Example 7.6.6.10 (03HT): Towers of Isofibrations

Example 7.6.6.10 (Towers of Isofibrations). Suppose we are given a collection of $\infty $-categories $\{ \operatorname{\mathcal{C}}(n) \} _{n \geq 0}$ and functors $F_ n: \operatorname{\mathcal{C}}(n+1) \rightarrow \operatorname{\mathcal{C}}(n)$, which we view as a tower

\[ \cdots \rightarrow \operatorname{\mathcal{C}}(4) \xrightarrow { F_3 } \operatorname{\mathcal{C}}(3) \xrightarrow { F_2 } \operatorname{\mathcal{C}}(2) \xrightarrow { F_1} \operatorname{\mathcal{C}}(1) \xrightarrow { F_0 } \operatorname{\mathcal{C}}(0) \]

If each of the functors $F_{n}$ is an isofibration, then the limit $\varprojlim _{n} \operatorname{\mathcal{C}}(n)$ (formed in the ordinary category of simplicial sets) is also an $\infty $-category, which can be also be viewed as a limit of the associated tower $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{QC}}$. This follows by combining Example 4.5.6.8, Example 7.5.5.3, and Proposition 7.5.5.7.