Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 7.2.1.16. The collection of left cofinal morphisms of simplicial sets is closed under the formation of filtered colimits (when regarded as a full subcategory of the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$).

Proof. For every morphism of simplicial sets $f: X \rightarrow Z$, let $X \xrightarrow {f'} Q(f) \xrightarrow {f''} Y$ be the factorization of Proposition 4.2.4.8, so that $f'$ is left anodyne, $f''$ is a left fibration, and the construction $f \mapsto Q(f)$ is a functor which commutes with filtered colimits. Using Propositions 7.2.1.5, 7.2.1.6, and Corollary 7.2.1.13, we see that $f$ is left cofinal if and only if the morphism $f'': Q(f) \rightarrow Z$ is a trivial Kan fibration. Since the collection of trivial Kan fibrations is closed under filtered colimits (Remark 1.4.5.3), it follows that the collection of left cofinal morphisms is also closed under filtered colimits. $\square$