Kerodon

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Definition 6.3.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We say that an object $Z \in \operatorname{\mathcal{C}}$ is $W$-local if, for every morphism $w: X \rightarrow Y$ belonging to $W$, precomposition with the homotopy class $[w]$ induces an isomorphism $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ [w] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. We say that $Z$ is $W$-colocal if, for every morphism $w: Y \rightarrow X$ belonging to $W$, postcomposition with the homotopy class $[w]$ induces an isomorphism $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( Z, Y) \xrightarrow { [w] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Z,X)$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.