Kerodon

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

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

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