Lemma 9.2.4.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small filtered colimits. Let $U$ be a collection of morphisms with the following property: for every object $D \in \operatorname{\mathcal{C}}$, there exists a morphism $u: D \rightarrow E$ which belongs to $U$. Then, for every object $C \in \operatorname{\mathcal{C}}$ and every (small) ordinal $\alpha $, there exists a diagram $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } ) \rightarrow \operatorname{\mathcal{C}}$ satisfying the following conditions:
- $(a)$
For every nonzero limit ordinal $\lambda \leq \alpha $, the restriction $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \lambda }) }$ is a colimit diagram.
- $(b)$
For every ordinal $\gamma < \alpha $, the morphism $F(\gamma ) \rightarrow F(\gamma +1)$ belongs to $U$.
- $(c)$
The object $F(0)$ coincides with $C$.
Proof.
Let $Q$ denote the collection of all diagrams $F_{\beta }: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) \rightarrow \operatorname{\mathcal{C}}$ satisfying conditions $(a)$, $(b)$, and $(c)$, where $\beta $ is an ordinal $\leq \alpha $. We regard $Q$ as a partially ordered set, where $F_{\beta } \leq F_{\beta '}$ if $\beta \leq \beta '$ and $F_{\beta } = F_{\beta '} |_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) }$. Note that $Q$ is nonempty: it has a least element given by the diagram $\operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq 0} ) \simeq \{ C\} \hookrightarrow \operatorname{\mathcal{C}}$ taking the value $C$. We claim that $Q$ satisfies the hypothesis of Zorn's lemma: that is, every linearly ordered set $Q' \subset Q$ admits an upper bound. Without loss of generality, we may assume that $Q'$ is nonempty and has no largest element. In this case, the elements of $Q'$ can be amalgamated to a diagram $F_{ < \lambda }: \operatorname{N}_{\bullet }( \mathrm{Ord}_{ < \lambda } ) \rightarrow \operatorname{\mathcal{C}}$, where $\lambda \leq \alpha $ is a nonzero limit ordinal. Our assumption on $\operatorname{\mathcal{C}}$ then guarantees that $F_{ < \lambda }$ can be extended to a colimit diagram
\[ F_{\lambda }: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \lambda } ) \simeq \operatorname{N}_{\bullet }( \mathrm{Ord}_{ < \lambda } )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}. \]
By construction, this diagram satisfies conditions $(a)$, $(b)$, and $(c)$, and is therefore an upper bound for $Q'$.
Applying Zorn's lemma, we deduce that $Q$ has a maximal element $F_{\beta }: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) \rightarrow \operatorname{\mathcal{C}}$. We will complete the proof by showing that $\beta < \alpha $. Assume otherwise, and set $X = F_{\beta }(\beta )$. By assumption, we can choose a morphism $u: X \rightarrow Y$ which belongs to $U$. Let us identify $u$ with an object $\widetilde{Y}$ of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{X/}$. Since the inclusion map $\{ \beta + 1\} \hookrightarrow \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta + 1} )$ is right anodyne (Example 4.3.7.11), the restriction map $\operatorname{\mathcal{C}}_{ F_{\beta } / } \rightarrow \operatorname{\mathcal{C}}_{ X/ }$ is a trivial Kan fibration (Corollary 4.3.6.14). We can therefore lift $\widetilde{Y}$ to an object of the $\infty $-category $\operatorname{\mathcal{C}}_{ F_{\beta } / }$, which we can identify with an extension of $F_{\beta }$ to a diagram $F_{\beta +1}: \operatorname{N}_{\bullet }( \mathrm{Ord}_{ \leq \beta +1 }) \rightarrow \operatorname{\mathcal{C}}$. By construction, this diagram carries the pair $(\beta , \beta +1)$ to the morphism $u$ of $\operatorname{\mathcal{C}}$. It follows that $F_{\beta +1}$ is also an element of $Q$, contradicting the maximality of $F_{\beta }$.
$\square$