Kerodon

Remark 9.2.3.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. It follows from Proposition 1.4.4.2 that an object $C \in \operatorname{\mathcal{C}}$ is weakly $W$-local if and only if, for every morphism $w: X \rightarrow Y$ which belongs to $W$, composition with the homotopy class $[w]$ induces a surjection $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, C ) \rightarrow \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,C)$. In other words, the object $C$ is weakly $W$-local (in the sense of Definition 9.2.3.5) if and only if it is weakly $[W]$-local when regarded as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (in the sense of Definition 9.2.3.1). Here $[W] = \{ [w]: w \in W \} $ denotes the collection of all homotopy classes of morphisms which belong to $W$.