Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 3.3.3.15. Let $X$ be a braced simplicial set. Then the morphism $f: X \rightarrow \operatorname{Ex}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ X }^{\mathrm{nd}} ) )$ of Construction 3.3.3.14 exhibits the nerve $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}} )$ as a subdivision of $X$, in the sense of Definition 3.3.3.1.

Proof of Proposition 3.3.3.15. Let $S$ be a semisimplicial set, and let $S^{+}$ denote the braced simplicial set given by Construction 3.3.1.6. Applying Construction 3.3.3.14, we obtain a comparison map of simplicial sets $u_{S}: \operatorname{Sd}( S^{+} ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd} })$. We wish to show that $u_ S$ is an isomorphism for every semisimplicial set $S$. Note that the functor $S \mapsto \operatorname{Sd}( S^{+} )$ preserves colimits (since it is a left adjoint) and the functor $S \mapsto \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd} })$ also preserves colimits (by Lemma 3.3.3.19). 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 $S \simeq ( \Delta ^{n} )^{\mathrm{nd} }$ is the semisimplicial set represented by an object $[n] \in \operatorname{{\bf \Delta }}_{\operatorname{inj}}$. In this case, the desired comparison is immediate from the definition of subdivision (see Examples 3.3.3.5 and 3.3.3.16). $\square$