Kerodon

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

Example 1.4.2.4. Let $\operatorname{\mathcal{C}}$ be a category. For each $n \geq 0$, we can identify $n$-simplices $\sigma $ of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ with diagrams

\[ C_0 \xrightarrow {f_1} C_1 \xrightarrow {f_2} \cdots \xrightarrow {f_{n-1}} C_{n-1} \xrightarrow {f_ n} C_ n \]

in the category $\operatorname{\mathcal{C}}$. Then $\sigma $ determines an $n$-simplex $\sigma '$ of $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{\operatorname{op}} )$, given by the diagram

\[ C_ n \xrightarrow {f_ n} C_{n-1} \xrightarrow { f_{n-1} } \cdots \xrightarrow {f_2} C_1 \xrightarrow {f_1} C_0 \]

in the opposite category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. The construction $\sigma \mapsto \sigma '$ determines an isomorphism of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\operatorname{op}} \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}^{\operatorname{op}} )$.