Kerodon

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

Variant 1.1.7.6. Let $\operatorname{\mathcal{C}}$ be any category and let $Q^{\bullet }$ be a cosimplicial object of $\operatorname{\mathcal{C}}$, which we view as a functor $Q: \operatorname{{\bf \Delta }}\rightarrow \operatorname{\mathcal{C}}$. For every object $X \in \operatorname{\mathcal{C}}$, the construction $( [n] \in \operatorname{{\bf \Delta }}) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Q( [n] ), X )$ determines a functor from $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ to the category of sets, which we can view as a simplicial set. We will denote this simplicial set by $\operatorname{Sing}^{Q}_{\bullet }(X)$, so that we have canonical bijections $\operatorname{Sing}^{Q}_{n}(X) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Q^{n}, X)$. We view the construction $X \mapsto \operatorname{Sing}^{Q}_{\bullet }(X)$ as a functor from the category $\operatorname{\mathcal{C}}$ to the category of simplicial sets, which we denote by $\operatorname{Sing}^{Q}_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$.