Lemma 11.9.8.6. For each nonnegative integer $n$, the last vertex map
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 11.9.8.4.
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 $\sigma (m)$. Let $G: [n] \rightarrow \operatorname{{\bf \Delta }}_{\Delta ^ n}$, be the functor which carries each element $m \in [n]$ to the map $\sigma : \Delta ^ m \rightarrow \Delta ^ n$ which is the identity on vertices. Note that the identity transformation $\lambda _{\Delta ^{n}}^{+} \circ G \xrightarrow {\sim } \operatorname{id}_{[n]}$ is the counit of an adjunction, so that $\lambda _{ \Delta ^{n} }$ is a reflective localization functor. The desired result is now a special case of Proposition 11.6.0.48. $\square$