Kerodon

Exercise 11.9.8.8. Let $S$ be a simplicial set and let $\operatorname{Sd}(S)$ denote the subdivision of $S$. Let $W$ be the collection of all edges $w$ of $\operatorname{Sd}(S)$ with the following property: there exists a simplex $\sigma : \Delta ^ n \rightarrow S$ such that $w = \operatorname{Sd}(\sigma )(\overline{w})$, where $\overline{w}$ is an edge of the subdivision $\operatorname{Sd}( \Delta ^ n ) \simeq \operatorname{N}_{\bullet }( \operatorname{Chain}[n] )$ corresponding to a pair of subsets $\emptyset \neq P \subseteq Q \subseteq [n]$ satisfying $\mathrm{max}(P) = \mathrm{max}(Q)$. Show that the last vertex map $\lambda _{S}: \operatorname{Sd}(S) \rightarrow S$ of Construction 3.3.4.3 exhibits $S$ as a localization of $\operatorname{Sd}(S)$ with respect to $W$.