Proposition 9.2.9.16. Let $\kappa \leq \lambda $ be regular cardinals and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a $(\kappa ,\lambda )$-cocomplete inner fibration of simplicial sets. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be a section of $U$ satisfying the following condition:
- $(\ast )$
For every edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the image $F(C)$ is $(\kappa ,\lambda )$-compact when viewed as an object of the $\infty $-category $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.
If the simplicial set $\operatorname{\mathcal{C}}$ is $\kappa $-small, then $F$ is $(\kappa ,\lambda )$-compact when viewed as an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.
Proof of Proposition 9.2.9.16.
We proceed as in the proof of Proposition 9.2.8.9. For every morphism of simplicial sets $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, Remark 9.2.9.12 guarantees that the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})$ is $(\kappa ,\lambda )$-cocomplete, and that the evaluation functors $\operatorname{ev}_{C}: \operatorname{Fun}_{ /\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}_ C$ are $(\kappa ,\lambda )$-finitary for each $C \in \operatorname{\mathcal{C}}'$. Let us say that $\operatorname{\mathcal{C}}'$ is good if the restriction $F|_{\operatorname{\mathcal{C}}'}$ is $(\kappa ,\lambda )$-compact when viewed as an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})$. We will complete the proof by showing that if $\operatorname{\mathcal{C}}'$ is $\kappa $-small, then $\operatorname{\mathcal{C}}'$ is good.
Note that $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})$ can be realized as the inverse limit of a tower of $\infty $-categories
\[ \cdots \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{sk}_{2}(\operatorname{\mathcal{C}}'), \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{sk}_{1}(\operatorname{\mathcal{C}}'), \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{sk}_{0}(\operatorname{\mathcal{C}}'), \operatorname{\mathcal{E}}), \]
where the transition maps are isofibrations (Corollary 4.4.5.12). Each of the $\infty $-categories appearing in this tower is $(\kappa ,\lambda )$-cocomplete and each of the transition functors is $(\kappa ,\lambda )$-finitary. It will therefore suffice to show that each of the skeleta $\operatorname{sk}_{n}(\operatorname{\mathcal{C}}')$ is good: this is immediate if $\kappa = \aleph _0$ (the assumption that $\operatorname{\mathcal{C}}'$ is finite guarantees that it coincides with $\operatorname{sk}_{n}(\operatorname{\mathcal{C}}')$ for $n \gg 0$), and follows from Corollary 9.2.8.7 when $\kappa > \aleph _0$. We may therefore replace $\operatorname{\mathcal{C}}'$ by $\operatorname{sk}_{n}(\operatorname{\mathcal{C}}')$ and thereby reduce to the case where the simplicial set $\operatorname{\mathcal{C}}$ has dimension $\leq n$, for some integer $n \geq -1$.
We now proceed by induction on $n$. If $n = -1$, then the simplicial set $\operatorname{\mathcal{C}}'$ is empty and the result is clear. Otherwise, let $\operatorname{\mathcal{C}}'' = \operatorname{sk}_{n-1}(\operatorname{\mathcal{C}}')$ be the $(n-1)$-skeleton of $\operatorname{\mathcal{C}}'$ and let $S$ be the collection of all nondegenerate $n$-simplices of $\operatorname{\mathcal{C}}$. Applying Proposition 1.1.4.12, we obtain a pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}}) \ar [r] \ar [d] & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}'', \operatorname{\mathcal{E}}) \ar [d] \\ \prod _{\sigma \in S} \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \Delta ^ n, \operatorname{\mathcal{E}}) \ar [r] & \prod _{\sigma \in S} \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{E}}). } \]
Since the horizontal maps in this diagram are isofibrations (Corollary 4.4.5.12), it is also a categorical pullback square (Corollary 4.5.3.28). Moreover, each of the $\infty $-categories in the diagram is $(\kappa ,\lambda )$-cocomplete of the transition functors is $(\kappa ,\lambda )$-finitary. Combining our inductive hypothesis with Corollary 9.2.8.5 and Proposition 9.2.8.6, we can reduce to the case where $\operatorname{\mathcal{C}}' = \Delta ^ n$ is a standard simplex.
If $n = 0$, then the map $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ can be identified with a vertex $C \in \operatorname{\mathcal{C}}$, and the desired result follows from the observation that $F(C)$ is $(\kappa ,\lambda )$-compact when viewed as an object of the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ (this is a special case of assumption $(\ast )$, applied to the degenerate edge $\operatorname{id}_{C}: C \rightarrow C$). If $n = 1$, then the desired result is a reformulation of Proposition 9.2.9.14. We may therefore assume that $n \geq 2$. Let $\operatorname{Spine}[n] \subseteq \Delta ^ n$ be as in Example 1.5.7.7. The inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne and therefore induces a trivial Kan fibration $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \Delta ^ n, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Spine}[n], \operatorname{\mathcal{E}})$. Consequently, to show that $\Delta ^ n$ is good, it will suffice to show that $\operatorname{Spine}[n]$ is good. This follows from our inductive hypothesis, since the simplicial set $\operatorname{Spine}[n]$ has dimension $1$.
$\square$