Kerodon

Remark 4.7.1.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. If $n \leq -2$, then an object $X \in \operatorname{\mathcal{C}}$ is $n$-truncated if and only if the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is contractible for every object $C \in \operatorname{\mathcal{C}}$ (in this case, we say that $X$ is a final object of $\operatorname{\mathcal{C}}$; see Definition 4.7.3.1). In particular, $X$ is $n$-truncated if and only if it is $(-2)$-truncated. Consequently, there is no loss of generality in restricting Definition 4.7.1.1 to the case $n \geq -2$.