Proposition 3.3.3.17. Let $X$ be a braced simplicial set. Then the comparison map $\psi _{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} ) \rightarrow \operatorname{Sd}(X)$ of Construction 3.3.3.10 restricts to an isomorphism $\psi _{X}^{\circ }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}) \xrightarrow {\sim } \operatorname{Sd}(X)$.
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Proof of Proposition 3.3.3.17. Let $X$ be a braced simplicial set. By virtue of Corollary 3.3.1.8, we can assume that $X = S^{+}$ for some semisimplicial set $S$. Let $\varphi _{S}$ denote the composite map
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}} ) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} ) \xrightarrow { \psi _{X} } \operatorname{Sd}(X) = \operatorname{Sd}(S^{+}); \]
we wish to show that $\varphi _{S}$ is an isomorphism. By virtue of Lemma 3.3.3.19, the construction $S \mapsto \varphi _{S}$ commutes with small colimits. Since every functor $S: \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ can be written as a colimit of representable functors (see ยง
), we may assume without loss of generality that $S$ is the semisimplicial set represented by an object $[n] \in \operatorname{{\bf \Delta }}_{\operatorname{inj}}$; that is, $X = \Delta ^{n}$ is the standard simplex. In this case, the conclusion follows immediately from the concrete description of $\psi _{X}$ given in Example 3.3.3.11.
$\square$