Remark 4.3.3.30. Let $X$ be a simplicial set. Then, for every nonnegative integer $n$, the $n$-skeleton of the cone $X^{\triangleright }$ fits into a pushout diagram
\[ \xymatrix { \operatorname{sk}_{n-1}(X) \ar [r] \ar [d] & \operatorname{sk}_{n-1}(X)^{\triangleright } \ar [d] \\ \operatorname{sk}_{n}(X) \ar [r] & \operatorname{sk}_{n}(X)^{\triangleright }; } \]
see Remark 4.3.3.18. In particular, $X^{\triangleright }$ has dimension $\leq n$ if and only if $X$ has dimension $\leq n-1$.