Definition 3.5.4.1. Let $n$ be an integer. We say that a simplicial set $X$ is weakly $n$-coskeletal if the restriction map
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, X) \]
is a bijection for $m \geq n+2$ and and injection for $m = n+1$ (provided that $n \geq -1$).