$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.2.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Assume that:
- $(1)$
Every identity morphism of $\operatorname{\mathcal{C}}$ belongs to $W$.
- $(2)$
The collection $W$ is closed under composition.
- $(3)$
The collection $W$ is closed under the formation of colimits in the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$.
Then $W$ is closed under transfinite composition.
Proof.
Let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which is a transfinite composition of morphisms of $W$; we wish to show that $f \in W$. Choose an ordinal $\alpha $ and a functor $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{ \leq \alpha } ) \rightarrow \operatorname{\mathcal{C}}$ which exhibits $f$ as a transfinite composition of morphisms of $W$. For each $\beta \leq \alpha $, the functor $F$ carries the ordered pair $(0 \leq \beta )$ to a morphism $f_{\beta }: F(0) \rightarrow F(\beta )$. We will complete the proof by showing that each the morphisms $f_{\beta }$ belongs to $W$. The proof proceeds by transfinite induction on $\beta $. If $\beta = 0$, then $f_{\beta } = \operatorname{id}_{X}$ and the desired result follows from assumption $(1)$. If $\beta $ is a nonzero limit ordinal, then the desired result follows from assumption $(3)$. Remark 9.2.1.9. It will therefore suffice to treat the case where $\beta = \gamma +1$ is a successor ordinal. In this case, the desired result follows by applying assumption $(2)$ to the diagram
\[ \xymatrix@R =50pt@C=50pt{ & F(\gamma ) \ar [dr] & \\ X \ar [ur]^{ f_{\gamma } } \ar [rr]^{ f_{\beta } } & & F(\beta ), } \]
since $f_{\gamma }$ belongs to $W$ by virtue of our inductive hypothesis.
$\square$