Proposition 10.2.6.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X/Y)$. Then the augmented simplicial object $\operatorname{\check{C}}_{\bullet }(X/Y)$ splits if and only if $f$ admits a right homotopy inverse.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Proposition 10.2.6.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X/Y)$. Applying Proposition 10.2.6.21 (in the case $n=0$), we see that precomposition with the inclusion map
\[ \operatorname{Ret}^{\operatorname{op}} \simeq \operatorname{Ret}\simeq \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq 1} \hookrightarrow \operatorname{{\bf \Delta }}_{\mathrm{min}} \]
of Example 10.2.6.18 induces a trivial Kan fibration
\[ \{ \operatorname{\check{C}}_{\bullet }(X/Y) \} \times _{ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}}) } \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}}), \operatorname{\mathcal{C}}) \rightarrow \{ f \} \times _{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) } \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{Ret}), \operatorname{\mathcal{C}}). \]
In particular, the left hand side is nonempty if and only if the right hand side is nonempty: that is, the Čechnerve $\operatorname{\check{C}}_{\bullet }(X/Y)$ splits if and only if $f$ has a right homotopy inverse. $\square$