Kerodon

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

Proposition 1.1.8.20. Let $\operatorname{\mathcal{C}}$ be a category, let $Q^{\bullet }$ be a cosimplicial object of $\operatorname{\mathcal{C}}$, and let $\operatorname{Sing}^{Q}_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be the functor of Variant 1.1.7.6. If the category $\operatorname{\mathcal{C}}$ admits small colimits, then the functor $\operatorname{Sing}^{Q}_{\bullet }$ admits a left adjoint $\operatorname{Set_{\Delta }}\rightarrow \operatorname{\mathcal{C}}$, which we will denote by $S_{\bullet } \mapsto | S_{\bullet } |^{Q}$.

Proof. Let us say that a simplicial set $S_{\bullet }$ is good if the functor

\[ (C \in \operatorname{\mathcal{C}}) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{Sing}^{Q}_{\bullet }(C) ) \]

is corepresentable by an object of the category $\operatorname{\mathcal{C}}$ (in which case we denote the corepresenting object by $|S_{\bullet }|^{Q}$). It follows from Yoneda's lemma that the standard $n$-simplex $\Delta ^ n$ is good for each $n \geq 0$, with $| \Delta ^{n} |^{Q} \simeq Q^{n}$. If $\operatorname{\mathcal{C}}$ admits small colimits, then the proof of Lemma 1.1.8.6 shows that the collection of good simplicial sets is closed under small colimits. It now suffices to observe that every simplicial set $S_{\bullet }$ can be written as a small colimit of simplices (Lemma 1.1.8.17). $\square$