Definition 3.5.1.13. Let $f: X \rightarrow Y$ be a morphism of Kan complexes. We say that $f$ is $n$-connective if it satisfies the following conditions:
If $n \geq 0$, then the underlying map of connected components $\pi _0(X) \rightarrow \pi _0(Y)$ is surjective.
If $n > 0$ and $x \in X$ is a vertex having image $y = f(x)$, the induced map $\pi _{m}(X,x) \rightarrow \pi _{m}(Y,y)$ is a bijection when $m < n$ and a surjection when $m = n$.