$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 6.3.2.7. Let $Q$ denote the quotient of the simplicial set $\Delta ^3$ obtained by collapsing the edges $e = \operatorname{N}_{\bullet }( \{ 0 < 2 \} )$ and $e' = \operatorname{N}_{\bullet }( \{ 1 < 3 \} )$, so that we have a pushout diagram
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \amalg \operatorname{N}_{\bullet }( \{ 1 < 3 \} ) \ar [r] \ar [d] & \Delta ^3 \ar [d] \\ \Delta ^0 \amalg \Delta ^0 \ar [r] & Q. } \]
Then the projection map $Q \rightarrow \Delta ^0$ is a categorical equivalence of simplicial sets.
Proof.
It follows from Corollary 6.3.2.5 that the quotient map $\Delta ^3 \twoheadrightarrow Q$ exhibits $Q$ as a localization of $\Delta ^3$ with respect to $W = \{ e, e' \} $. It will therefore suffice to show that the projection map $q: \Delta ^3 \rightarrow \Delta ^0$ has the same property (Corollary 6.3.1.20). Note that $q$ is a weak homotopy equivalence, and therefore exhibits $\Delta ^0$ as a localization of $\Delta ^3$ with respect to the collection $W'$ of all edges of $\Delta ^3$. To complete the proof, it will suffice to show that these localizations are the same: that is, a functor of $\infty $-categories $F: \Delta ^3 \rightarrow \operatorname{\mathcal{C}}$ carries each edge of $W$ to an isomorphism if and only if it carries each edge of $W'$ to an isomorphism. This is a restatement of the two-out-of-six property for isomorphisms (Proposition 5.4.6.5).
$\square$