# Kerodon

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Proposition 1.2.5.4. Let $S_{\bullet }$ be a simplicial set. Then there exists a category $\operatorname{\mathcal{C}}$ and a map of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{\mathcal{C}}$ as a homotopy category of $S_{\bullet }$.

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 $S_{\bullet }$ can be identified with a realization $| S_{\bullet } |^{Q}$, whose existence is a special case of Proposition 1.1.8.22. Alternatively, we can give a direct construction of the homotopy category $\mathrm{h} \mathit{S}_{\bullet }$:

• The objects of $\mathrm{h} \mathit{S}_{\bullet }$ are the vertices of $S_{\bullet }$.

• Every edge $e$ of $S_{\bullet }$ determines a morphism $[e]$ in $\mathrm{h} \mathit{S}_{\bullet }$, whose source is the vertex $d_1(e)$ and whose target is the vertex $d_0(e)$.

• The collection of morphisms in $\mathrm{h} \mathit{S}_{\bullet }$ is generated under composition by morphisms of the form $[e]$, subject only to the relations

$[ s_0(x) ] = \operatorname{id}_ x \text{ for x \in S_0 } \quad \quad [ d_1(\sigma ) ] = [ d_0(\sigma ) ] \circ [ d_2(\sigma ) ] \text{ for \sigma \in S_2. }$
$\square$