Variant 7.6.6.12 (03HV): Limits of General Towers

Variant 7.6.6.12 (Limits of General Towers). Suppose we are given a sequence 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

7.71

\begin{equation} \label{equation:sequential-limits-in-QCat} \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) \end{equation}

If the functors $F_{n}$ are not assumed to be isofibrations, then the limit $\varprojlim _{n} \operatorname{\mathcal{C}}(n)$ (formed in the ordinary category of simplicial sets) might not be a limit of the associated tower in $\operatorname{\mathcal{QC}}$ (for example, $\varprojlim _{n} \operatorname{\mathcal{C}}(n)$ might fail to be an $\infty $-category). Nevertheless, we can always compute the relevant limit in $\operatorname{\mathcal{QC}}$ by replacing (7.71) by a levelwise equivalent diagram of $\infty $-categories in which the transition functors are isofibrations. For example, we can replace (7.71) by the isofibrant tower of iterated homotopy fiber products

\[ \cdots \rightarrow \operatorname{\mathcal{C}}(2) \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}(1)} (\operatorname{\mathcal{C}}(1) \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}(0)} \operatorname{\mathcal{C}}(0)) \rightarrow \operatorname{\mathcal{C}}(1) \times _{\operatorname{\mathcal{C}}(0)}^{\mathrm{h}} \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(0). \]

Let us denote the limit of this tower (in the category of simplicial sets) by

\[ \cdots \times _{ \operatorname{\mathcal{C}}(3) }^{\mathrm{h}} \operatorname{\mathcal{C}}(3) \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}(2)} \operatorname{\mathcal{C}}(2) \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}(1) } \operatorname{\mathcal{C}}(1) \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}(0)} \operatorname{\mathcal{C}}(0). \]

It is an $\infty $-category whose objects can be identified with sequences of pairs $\{ (C_ n, \alpha _ n) \} _{n \geq 0}$, where each $C_{n}$ is an object of the $\infty $-category $\operatorname{\mathcal{C}}(n)$ and each $\alpha _{n}: F_{n}( C_{n+1} ) \xrightarrow {\sim } C_ n$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}(n)$. Combining Example 7.6.6.10 with Remark 7.1.1.8, we see that it can be identified with a limit of the diagram (7.71) in the $\infty $-category $\operatorname{\mathcal{QC}}$.