Proposition 9.1.6.12 (02QC)

Proposition 9.1.6.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa $ be an infinite cardinal. The following conditions are equivalent:

$(1)$: There exists a $\kappa $-filtered diagram of simplicial sets $\{ \operatorname{\mathcal{C}}_{\alpha } \} $, where each $\operatorname{\mathcal{C}}_{\alpha }$ is an $\infty $-category with a final object, and an equivalence of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \varinjlim _{\alpha } \operatorname{\mathcal{C}}_{\alpha }$.
$(2)$: There exists a $\kappa $-filtered diagram of simplicial sets $\{ \operatorname{\mathcal{C}}_{\alpha } \} $, where each $\operatorname{\mathcal{C}}_{\alpha }$ is a $\kappa $-filtered $\infty $-category, and an equivalence of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \varinjlim _{\alpha } \operatorname{\mathcal{C}}_{\alpha } $.
$(3)$: There exists an equivalence of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$, where $\operatorname{\mathcal{C}}'$ is $\kappa $-filtered
$(4)$: The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-filtered.

Proof. The implication $(1) \Rightarrow (2)$ follows from Example 9.1.1.6, the implication $(2) \Rightarrow (3)$ from Remark 9.1.6.11, and the implication $(3) \Rightarrow (4)$ from Corollary 9.1.1.15. We will complete the proof by showing that every $\kappa $-filtered $\infty $-category $\operatorname{\mathcal{C}}$ satisfies condition $(1)$. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ satisfies condition $(\ast )$ of Lemma 9.1.6.4. Let $A$ be the $\kappa $-directed partially ordered set defined in the proof of Theorem 9.1.6.2. For each $\alpha \in A$, let $L_{\alpha } \subseteq \operatorname{\mathcal{C}}$ denote the corresponding subset of $\operatorname{\mathcal{C}}$. Using Corollary 4.1.3.3, we can choose an $\infty $-category $\operatorname{\mathcal{C}}_{\alpha }$ and an inner anodyne morphism $F_{\alpha }: L_{\alpha } \hookrightarrow \operatorname{\mathcal{C}}_{\alpha }$, depending functorially on $\alpha $. Applying Corollary 4.5.7.2, we see that the morphisms $F_{\alpha }$ induce an equivalence of $\infty $-categories

\[ \operatorname{\mathcal{C}}\simeq \varinjlim _{\alpha \in A} L_{\alpha } \xrightarrow { \{ F_{\alpha } \} _{\alpha \in A} } \varinjlim _{\alpha \in A} \operatorname{\mathcal{C}}_{\alpha }. \]

To complete the proof, it will suffice to show that each of the $\infty $-categories $\operatorname{\mathcal{C}}_{\alpha }$ contains a final object. By construction, there exists an isomorphism of simplicial sets $u: L_{\alpha } \simeq K^{\triangleright }$, for some $\kappa $-small simplicial set $K$. Using Corollary 4.1.3.3, we can choose a categorical equivalence $v: K \rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is an $\infty $-category. Applying Corollary 4.5.8.9, we deduce that the map $v^{\triangleright }: K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}^{\triangleright }$ is also a categorical equivalence of simplicial sets. Since $F_{\alpha }$ is inner anodyne, there exists a functor $G: \operatorname{\mathcal{C}}_{\alpha } \rightarrow \operatorname{\mathcal{D}}^{\triangleright }$ satisfying $G \circ F_{\alpha } = v^{\triangleright } \circ u$. Applying the two-out-of-three property (Remark 4.5.3.5), we see that $G$ is an equivalence of $\infty $-categories. Since the $\infty $-category $\operatorname{\mathcal{D}}^{\triangleright }$ has a final object (given by the cone point; see Example 4.6.7.5), it follows that $\operatorname{\mathcal{C}}_{\alpha }$ also has a final object (Corollary 4.6.7.21). $\square$