Kerodon

Proof of Theorem 2.3.2.1 from Theorem 2.3.2.5. Let $\operatorname{\mathcal{C}}$ be a $2$-category. If the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }( \operatorname{\mathcal{C}})$ is an $\infty $-category, then every $2$-simplex of $\operatorname{N}^{\operatorname{D}}_{\bullet }( \operatorname{\mathcal{C}})$ is thin (Example 2.3.2.4), so that every $2$-morphism in $\operatorname{\mathcal{C}}$ is invertible by virtue of Theorem 2.3.2.5. Conversely, if $\operatorname{\mathcal{C}}$ is a $(2,1)$-category, then every $2$-simplex of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ is thin (Theorem 2.3.2.5). Consequently, to show that $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ is an $\infty $-category, it will suffice to show that every map of simplicial sets $u_0: \Lambda ^2_1 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ can be extended to a $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$. Note that we can identify $u_0$ with a composable pair of $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ in $\operatorname{\mathcal{C}}$. To extend this to a $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$, it suffices to choose a $1$-morphism $h: X \rightarrow Z$ and a $2$-morphism $\gamma : g \circ f \Rightarrow h$. This is always possible: for example, we can take $h = g \circ f$ and $\gamma $ to be the identity $2$-morphism. $\square$