Theorem 6.3.7.1. Let $S$ be a simplicial set. Then there exists a partially ordered set $(A,\leq )$ and a universally localizing morphism $\operatorname{N}_{\bullet }(A) \rightarrow S$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Theorem 6.3.7.1. Let $S$ be a simplicial set. Applying Proposition 6.3.7.14, we can choose a universally localizing morphism $\varphi : \widetilde{S} \rightarrow S$, where $\widetilde{S}$ is a nonsingular simplicial set. Let $A = \operatorname{Sub}_{\operatorname{{\bf \Delta }}}( \widetilde{S} )$ denote the partially ordered set of simplicial subsets of $\widetilde{S}$ which are isomorphic to a standard simplex, so that Corollary 6.3.7.13 supplies a universally localizing morphism $\lambda _{ \widetilde{S} }: \operatorname{N}_{\bullet }(A) \rightarrow \widetilde{S}$. Applying Proposition 6.3.6.10, we deduce that the composite morphism
\[ \operatorname{N}_{\bullet }(A) \xrightarrow { \lambda _{ \widetilde{S} } } \widetilde{S} \xrightarrow { \varphi } S \]
is also universally localizing. $\square$