Definition 9.1.1.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $w: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We say that an object $C \in \operatorname{\mathcal{C}}$ is $w$-local if precomposition with the homotopy class $[w]$ induces a homotopy equivalence of mapping spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,C) \xrightarrow {\circ [w]} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,C)$. We say that $C$ is $w$-colocal if postcomposition with $[w]$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \xrightarrow {[w] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)$.
If $W$ is a collection of morphisms of $\operatorname{\mathcal{C}}$, we say that an object $C \in \operatorname{\mathcal{C}}$ is $W$-local if it is $w$-local for each $w \in W$. Similarly, we say that $C$ is $W$-colocal if it is $w$-colocal for each $w \in W$.