Remark 3.5.1.19. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. Then $f$ is a weak homotopy equivalence if and only if it is $n$-connective for every integer $n$. To see this, we can assume without loss of generality that $X$ and $Y$ are Kan complexes, in which case it is a restatement of Theorem 3.2.7.1.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$