Corollary 3.5.7.12 (Homotopy Invariance). Let $n \geq -2$ and let $X$ and $Y$ be Kan complexes which are homotopy equivalent. Then $X$ is $n$-truncated if and only if $Y$ is $n$-truncated.
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Corollary 3.5.7.12 (Homotopy Invariance). Let $n \geq -2$ and let $X$ and $Y$ be Kan complexes which are homotopy equivalent. Then $X$ is $n$-truncated if and only if $Y$ is $n$-truncated.
Proof. For $n \geq 0$ this follows from the criterion of Proposition 3.5.7.7. The case $n < 0$ follows from Examples 3.5.7.4 and 3.5.7.3. $\square$