mathscr {F}(n+1) \rightarrow \mathscr {F}(n)$ is an isofibration of $\infty $-categories.