Kerodon

Proof. We will show that the right-pinched morphism space $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y)$ is an $(n-1)$-groupoid; the analogous statement for the left-pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ follows by a similar argument. If $n = 1$, then $\operatorname{\mathcal{C}}$ can be identified with the nerve of an ordinary category $\operatorname{\mathcal{C}}_0$ (Example 4.8.1.3) and the desired result follows from Example 4.6.5.12. We may therefore assume that $n > 1$. Since $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ is a Kan complex, it will suffice to show that it is an $(n-1,1)$-category (Example 4.8.1.11). Let $m \geq n$ and let $\sigma _0$ and $\sigma _1$ be $m$-simplices of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ which satisfy $\sigma _0|_{ \Lambda ^{m}_{i} } = \sigma _1|_{ \Lambda ^{m}_{i} }$ for some $0 < i < m$; we wish to show that $\sigma _0 = \sigma _1$. Let us identify $\sigma _0$ and $\sigma _1$ with morphisms $\tau _0, \tau _1: \Delta ^{m+1} \rightarrow \operatorname{\mathcal{C}}$ which carry the simplicial subset $\Delta ^{m} \subset \Delta ^{m+1}$ to the object $X$ and the final vertex of $\Delta ^{m+1}$ to the object $Y$. Our assumptions then guarantee that $\tau _0$ and $\tau _1$ have the same restriction to $\Lambda ^{m+1}_{i}$. Since $\operatorname{\mathcal{C}}$ is an $(n,1)$-category, it follows that $\tau _0 = \tau _1$. $\square$