Kerodon

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

Proposition 5.4.5.6. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. Then $\operatorname{Pith}(\operatorname{\mathcal{C}})$ is an $\infty $-category.

Proof of Proposition 5.4.5.6. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. Suppose we are given integers $0 < i < n$ and a morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$; we wish to show that $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$. If $n = 2$, then condition $(1)$ of Definition 5.4.1.3 guarantees that we can extend $\sigma _0$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$, which then belongs to $\operatorname{Pith}(\operatorname{\mathcal{C}})$ by virtue of Remark 5.4.5.2. We may therefore assume that $n \geq 3$. In this case, we observe that the composite map

\[ \Delta ^2 \simeq \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) \hookrightarrow \Lambda ^{n}_{i} \xrightarrow {\sigma _0} \operatorname{Pith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$, so that we can extend $\sigma _0$ to an $n$-simplex $\sigma : \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$. To complete the proof, it will suffice to show that $\sigma $ carries each $2$-simplex of $\Delta ^ n$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$. If $n \geq 4$, this is automatic (since every $2$-simplex of $\Delta ^ n$ is contained in the horn $\Lambda ^{n}_{i}$). In the case $n=3$, it follows from our assumption that the collection of thin $2$-simplices of $\operatorname{\mathcal{C}}$ has the inner exchange property (Proposition 5.4.5.10). $\square$