Kerodon

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

Corollary 4.3.3.24. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Then the join $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is an $\infty $-category.

Proof. Since $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, the projection maps $u: \operatorname{\mathcal{C}}\rightarrow \Delta ^0$ and $v: \operatorname{\mathcal{D}}\rightarrow \Delta ^0$ are inner fibrations (Example 4.1.1.2). Applying Proposition 4.3.3.23, we deduce that the join

\[ (u \star v): \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\rightarrow \Delta ^0 \star \Delta ^0 \simeq \Delta ^1 \]

is also an inner fibration. Since $\Delta ^1$ is an $\infty $-category, it follows that $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is an $\infty $-category (Remark 4.1.1.9). $\square$