Proposition 6.3.7.2. Let $S$ be a simplicial set, let $\operatorname{Sd}(S)$ denote the subdivision of $S$ (Definition 3.3.3.1), and let $\lambda _{S}: \operatorname{Sd}(S) \rightarrow S$ denote the last vertex map (Construction 3.3.4.3). Then $\lambda _{S}$ is universally localizing.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Proposition 6.3.7.2. By virtue of Proposition 6.3.6.12, we may assume without loss of generality that the simplicial set $S$ is finite. If $S$ is empty, there is nothing to prove. We may therefore assume that $S$ has dimension $n$ for some integer $n \geq 0$. We proceed by induction on $n$ and on the number of nondegenerate $n$-simplices of $S$. Fix a nondegenerate $n$-simplex $\sigma : \Delta ^ n \rightarrow S$. Using Proposition 1.1.4.12, we see that there is a pushout square of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & \Delta ^ n \ar [d]^{\sigma } \\ S' \ar [r] & S, } \]
where $S'$ is a simplicial set of dimension $\leq n$ having fewer nondegenerate $n$-simplices than $S$. Applying Proposition 6.3.6.13 to the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Sd}( \operatorname{\partial \Delta }^ n) \ar [dr]^{ \lambda _{\operatorname{\partial \Delta }^ n} } \ar [rr] \ar [dd] & & \operatorname{Sd}(\Delta ^ n) \ar [dr]^{ \lambda _{\Delta ^ n}} \ar [dd] & \\ & \operatorname{\partial \Delta }^ n \ar [rr] \ar [dd] & & \Delta ^ n \ar [dd] \\ \operatorname{Sd}(S') \ar [dr]^{ \lambda _{S'} } \ar [rr] & & \operatorname{Sd}(S) \ar [dr]^{ \lambda _{S} } & \\ & S' \ar [rr] & & S, } \]
we are reduced to showing that the morphisms $\lambda _{S'}$, $\lambda _{\operatorname{\partial \Delta }^{n}}$, and $\lambda _{\Delta ^{n}}$ are universally localizing. In the first two cases, this follows from our inductive hypothesis. We are therefore reduced to proving Proposition 6.3.7.2 in the special case where $S = \Delta ^ n$ is a standard simplex. Using Example 3.3.3.5, we can identify the subdivision $\operatorname{Sd}(S) = \operatorname{Sd}( \Delta ^ n )$ with the nerve of the partially ordered set $\operatorname{Chain}[n]$ of nonempty subsets $P \subseteq [n]$. Under this identification, $\lambda _{S}$ is obtained from the map of partially ordered sets
\[ \operatorname{Chain}[n] \rightarrow [n] \quad \quad (S \subseteq [n] ) \mapsto \mathrm{max}(S). \]
We now observe that this map admits section $\mu : [n] \rightarrow \operatorname{Chain}[n]$, given by the construction $i \mapsto \{ 0 < 1 < \cdots < i \} $, and there is a (unique) natural transformation $\operatorname{id}_{ \operatorname{Sd}(S) } \rightarrow \mu \circ \lambda _{S}$ which belongs to $\operatorname{Fun}_{/S}( \operatorname{Sd}(S), \operatorname{Sd}(S) )$. The desired result now follows from the criterion of Proposition 6.3.6.8. $\square$