Kerodon

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

Warning 10.2.6.12. The terminology of Variant 10.2.6.11 (and Definition 10.2.6.3) is potentially confusing. We will use the term split simplicial object to refer to a simplicial object $X_{\bullet }$ of an $\infty $-category $\operatorname{\mathcal{C}}$ for which there exists a splitting $\overline{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ \mathrm{min}}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$. Unless otherwise specified, we do not assume that a particular splitting has been chosen. Beware that $\overline{X}$ is not uniquely determined by $X_{\bullet }$. However, the underlying augmented simplicial object

\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \xrightarrow { \operatorname{N}_{\bullet }( C_{+}^{\operatorname{op}} ) } \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}} ) \xrightarrow { \overline{X} } \operatorname{\mathcal{C}} \]

is determined up to isomorphism by $X_{\bullet }$: by virtue of Proposition 10.2.6.9, it is an extension of $X_{\bullet }$ to a colimit diagram in $\operatorname{\mathcal{C}}$.