Definition 3.5.4.14. Let $X$ be a simplicial set and let $n$ be an integer. We will say that a morphism of simplicial sets $f: X \rightarrow Y$ exhibits $Y$ as a weak $n$-coskeleton of $X$ if the following conditions are satisfied:
The simplicial set $Y$ is weakly $n$-coskeletal.
The morphism $f$ is bijective on simplices of dimension $\leq n$ and surjective on $(n+1)$-simplices (provided that $n \geq -1$).