Corollary 3.5.2.5. Let $X$ be a Kan complex and let $n$ be an integer. The following conditions are equivalent:
- $(1)$
The Kan complex $X$ is $n$-connective.
- $(2)$
There exists a monomorphism of Kan complexes $f: X \hookrightarrow Y$ where $Y$ is contractible and $f$ is bijective on $k$-simplices for $0 \leq k \leq n$.
- $(3)$
There exists a morphism of simplicial sets $f: X \rightarrow Y$ where $Y$ is $n$-connective, $f$ is bijective on $k$-simplices for $k < n$, and $f$ is surjective on $n$-simplices.