Kerodon

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

Example 1.4.2.8 (Commutative Squares in a Category). Let $K_{\bullet } = \operatorname{\partial }( \Delta ^1 \times \Delta ^1 )$ be as in Example 1.4.2.4. For any ordinary category $\operatorname{\mathcal{C}}$, we can display a diagram $\sigma : K_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ pictorially as

\[ \xymatrix { C_{00} \ar [r]^{f} \ar [d]^{g} & C_{01} \ar [d]^{g'} \\ C_{10} \ar [r]^{ f'} & C_{11}. } \]

The diagram $\sigma $ is commutative if and only if we have $g' \circ f = f' \circ g$ in $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_{00}, C_{11} )$. In this case, Proposition 1.4.2.6 ensures that $\sigma $ extends uniquely to a diagram $\overline{\sigma }: \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, or equivalently to a functor of ordinary categories $[1] \times [1] \rightarrow \operatorname{\mathcal{C}}$.