Kerodon

Warning 8.1.8.1. Let $\operatorname{\mathcal{C}}$ be a $2$-category. If every $2$-morphism in $\operatorname{\mathcal{C}}$ is invertible, then the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.3.2.1). It follows that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the simplicial set $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }( X,Y )$ is a Kan complex. Beware that, in the case where $\operatorname{\mathcal{C}}$ contains non-invertible $2$-morphisms, the simplicial set $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }( X,Y )$ is generally not an $\infty $-category (in fact, it is not even an $(\infty ,2)$-category unless the category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ admits pushouts: see Proposition 8.1.4.1). In such cases, it may be more useful to consider the pinched morphism spaces of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$: see Example 4.6.5.13 and Remark 8.1.8.8.