$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 9.4.4.7 (Compact Generation of Inverse Limits). Let $\kappa \trianglelefteq \lambda $ be regular cardinals and suppose we are given a tower of $\infty $-categories
\[ \cdots \rightarrow \operatorname{\mathcal{C}}_{3} \xrightarrow { U_2} \operatorname{\mathcal{C}}_2 \xrightarrow { U_1 } \operatorname{\mathcal{C}}_1 \xrightarrow {U_0} \operatorname{\mathcal{C}}_0. \]
Assume that each of the $\infty $-categories $\operatorname{\mathcal{C}}_{n}$ is $(\kappa ,\lambda )$-compactly generated, and that each of the functors $U_{n}$ is a $(\kappa ,\lambda )$-compact isofibration, and that the following additional condition is satisfied:
- $(\star )$
There exists a regular cardinal $\kappa _0 < \kappa $ such that, for each $n \geq 0$, the full subcategory $\operatorname{\mathcal{C}}_{n}^{0} \subseteq \operatorname{\mathcal{C}}_ n$ spanned by the $(\kappa ,\lambda )$-compact objects admits $\kappa _0$-sequential colimits, which are preserved by functors $U_{n}|_{ \operatorname{\mathcal{C}}_{n+1}}: \operatorname{\mathcal{C}}_{n+1}^{0} \rightarrow \operatorname{\mathcal{C}}_{n}^{0}$.
Then the inverse limit $\varprojlim ( \operatorname{\mathcal{C}}_{n} )$ is $(\kappa ,\lambda )$-compactly generated. Moreover, an object $X = \{ X_ n \} _{n \geq 0} \in \varprojlim ( \operatorname{\mathcal{C}}_{n} )$ is $(\kappa ,\lambda )$-compact if and only if each $X_ n$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{n}$.
Proof.
Set $\operatorname{\mathcal{C}}= \prod _{n} \operatorname{\mathcal{C}}_{n}$. Since $\kappa $ is uncountable, Proposition 9.4.3.13 implies that $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated, and that an object $\{ X_ n \} _{n \geq 0} \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact if and only if each $X_{n}$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{n}$. Moreover, an analogous statement holds for the product $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}$. Let $S: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ be the shift functor (given on objects by the construction $\{ X_ n \} _{n \geq 0} \mapsto \{ U_{n}( X_{n+1} ) \} _{n \geq 0}$). The desired result now follows by applying Theorem 9.4.4.4 to the categorical pullback square
\[ \xymatrix@R =50pt@C=50pt{ \varprojlim (\operatorname{\mathcal{C}}_{n}) \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{ (\operatorname{id}, \operatorname{id}) } \\ \operatorname{\mathcal{C}}\ar [r]^-{ (\operatorname{id},S) } & \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}} \]
supplied by Corollary 4.5.7.19.
$\square$