Kerodon

Warning 9.6.1.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, let $\kappa $ be a regular cardinal, and let $f$ be a morphism of $\operatorname{\mathcal{C}}$. If $f$ is a $\kappa $-small transfinite composition of morphisms in $W$, then it belongs to the $\kappa $-small transfinite closure of $W$. Beware that, if we strictly adhere to the terminology of Definition 9.6.1.1, then the converse need not be true. For example, if $W = \emptyset $ and $f$ is an isomorphism, then $f$ belongs to the $\kappa $-transfinite closure of $W$ for every uncountable regular cardinal $\kappa $ (Example 9.6.1.14). However, $f$ is a transfinite composition of morphisms in $W$ if and only if it is an identity morphism.