Corollary 3.5.7.13. Let $n$ be an integer and let $f: X \rightarrow Y$ be a morphism of $n$-truncated Kan complexes. Then $f$ is a homotopy equivalence if and only if it is $(n+1)$-connective.
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Corollary 3.5.7.13. Let $n$ be an integer and let $f: X \rightarrow Y$ be a morphism of $n$-truncated Kan complexes. Then $f$ is a homotopy equivalence if and only if it is $(n+1)$-connective.
Proof. If $n \leq -2$, then $X$ and $Y$ are contractible and there is nothing to prove. For $n \geq -1$, the desired result follows by combining Proposition 3.5.7.7 (and Remark 3.5.7.8) with Theorem 3.2.7.1. $\square$