Remark 4.3.3.19. Let $X_{\bullet }$ and $Y_{\bullet }$ be simplicial sets. By virtue of Remark 4.3.3.17, the join $(X \star Y)_{\bullet }$ can be described explicitly by the formula
\[ (X \star Y)_{n} = X_{n} \amalg ( \coprod _{ p+1+q = n } X_{p} \times Y_ q ) \amalg Y_{n}. \]
In these terms, the face and degeneracy operators $\{ d^{n}_ i: (X \star Y)_{n} \rightarrow (X \star Y)_{n-1} \} _{0 \leq i \leq n}$ and $\{ s^{n}_ i: (X \star Y)_{n} \rightarrow ( X \star Y)_{n+1} \} $ are given on the first and third summand by the analogous operators for $X_{\bullet }$ and $Y_{\bullet }$, and on elements $(\sigma , \tau ) \in X_{p} \times Y_{q}$ by the formula
\[ d^{n}_ i( \sigma , \tau ) = \begin{cases} ( d^{p}_ i(\sigma ), \tau ) & \textnormal{ if $i \leq p$ } \\ (\sigma , d^{q}_{i-1-p}(\tau )) & \textnormal{ if } i > p. \end{cases} \quad \quad s^{n}_ i(\sigma , \tau ) = \begin{cases} (s^{p}_ i(\sigma ), \tau ) & \textnormal{ if } i \leq p \\ (\sigma , s^{q}_{i-1-p}(\tau )) & \textnormal{ if } i > p. \end{cases} \]