Kerodon

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

Remark 1.4.4.17. Let $\operatorname{\mathcal{C}}$ be a category and let $T_0$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Then there exists a smallest collection of morphisms $T$ of $\operatorname{\mathcal{C}}$ such that $T_0 \subseteq T$ and $T$ is weakly saturated (for example, we can take $T$ to be the intersection of all the weakly saturated collections of morphisms containing $T_0$). We will refer to $T$ as the weakly saturated collection of morphisms generated by $T_0$. It follows from Proposition 1.4.4.16 that if every morphism of $T_0$ has the left lifting property with respect to some collection of morphisms $S$, then every morphism of $T$ also has the left lifting property with respect to $S$.