Example 4.7.2.6. Let $f: X \rightarrow Y$ be a morphism of Kan complexes and let $n$ be an integer. The following conditions are equivalent:
- $(1)$
The morphism $f$ is $n$-truncated, in the sense of Definition 3.5.9.1.
- $(2)$
For every Kan complex $K$, composition with $f$ induces an $n$-truncated morphism of Kan complexes $\operatorname{Fun}(K,X) \rightarrow \operatorname{Fun}(K,Y)$.
- $(3)$
For every simplicial set $K$, composition with $f$ induces an $n$-truncated morphism $\operatorname{Fun}(K,X) \rightarrow \operatorname{Fun}(K,Y)$.
- $(4)$
The morphism $f$ is $n$-truncated when regarded as a morphism in the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces, in the sense of Definition 4.7.2.1.
The implications $(3) \Rightarrow (2) \Rightarrow (1)$ are immediate, the implication $(1) \Rightarrow (3)$ follows from Corollary 3.5.9.30, and the equivalence $(2) \Leftrightarrow (4)$ follows from Example 4.6.7.7.