Kerodon

Proof of Proposition 9.2.2.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $\overline{W}$ be the collection of morphisms which can be written as transfinite compositions of morphisms belonging to $W$. Suppose we are given a diagram $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } ) \rightarrow \operatorname{\mathcal{C}}$ which exhibits the underlying map $f: F(0) \rightarrow F(\alpha )$ as a transfinite composition of morphisms of $\overline{W}$. We wish to show that $f$ also belongs to $\overline{W}$.

For each ordinal $\beta < \alpha $, let $u_{\beta }: F(\beta ) \rightarrow F(\beta +1)$ denote the morphism of $\operatorname{\mathcal{C}}$ obtained by evaluating $F$ on the pair $(\beta , \beta +1)$. By assumption, $u_{\beta }$ belongs to $\overline{W}$. We can therefore choose a well-ordered set $(B(\beta ), \leq _{\beta } )$ and a diagram $G_{\beta }: \operatorname{N}_{\bullet }( B(\beta ))^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which exhibits $u_{\beta }$ as a transfinite composition of morphisms of $W$, in the sense of Variant 9.2.2.3. Since $W$ contains all isomorphisms in $\operatorname{\mathcal{C}}$, we can assume without loss of generality that each $B(\beta )$ is nonempty (see Examples 9.2.2.4 and 9.2.2.5), and therefore contains a smallest element $a_{\beta }$. Let $a_{\alpha }$ be an auxiliary symbol, set $B(\alpha )= \{ a_{\alpha } \} $, and let $B$ denote the disjoint union $\coprod _{ \beta \leq \alpha } B(\beta )$. Given elements $b \in B(\beta )$ and $b' \in B(\beta ')$, we write $b \leq b'$ if either $\beta < \beta '$, or $\beta = \beta '$ and $b \leq _{\beta } b'$. Set $A = \{ a_{\beta } \} _{\beta \leq \alpha } \subseteq B$. The construction $\beta \mapsto a_{\beta }$ determines an order-preserving bijection $\mathrm{Ord}_{\leq \alpha } \xrightarrow {\sim } A$, so that the diagram $F$ can be identified with a functor from $\operatorname{N}_{\bullet }(A)$ to $\operatorname{\mathcal{C}}$. For each $\beta < \alpha $, let us identify $G_{\beta }$ with a functor from $\operatorname{N}_{\bullet }( B(\beta ) \cup \{ a_{ \beta +1} \} )$ to $\operatorname{\mathcal{C}}$. Then the functors $F$ and $\{ G_{\beta } \} _{\beta < \alpha }$ determine a morphism of simplicial sets $H_0: K(A,B) \rightarrow \operatorname{\mathcal{C}}$, where $K(A,B) \subseteq \operatorname{N}_{\bullet }(B)$ is the simplicial subset appearing in the statement of Lemma 9.2.2.17. Since the inclusion map $K(A,B) \hookrightarrow \operatorname{N}_{\bullet }(B)$ is a categorical equivalence of simplicial sets, we can extend $H_0$ to a diagram $H: \operatorname{N}_{\bullet }( B ) \rightarrow \operatorname{\mathcal{C}}$.

Note that the linear ordering on $B$ is a well-ordering, with largest element $a_{\alpha }$. We claim that $H$ exhibits $f$ as a transfinite composition of morphisms of $W$, in the sense of Variant 9.2.2.3. It follows immediately from the construction that $H$ carries the pair $(a_0 \leq a_{\alpha } )$ to the morphism $f$. Moreover, if an element $b \in B$ has an immediate predecessor $b' \in B$, then there is a (unique) ordinal $\beta < \alpha $ such that both $b$ and $b'$ belong to $B(\beta ) \cup \{ a_{\beta +1} \} $; our assumption on $G_{\beta }$ then guarantees that the morphism $H(b') \rightarrow H(b)$ belongs to $W$. To complete the proof, it will suffice to show that if $b \neq a_0$ is an element of $B$ which does not have an immediate predecessor, then the restriction $H|_{ \operatorname{N}_{\bullet }( B_{\leq b} ) }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Note that $b$ belongs to $B(\beta )$ for some unique ordinal $0 \leq \beta \leq \alpha $. We consider three cases:

• Suppose that $b$ is not equal to $a_{\beta }$. In this case, the inclusion map $B(\beta )_{ < b } \hookrightarrow B_{< b }$ is cofinal (in the sense of Definition 4.7.1.26), and therefore induces a right cofinal morphism of simplicial sets $\operatorname{N}_{\bullet }( B(\beta )_{< b} ) \hookrightarrow \operatorname{N}_{\bullet }( B_{7.2.3.4). Using Corollary 7.2.2.2, we are reduced to showing that the diagram $H|_{ \operatorname{N}_{\bullet }( B(\beta )_{\leq b} )}$ is a colimit diagram in $\operatorname{\mathcal{C}}$, which follows from our assumption on $G_{\beta }$.

• Suppose that $b = a_{\beta }$ and that $\beta = \gamma +1$ is a successor ordinal. In this case, the inclusion map $B_{ \gamma } \hookrightarrow B_{< b}$ is cofinal, and therefore induces a right cofinal morphism of simplicial sets $\operatorname{N}_{\bullet }( B_{\beta } ) \hookrightarrow \operatorname{N}_{\bullet }( B_{< b })$. The desired result now follows again Corollary 7.2.2.2, since the restriction $H|_{ \operatorname{N}_{\bullet }( B_{\gamma } \cup \{ b\} ) }$ can be identified with $G_{\gamma }$ and is therefore a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.

• Suppose that $b = a_{\beta }$, where $\beta $ is a nonzero limit ordinal. In this case, the inclusion map $A_{ < b } \hookrightarrow B_{ < b }$ is cofinal and therefore induces a right cofinal morphism of simplicial sets $\operatorname{N}_{\bullet }( A_{7.2.2.2, we are reduced to showing that the restriction $H|_{ \operatorname{N}_{\bullet }( A_{\leq b} ) }$ is a colimit diagram in $\operatorname{\mathcal{C}}$. We conclude by observing that this restriction identifies with $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) }$.