Kerodon

Warning 9.2.2.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $f$ be a morphism of $\operatorname{\mathcal{C}}$. If $f$ is a transfinite composition of morphisms in $W$, then it belongs to the transfinite closure of $W$. Beware that, if we strictly adhere to the terminology of Definition 9.2.2.1, then the converse need not be true. For example, if $W = \emptyset $ and $f$ is an isomorphism, then $f$ belongs to the transfinite closure of $W$ (Example 9.2.2.14). However, $f$ is a transfinite composition of morphisms in $W$ if and only if it is an identity morphism (Example 9.2.2.4).