Theorem 11.9.8.4. Let $S$ be a simplicial set, and let $W_{S}$ be the collection of all morphisms $([n], \sigma ) \rightarrow ([n'], \sigma ')$ in the category of simplices $\operatorname{{\bf \Delta }}_{S}$ for which the underlying map of linearly ordered sets $\alpha : [n] \rightarrow [n']$ satisfies $\alpha (n) = n'$. Then the last vertex map $\lambda _{S}^{+}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \rightarrow S$ exhibits $S$ as a localization of $\operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{S} )$ with respect to $W_{S}$.
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Proof of Theorem 11.9.8.4. Let $S$ be a simplicial set and let $W_{S}$ be the collection of morphisms in $\operatorname{{\bf \Delta }}_{S} $ appearing in the statement of Theorem 11.9.8.4; we wish to show that the last vertex map $\lambda _{S}^{+}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \rightarrow S$ exhibits $S$ as a localization of $\operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{S} )$ with respect to $W_{S}$. Using Lemma 11.9.8.7, we can realize $\lambda _{S}^{+}$ as a filtered colimit of morphisms $\lambda _{S_{\alpha }}^{+}$ (and $W_{S}$ with a filtered colimit of $W_{S_{\alpha } }$), where $\{ S_{\alpha } \} $ is the collection of all finite simplicial subsets of $S$ (Remark 3.6.1.8). By virtue of Proposition 6.3.4.1, it will suffice to show that each $\lambda _{S_{\alpha }}^{+}$ exhibits $S_{\alpha }$ as a localization of $\operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{S_{\alpha }} )$ with respect to $W_{S_{\alpha }}$. We may therefore replace $S$ by $S_{\alpha }$, and thereby reduce to the case where the simplicial set $S$ is finite.
Since $S$ is a finite simplicial set, it has dimension $\leq n$ for some integer $n \geq -1$. We proceed by induction on $n$. If $n = -1$, then both $S$ and $\operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{S} )$ are empty, and the desired result is clear. We may therefore assume that $n \geq 0$ and that Theorem 11.9.8.4 holds for all finite simplicial sets of dimension $< n$. We now proceed by induction on the number $m$ of nondegenerate $n$-simplices of $S$. If $m = 0$, then $S$ has dimension $\leq n-1$ and the desired result holds by virtue of our first inductive hypothesis. We may therefore assume that $S$ has at least one nondegenerate $n$-simplex $\sigma : \Delta ^ n \rightarrow S$. Using Proposition 1.1.4.12, we see that there is a pushout diagram 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 $\leq n$ with exactly $(m-1)$-nondegenerate $m$-simplices. It follows from our inductive hypotheses together with Lemma 11.9.8.6 that the simplicial sets $\operatorname{\partial \Delta }^ n$, $\Delta ^ n$, and $S'$ satisfy the conclusion of Theorem 11.9.8.4. Moreover, the last vertex map $\lambda _{S}^{+}$ fits into a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ \operatorname{\partial \Delta }^ n } ) \ar [dr]^{ \lambda _{\operatorname{\partial \Delta }^ n}^{+} } \ar [rr] \ar [dd] & & \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\Delta ^ n} ) \ar [dr]^{ \lambda _{\Delta ^ n}^{+} } \ar [dd] & \\ & \operatorname{\partial \Delta }^ n \ar [rr] \ar [dd] & & \Delta ^ n \ar [dd] \\ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S'} ) \ar [dr]^{ \lambda _{S'}^{+} } \ar [rr] & & \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \ar [dr]^{ \lambda _{S}^{+} } & \\ & S' \ar [rr] & & S, } \]
where the front and back faces are pushout squares (Lemma 11.9.8.7) in which the horizontal maps are monomorphisms, hence categorical pushout squares (Example 4.5.4.12). Since $W_{S}$ is the union of the images of $W_{S'}$ and $W_{ \Delta ^ n }$, it follows from Proposition 6.3.4.2 that $\lambda _{S}^{+}$ exhibits $S$ as a localization of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} )$ with respect to $W_{S}$. $\square$