$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 6.3.2.8. Let $Q$ be the quotient of $\Delta ^3$ described in Corollary 6.3.2.7. Then the morphism
\[ e: \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \hookrightarrow \Delta ^3 \twoheadrightarrow Q \]
exhibits $Q$ as a localization of $\Delta ^1$ with respect to $\{ \operatorname{id}_{\Delta ^1} \} $.
Proof.
By virtue of Corollary 6.3.2.7, this is equivalent to the statement that the projection map $\Delta ^1 \twoheadrightarrow \Delta ^0$ exhibits $\Delta ^0$ as a localization of $\Delta ^1$ with respect to $\{ \operatorname{id}_{\Delta ^1} \} $ (Remark 6.3.1.20), which follows from Example 6.3.1.14.
$\square$