Definition 9.3.1.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. We say that an object $X \in \operatorname{\mathcal{C}}$ is $n$-truncated if, for every object $Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)$ is an $n$-truncated Kan complex.
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