Proposition 1.3.6.4. Let $S = S_{\bullet }$ be a simplicial set. Then there exists a category $\operatorname{\mathcal{C}}$ and a map of simplicial sets $u: S \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{\mathcal{C}}$ as a homotopy category of $S$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $Q^{\bullet }$ denote the cosimplicial object of $\operatorname{Cat}$ given by the inclusion $\operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Cat}$. Unwinding the definitions, we see that a homotopy category of $S$ can be identified with a realization $| S |^{Q}$, whose existence is a special case of Proposition 1.2.3.15. Alternatively, we can give a direct construction of the homotopy category $\mathrm{h} \mathit{S}$:
The objects of $\mathrm{h} \mathit{S}$ are the vertices of $S$.
Every edge $e$ of $S$ determines a morphism $[e]$ in $\mathrm{h} \mathit{S}$, whose source is the vertex $d^{1}_1(e)$ and whose target is the vertex $d^{1}_0(e)$.
The collection of morphisms in $\mathrm{h} \mathit{S}$ is generated under composition by morphisms of the form $[e]$, subject only to the relations
\[ [ s^{0}_0(x) ] = \operatorname{id}_ x \text{ for $x \in S_0$ } \quad \quad [ d^{2}_1(\sigma ) ] = [ d^{2}_0(\sigma ) ] \circ [ d^{2}_2(\sigma ) ] \text{ for $\sigma \in S_2$. } \]