Proposition 9.6.3.9. Let $\lambda $ be a regular cardinal, let $\operatorname{\mathcal{C}}$ be a $\lambda $-cocomplete $\infty $-category and let $W = \{ w_ s: C_ s \rightarrow D_ s \} _{s \in S}$ be a $\lambda $-small collection of morphisms of $\operatorname{\mathcal{C}}$. Assume that:
- $(1)$
There exists a regular cardinal $\kappa < \lambda $ such that each of the objects $C_ s$ is $(\kappa ,\lambda )$-compact.
- $(2)$
For every object $E \in \operatorname{\mathcal{C}}$ and every $s \in S$, the set of homotopy classes $\operatorname{Hom}_{\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}}}( C_ s, E )$ is $\lambda $-small.
For every object $X \in \operatorname{\mathcal{C}}$, there exists a morphism $f: X \rightarrow Y$ where $Y$ is weakly $W$-local and $f$ is a $\lambda $-small transfinite pushout of morphisms of $W$.
Proof.
Without loss of generality, we may assume that $X$ is not weakly $W$-local (otherwise, we can take $f = \operatorname{id}_ X$). Let $W'$ be the collection of all morphisms of $\operatorname{\mathcal{C}}$ which are pushouts of morphisms of $W$, and let $\overline{W}$ denote the transfinite closure of $W'$ (Definition 9.6.1.13). Let $U \subseteq \overline{W}$ denote the subcollection consisting of those morphisms $u$ which satisfy the requirement of Lemma 9.6.3.7. Using Lemma 9.6.3.8 we deduce that there exists a diagram $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \kappa } ) \rightarrow \operatorname{\mathcal{C}}$ where $F(0) = X$ and $F$ exhibits the induced map $F(0) \xrightarrow {f} F(\kappa )$ as a transfinite composition of morphisms of $U$.
We first claim that the object $Y = F(\kappa )$ is weakly $W$-local. Let $w: C \rightarrow D$ be a morphism which belongs to $W$. We wish to show that every morphism $[ \overline{e} ]: C \rightarrow Y$ in the homotopy category $\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}}$ factors through the homotopy class $[w]$. Since $F$ is a colimit diagram and the object $C$ is $(\kappa ,\lambda )$-compact, the morphism $[ \overline{e} ]$ factors as a composition $C \xrightarrow { [e] } F(\alpha ) \rightarrow F(\kappa ) = Y$ for some ordinal $\alpha < \kappa $ and some morphism $e: C \rightarrow F(\alpha )$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Since the transition map $F( \alpha ) \rightarrow F(\alpha +1)$ belongs to $U$, we can choose a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ C \ar [d] \ar [r]^-{w} \ar [d]^{e} & D \ar [d]^{e'} \\ F(\alpha ) \ar [r] & F(\alpha +1). } \]
It follows that $[ \overline{e} ]$ factors as a composition $C \xrightarrow {[w]} D \xrightarrow { [e'] } F(\alpha +1) \rightarrow F(\kappa ) = Y$.
To complete the proof, it will suffice to show that $f$ is a $\lambda $-small transfinite composition of morphisms belonging to $W'$. By construction, $f$ is a $\lambda $-small transfinite pushout of morphisms of $U \subseteq \overline{W}$, and therefore belongs to $\overline{W}$. The desired result now follows from Corollary 9.6.1.19, since $f$ is not an isomorphism (otherwise, the object $X$ would also be weakly $W$-local, contrary to our initial assumption).
$\square$