Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 4.3.3.20. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories. Using Example 4.3.3.7, we obtain a canonical isomorphism of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \star \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}})$, where $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ denotes the join of the categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$.

In particular, for integers $p, q \geq 0$, there is a unique isomorphism of simplicial sets

\[ \Delta ^{p} \star \Delta ^{q} \simeq \Delta ^{p+1+q}, \]

which is given on vertices of $\Delta ^{p}$ by the construction $i \mapsto i$ and on vertices of $\Delta ^{q}$ by $j \mapsto p+1+j$.