Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 5.3.4.6. For each nonnegative integer $n$, the last vertex map

\[ \lambda ^{+}_{\Delta ^ n}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ \Delta ^{n} } ) \rightarrow \Delta ^ n \]

exhibits $\Delta ^ n$ as a localization of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\Delta ^{n} } )$ with respect to the collection of morphisms $W_{\Delta ^ n}$ appearing in the statement of Theorem 5.3.4.4.

Proof. Note that we can identify $\lambda _{\Delta ^ n}^{+}$ with a functor of ordinary categories $\operatorname{{\bf \Delta }}_{ \Delta ^ n } \rightarrow [n]$, which carries each simplex $\sigma : \Delta ^ m \rightarrow \Delta ^ n$ to its last vertex $\sigma (m)$. This functor has a right adjoint $G: [n] \rightarrow \operatorname{{\bf \Delta }}_{\Delta ^ n}$, which carries each element $m \in [n]$ to the map $\sigma : \Delta ^ m \rightarrow \Delta ^ n$ which is the identity on vertices. We observe that the composition $\lambda _{\Delta ^ n}^{+} \circ G$ is equal to the identity, and the unit transformation $u: \operatorname{id}_{\operatorname{{\bf \Delta }}_{\Delta ^ n}} \rightarrow G \circ \lambda _{\Delta ^ n}^{+}$ carries each object of $\operatorname{{\bf \Delta }}_{\Delta ^ n}$ to a morphism which belongs to $W_{ \Delta ^ n}$. Applying Proposition 5.3.4.5, we conclude that $\lambda _{\Delta ^ n}^{+}$ exhibits $\Delta ^ n$ as a localization of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\Delta ^{n} } )$ with respect to $W_{\Delta ^ n}$. $\square$