Kerodon

Proof. Choose a diagram $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } ) \rightarrow \operatorname{\mathcal{C}}$ which exhibits $f$ as a transfinite composition of morphisms of $W$. For every pair of ordinals $0 \leq \gamma \leq \beta \leq \alpha $, let $u_{\beta , \gamma }: F(\beta ) \rightarrow F(\gamma )$ denote the morphism of $\operatorname{\mathcal{C}}$ obtained by evaluating $F$ on the edge $(\gamma \leq \beta )$ of $\operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } )$. We write $\beta \sim \gamma $ if, for every ordinal $\lambda $ satisfying $\gamma \leq \lambda \leq \beta $, the morphisms $u_{\beta , \lambda }$ and $u_{\lambda , \gamma }$ are both isomorphisms. It is not difficult to see that this is an equivalence relation on the set $\mathrm{Ord}_{\leq \alpha }$. For every ordinal $\beta \leq \alpha $, the equivalence class of $\beta $ contains a smallest element which we will denote by $\beta _{-}$ (since $\mathrm{Ord}_{\leq \alpha }$ is well-ordered), and a largest element which we will denote by $\beta _{+}$ (since the collection of isomorphisms is closed under transfinite composition; see Corollary 9.2.2.12).

Choose a subset $A \subseteq \mathrm{Ord}_{\leq \alpha }$ which contains exactly one representative of each $\sim $-equivalence class. Our assumption that $f = u_{\alpha ,0}$ is not an isomorphism guarantees that $0$ and $\alpha $ belong to different equivalence classes; we can therefore arrange that both $0$ and $\alpha $ are contained in $A$. We will complete the proof by showing that the diagram $F|_{ \operatorname{N}_{\bullet }(A) }$ exhibits $f$ as a transfinite composition of morphisms of $W$ which are not isomorphisms (in the sense of Variant 9.2.2.3).

For any pair of ordinals $\gamma < \beta $ which belong to $A$, we have inequalities $\gamma \leq \gamma _{+} < \beta _{-} \leq \beta $. Then $u_{\beta , \gamma }$ factors as a composition

where the maps on the left and right are isomorphisms. In particular, $u_{ \beta , \gamma }$ is isomorphic to $u_{ \beta _{-}, \gamma _{+} }$ as an object of the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$. If $\beta $ is an immediate successor of $\gamma $ in $A$, then $\beta _{-}$ is an immediate successor of $\gamma _{+}$ in $\mathrm{Ord}_{\leq \alpha }$. Our assumption on $F$ then guarantees that $u_{ \beta _{-}, \gamma _{+} }$ is contained in $W$. Since $W$ is closed under isomorphism, it follows that $u_{\beta , \gamma }$ is also contained in $W$. Moreover, $u_{\beta , \gamma }$ cannot be an isomorphism (otherwise we would have $\beta \sim \gamma $, contradicting our assumption that $A$ contains exactly one representative of each equivalence class).

For each element $\beta \in A$, set $A_{\leq \beta } = \{ \gamma \in A: \gamma \leq \beta \} $ and $A_{ < \beta } = \{ \gamma \in A: \gamma < \beta \} $. To complete the proof, it will suffice to show that if $\beta \neq 0$ is not the immediate successor of another element of $A$, then the restriction $F|_{ \operatorname{N}_{\bullet }( A_{\leq \beta } ) }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $u_{\beta , \beta _{-} }$ is an isomorphism, it will suffice to show that $G = F|_{ \operatorname{N}_{\bullet }( A_{ < \beta } \cup \{ \beta _{-} \} )}$ is a colimit diagram (Corollary 7.1.3.14). Our assumption that $\beta $ has no immediate predecessor in $A$ guarantees that $\beta _{-}$ is a limit ordinal and that $A_{ < \beta }$ is a cofinal subset of in $\mathrm{Ord}_{< \beta _{-} }$. It follows that the inclusion map $\operatorname{N}_{\bullet }( A_{< \beta } ) \hookrightarrow \operatorname{N}_{\bullet }( \mathrm{Ord}_{< \beta } )$ is right cofinal (Corollary 7.2.3.4). The desired result now follows from Corollary 7.2.2.2, since the restriction $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta _{-} } ) }$ is a colimit diagram in $\operatorname{\mathcal{C}}$. $\square$