Kerodon

Example 6.3.1.14. Let $W = \{ \operatorname{id}_{\Delta ^1} \} $ consist of the single nondegenerate edge of the standard $1$-simplex $\Delta ^1$. Then the projection map $\Delta ^1 \rightarrow \Delta ^0$ exhibits $\Delta ^0$ as a localization of $\Delta ^1$ with respect to $W$. To prove this, it will suffice to show that for every $\infty $-category $\operatorname{\mathcal{E}}$, the construction $X \mapsto \operatorname{id}_{X}$ induces an equivalence of $\infty $-categories $\operatorname{\mathcal{E}}= \operatorname{Fun}( \Delta ^0, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \Delta ^1[W^{-1}], \operatorname{\mathcal{E}}) = \operatorname{Isom}(\operatorname{\mathcal{E}})$, which follows from Corollary 4.5.3.13.