Kerodon

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

Lemma 5.3.4.7. The functor

\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}\quad \quad S \mapsto \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \]

commutes with the formation of colimits.

Proof. Fix an integer $n \geq 0$; we wish to show that the set-valued functor $S \mapsto \operatorname{N}_{n}( \operatorname{{\bf \Delta }}_{S} )$ commutes with colimits. For every $n$-simplex $\tau $ of $\operatorname{N}_{n}(\operatorname{{\bf \Delta }})$, let us write $\operatorname{N}_{n}(\operatorname{{\bf \Delta }}_{S})_{\tau }$ for the inverse image of $\tau $ under the natural map $\operatorname{N}_{n}( \operatorname{{\bf \Delta }}_{S} ) \rightarrow \operatorname{N}_{n}( \operatorname{{\bf \Delta }})$, so that we have a decomposition

\[ \operatorname{N}_{n}( \operatorname{{\bf \Delta }}_{S} ) \simeq \coprod _{\tau \in \operatorname{N}_{n}(\operatorname{{\bf \Delta }})} \operatorname{N}_{n}(\operatorname{{\bf \Delta }}_{S})_{\tau } \]

depending functorially on $S$. It will therefore suffice to show that, for each $\tau $, the functor

\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}\quad \quad S \mapsto \operatorname{N}_{n}(\operatorname{{\bf \Delta }}_{S})_{\tau } \]

commutes with colimits. We now observe that, if $\tau $ corresponds to a diagram $[m_0] \rightarrow [m_1] \rightarrow \cdots \rightarrow [m_ n]$ in the category $\operatorname{{\bf \Delta }}$, then $\operatorname{N}_{n}(\operatorname{{\bf \Delta }}_{S})_{\tau }$ can be identified with the set of $m_ n$-simplices of $S$. $\square$