Corollary 3.5.1.29. Let $f: X \rightarrow Y$ be a Kan fibration of simplicial sets and let $n$ be a nonnegative integer. Then $f$ is $n$-connective if and only if it satisfies the following pair of conditions:
The map of connected components $\pi _0(f): \pi _0(X) \rightarrow \pi _0(Y)$ is surjective.
The diagonal map $\delta _{X/Y}: X \rightarrow X \times _{Y} X$ is $(n-1)$-connective.
Proof. Without loss of generality, we may assume that condition $(a)$ is satisfied. Since $f$ is a Kan fibration, every vertex $y \in Y$ has the form $f(x)$ for some vertex $x \in X$ (Corollary 3.2.6.3). It follows that every fiber of $f$ can also be viewed as a fiber of the map $q: X \times _{Y} X \rightarrow X$ given by projection onto the first factor. Using the criterion of Proposition 3.5.1.22, we see that $f$ is $n$-connective if and only if $q$ is $n$-connective. The desired result now follows by applying Corollary 3.5.1.28 to the morphisms $X \xrightarrow {\delta _{X/Y}} X \times _{Y} X \xrightarrow {q} X$, since the composite map $q \circ \delta _{X/Y} = \operatorname{id}_{X}$ is $n$-connective. $\square$