Remark 4.3.3.20. Let $X$ and $Y$ be simplicial sets, and let $\sigma : \Delta ^{m} \rightarrow X \star Y$ be an $m$-simplex which factors as a composition
\[ \Delta ^{m} \simeq \Delta ^{m_{-}} \star \Delta ^{m_{+}} \xrightarrow { \sigma _{-} \star \sigma _{+} } X \star Y \]
for some integers $m_{-}, m_{+} \geq 0$ satisfying $m_{-} + 1 + m_{+} = m$. Then $\sigma $ is nondegenerate if and only if both $\sigma _{-}$ and $\sigma _{+}$ are nondegenerate. It follows that, for every integer $n$, the $n$-skeleton of $X \star Y$ is given by the union
\[ \operatorname{sk}_{n}(X) \cup (\bigcup _{p+1+q =n} \operatorname{sk}_{p}(X) \star \operatorname{sk}_{q}(Y) ) \cup \operatorname{sk}_{n}(Y). \]
In particular, we have an equality
\[ \mathrm{dim}(X \star Y) = \mathrm{dim}(X) + 1 + \mathrm{dim}(Y), \]
provided that we adopt the convention that an empty simplicial set has dimension $-1$.