Kerodon

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

Exercise 1.4.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K_{\bullet } \subseteq \Delta ^1 \times \Delta ^1$ be the simplicial subset appearing in Example 1.4.2.8. Suppose we are given a diagram $\sigma : K_{\bullet } \rightarrow \operatorname{\mathcal{C}}$, which we depict graphically as

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

Composing with the unit map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$, we obtain a diagram $\sigma '$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, which we can depict as

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

Show that the diagram $\sigma '$ is commutative if and only if $\sigma $ can be extended to a map $\overline{\sigma }: \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$. Beware, however, that this extension is generally not unique.