Proposition 3.5.7.15. Let $X$ be a Kan complex and let $n$ be an integer. The following conditions are equivalent:
The Kan complex $X$ is $n$-truncated.
There exists an $n$-truncated Kan complex $Y$ which is homotopy equivalent to $X$.
There exists an $(n+1)$-coskeletal Kan complex $Y$ which is homotopy equivalent to $X$.
The tautological map $X \rightarrow \operatorname{cosk}_{n+1}(X)$ is a homotopy equivalence.
Proof. The implication $(2) \Rightarrow (1)$ follows from Corollary 3.5.7.12, the implication $(3) \Rightarrow (2)$ from Example 3.5.7.2, and the implication $(4) \Rightarrow (3)$ from the observation that $\operatorname{cosk}_{n+1}(X)$ is a Kan complex (Proposition 3.5.3.23). We will complete the proof by showing that $(1)$ implies $(4)$. Assume that $X$ is $n$-truncated; we wish to show that the tautological map $u: X \rightarrow \operatorname{cosk}_{n+1}(X)$ is a homotopy equivalence. Since $\operatorname{cosk}_{n+1}(X)$ is also $n$-truncated (Example 3.5.7.2), it will suffice to show that $u$ is $(n+1)$-connective (Corollary 3.5.7.13). This is a special case of Remark 3.5.3.22. $\square$