Kerodon

Warning 4.8.1.21. Let $n \geq 1$ be an integer and let $\operatorname{\mathcal{C}}$ be an $(n,1)$-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, Corollary 4.8.1.20 guarantees that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $(n-1)$-truncated, and is therefore homotopy equivalent to an $(n-1)$-groupoid (for example, it is homotopy equivalent to the pinched morphism spaces $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y)$). Beware that, for $n \geq 2$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ itself is generally not an $(n-1)$-groupoid. For example, this usually fails in the case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}_0)$ arises as the Duskin nerve of a $2$-category $\operatorname{\mathcal{C}}_0$: see Remark 8.1.8.8. However, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is always weakly $(n-1)$-coskeletal: see Corollary 4.8.3.5.