Theorem 9.4.5.13. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets which is $(\kappa ,\lambda )$-compactly generated, and let $\operatorname{\mathcal{E}}_{0} \subseteq \operatorname{\mathcal{E}}$ be the full simplicial subset spanned by those vertices $X$ which are $(\kappa ,\lambda )$-compact when viewed as objects of the fiber $\operatorname{\mathcal{E}}_{C}$ for $C = U(X)$. Assume that $\operatorname{\mathcal{C}}$ is $\kappa $-small and that the inner fibration $(U_0 = U|_{ \operatorname{\mathcal{E}}_0}): \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}$ is $\kappa _0$-sequentially cocomplete, for some regular cardinal $\kappa _0 < \kappa $ (see Variant 9.2.9.5). Then:
- $(1)$
The $\infty $-category of sections $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is $(\kappa ,\lambda )$-compactly generated.
- $(2)$
An object $F \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is $(\kappa ,\lambda )$-compact if and only if it factors through $\operatorname{\mathcal{E}}_0 \subseteq \operatorname{\mathcal{E}}$.
In particular, the natural map $\operatorname{Ind}_{\kappa }^{\lambda }( \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}_0 ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is an equivalence of $\infty $-categories.
Proof.
We proceed as in the proof of Proposition 9.2.8.9. We say that a simplicial set $\operatorname{\mathcal{C}}'$ equipped with a map $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ is good if the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})$ is $(\kappa ,\lambda )$-compactly generated, and an object $F \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})$ is $(\kappa ,\lambda )$-compact if and only if $F(C) \in \operatorname{\mathcal{E}}_{C}$ is $(\kappa ,\lambda )$-compact for each vertex $C \in \operatorname{\mathcal{C}}'$. We will complete the proof by showing that if $\operatorname{\mathcal{C}}'$ is $\kappa $-small, then it 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). It follows from Corollary 9.4.4.7 that if each $\operatorname{sk}_{n}(\operatorname{\mathcal{C}}')$ is good, then $\operatorname{\mathcal{C}}'$ is good. 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
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}}) \ar [r] \ar [d] & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}'', \operatorname{\mathcal{E}}) \ar [d] \\ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \coprod _{\sigma \in S} \Delta ^ n, \operatorname{\mathcal{E}}) \ar [r] & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \coprod _{\sigma \in S} \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{E}}). } \]
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). Our inductive hypothesis guarantees that $\operatorname{\mathcal{C}}''$ and $\coprod _{\sigma \in S} \operatorname{\partial \Delta }^ n$ are good. By virtue of Theorem 9.4.4.4, to prove that $\operatorname{\mathcal{C}}'$ is good, it will suffice to show that the disjoint union $\coprod _{\sigma \in S} \Delta ^ n$ is good. Using Proposition 9.4.3.13, can further reduce to the case where $\operatorname{\mathcal{C}}' = \Delta ^ n$ is the standard $n$-simplex.
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. For $n \geq 2$, this follows from our inductive hypothesis. 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 our assumption that the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ is $(\kappa ,\lambda )$-compactly generated. We may therefore assume that $n = 1$. Replacing $U$ by the projection map $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}'$, we are reduced to proving Theorem 9.4.5.13 in the special case where $\operatorname{\mathcal{C}}= \Delta ^1$ is the standard $1$-simplex. Set $\operatorname{\mathcal{E}}_0 = \{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}_{1} = \{ 1\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Then $\operatorname{Fun}_{ / \Delta ^1}( \Delta ^1, \operatorname{\mathcal{C}})$ can be identified with the oriented fiber product $\operatorname{\mathcal{E}}_0 \vec{\times }_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}_1$, and the desired result follows from Theorem 9.4.3.4.
$\square$