Example 9.3.1.4. Let $X$ be a Kan complex and let $n$ be an integer. The following conditions are equivalent:
- $(1)$
The Kan complex $X$ is $n$-truncated, in the sense of Definition 3.5.7.1.
- $(2)$
For every Kan complex $Y$, the Kan complex $\operatorname{Fun}(Y,X)$ is $n$-truncated.
- $(3)$
For every simplicial set $Y$, the Kan complex $\operatorname{Fun}(Y, X)$ is $n$-truncated.
- $(4)$
The Kan complex $X$ is $n$-truncated when regarded as an object of the $\infty $-category $\operatorname{\mathcal{S}}$ (in the sense of Definition 9.3.1.1).
The implications $(3) \Rightarrow (2) \Rightarrow (1)$ are immediate, the implication $(1) \Rightarrow (3)$ follows from Corollary 3.5.9.28, and the equivalence $(2) \Leftrightarrow (4)$ follows from the homotopy equivalence $\operatorname{Fun}(Y,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}(Y,X)$ of Remark 5.5.1.5.