Corollary 3.5.9.30 (056V)

Corollary 3.5.9.30. Let $X$ be a Kan complex and let $n \geq -2$ be an integer. The following conditions are equivalent:

$(1)$: The Kan complex $X$ is $n$-truncated.
$(2)$: For $m \geq n+2$, every morphism $f: \operatorname{\partial \Delta }^{m} \rightarrow X$ is nullhomotopic.
$(3)$: If $A$ is an $(n+1)$-connective simplicial set, then every morphism $f: A \rightarrow X$ is nullhomotopic.
$(4)$: If $A$ is an $(n+1)$-connective simplicial set, then the diagonal map $X \rightarrow \operatorname{Fun}(A,X)$ is a homotopy equivalence.
$(5)$: For every $(n+1)$-connective morphism of simplicial sets $A \rightarrow B$, the induced map $\operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X)$ is a homotopy equivalence.

Proof. The equivalence of $(1) \Leftrightarrow (2)$ follows from Variant 3.2.4.12. The implication $(3) \Rightarrow (2)$ follows from Corollary 3.5.2.6 and the implications $(5) \Rightarrow (4) \Rightarrow (3)$ are immediate (see Example 3.5.1.18). To complete the proof, it will suffice to show that $(1)$ implies $(5)$. This follows by applying Corollary 3.5.9.29 in the special case $Y = \Delta ^0$. $\square$