Kerodon

Proof. Combine Proposition 7.1.10.1 with Example 7.1.7.10. $\square$

Proof. Combine Proposition 7.1.10.1 with Corollary 7.1.7.16. $\square$

Proof. Apply Proposition 7.1.10.1 to every diagram $q: K \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. $\square$

Proof. We first consider the case where $\operatorname{\mathcal{C}}$ is an $\infty $-category. In this case, we can use Proposition 7.1.7.14 to reformulate condition $(\ast )$ as follows:

$(\ast ')$

For each object $C \in \operatorname{\mathcal{C}}$, the diagram

\[ \overline{q}_{C}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C} \hookrightarrow \operatorname{\mathcal{E}} \]

is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$.

Applying Corollary 7.1.8.14, we see that $\overline{q}$ is a $U^{\operatorname{\mathcal{C}}}$-colimit diagram, where $U^{\operatorname{\mathcal{C}}}: \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ is the functor given by composition with $U$. The desired result now follows by applying Corollary 7.1.7.8 to the pullback square

We now treat the general case. For every morphism of simplicial sets $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, let $\overline{q}_{\operatorname{\mathcal{C}}'}$ denote the composition of $\overline{q}$ with the restriction functor

Let us say that $\operatorname{\mathcal{C}}'$ is good if $\overline{q}_{\operatorname{\mathcal{C}}'}$ is a colimit diagram in the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})$. We wish to show that the simplicial set $\operatorname{\mathcal{C}}$ 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

where the transition maps are isofibrations (Corollary 4.4.5.12). By virtue of Proposition 7.1.9.9, it will suffice to show each skeleton $\operatorname{sk}_{n}(\operatorname{\mathcal{C}})$ 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 there is nothing to prove. 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

Since the horizontal maps in this diagram are isofibrations (Corollary 4.4.5.12), it is also a categorical pullback square (Corollary 4.5.2.28). Our inductive hypothesis guarantees that the simplicial sets $\operatorname{\mathcal{C}}'$ and $\coprod _{\sigma \in S} \operatorname{\partial \Delta }^ n$ are good. By virtue of Proposition 7.1.9.8, to show that $\operatorname{\mathcal{C}}$ is good, it will suffice to show that the coproduct $\coprod _{\sigma \in S} \Delta ^ n$ is good. This follows from the first part of the proof, since the disjoint union $\coprod _{\sigma in S} \Delta ^ n$ is an $\infty $-category. $\square$

Proof of Proposition 7.1.10.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let $q: K \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ be a diagram. Assume that, for every vertex $C \in \operatorname{\mathcal{C}}$, the induced map $q_ C: K \rightarrow \operatorname{\mathcal{E}}_{C}$ can be extended to a colimit diagram $K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$, which is also an edgewise $U$-colimit diagram in $\operatorname{\mathcal{E}}$. We will show that $q$ admits an extension $\overline{q}: K^{\triangleright } \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ having the property that, for each vertex $C \in \operatorname{\mathcal{C}}$, the induced map $\overline{q}_{C}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$. Lemma 7.1.10.8 guarantees that the extension $\overline{q}$ is automatically a colimit diagram in the $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. It then follows that any other extension of $q$ to a colimit diagram $\overline{q}': K^{\triangleright } \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is isomorphic to $\overline{q}$, and therefore also induces colimit diagrams $\overline{q}'_{C}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ for each $C \in \operatorname{\mathcal{C}}$.

We proceed as in the proof of Lemma 7.1.10.8. For every morphism of simplicial sets $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, let $q_{\operatorname{\mathcal{C}}'}$ denote the composition of $q$ with the restriction functor

Let us say that $\operatorname{\mathcal{C}}'$ is good if $q_{\operatorname{\mathcal{C}}'}$ admits a colimit in the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})$ which is preserved by the evaluation functors

for each vertex $C \in \operatorname{\mathcal{C}}'$. If this condition is satisfied, then Lemma 7.1.10.8 guarantees that the colimit of $q_{\operatorname{\mathcal{C}}'}$ is preserved by the restriction functor

associated to any morphism of simplicial sets $\operatorname{\mathcal{C}}'' \rightarrow \operatorname{\mathcal{C}}'$.

We wish to show that the simplicial set $\operatorname{\mathcal{C}}$ 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

where the transition maps are isofibrations (Corollary 4.4.5.12). By virtue of Proposition 7.1.9.9, it will suffice to show each skeleton $\operatorname{sk}_{n}(\operatorname{\mathcal{C}})$ 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 $n \geq 0$.

We now proceed by induction on $n$. If $n = 0$, then $\operatorname{\mathcal{C}}$ is discrete and the desired result follows from our assumption that each of the diagrams $q_{C}$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$. 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

Since the horizontal maps in this diagram are isofibrations (Corollary 4.4.5.12), it is also a categorical pullback square Corollary 4.5.2.28. Our inductive hypothesis guarantees that the simplicial sets $\operatorname{\mathcal{C}}'$ and $\coprod _{\sigma \in S} \operatorname{\partial \Delta }^ n$ are good. Using Proposition 7.1.9.8, we can replace $\operatorname{\mathcal{C}}$ by $\coprod _{\sigma \in S} \Delta ^ n$ and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is a disjoint union of simplices. Decomposing $\operatorname{\mathcal{C}}$ into connected components, we may further assume that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex.

Let $\operatorname{Spine}[n]$ denote the spine of the standard $n$-simplex, so that the inclusion map $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne (Example 1.5.7.7). It follows that the restriction map

is a trivial Kan fibration. Consequently, to show that $\Delta ^ n$ is good, it will suffice to show that $\operatorname{Spine}[n]$ is good. If $n \geq 2$, this follows from our inductive hypothesis. We may therefore assume that $n = 1$. Since $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}= \Delta ^1$ is an inner fibration, the simplicial set $\operatorname{\mathcal{E}}$ is an $\infty $-category.

Let $\operatorname{\mathcal{E}}_{0} = \{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}_{1} = \{ 1\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ denote the fibers of $U$, so we can identify $q$ with a pair of diagrams $q_0: K \rightarrow \operatorname{\mathcal{E}}_0$ and $q_1: K \rightarrow \operatorname{\mathcal{E}}_1$ together with a natural transformation $\alpha : q_0 \rightarrow q_1$ between $K$-indexed diagrams in $\operatorname{\mathcal{E}}$. Invoking assumption $(\ast )$, we deduce that $q_0$ and $q_1$ can be extended to colimit diagrams $\overline{q}_0: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_0$ and $\overline{q}_1: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_1$, respectively. Moreover, the extension $\overline{q}_0$ is also a colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$. Applying Corollary 7.1.6.20, we deduce that the restriction map

is a trivial Kan fibration. It follows that $\alpha $ can be extended to a natural transformation $\overline{\alpha }: \overline{q}_0 \rightarrow \overline{q}_1$ of $K^{\triangleright }$-indexed diagrams in $\operatorname{\mathcal{E}}$. The triple $( \overline{q}_0, \overline{q}_1, \overline{\alpha } )$ then determines an extension of $q$ to a diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ having the desired properties. $\square$