Kerodon

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

Remark 3.5.2.10. Let $X$ be a simplicial set, which we view as a functor from $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ to the category of sets. Then $X$ admits a canonical extension to a functor $\operatorname{Lin}_{\neq \emptyset }^{\operatorname{op}} \rightarrow \operatorname{Set}$, given on objects by the construction $(I = \{ i_0 < i_1 < \cdots < i_ n \} ) \mapsto X_{n}$. Let us write $X(I)$ for the value of this extension on an object $I \in \operatorname{Lin}_{\neq \emptyset }$. Arguing as in the proof of Theorem 3.5.2.1, we obtain a canonical bijection

\[ \varinjlim _{S} X( [0,1] \setminus S) \simeq \varinjlim _{S} |X|_{S} \xrightarrow {\sim } X, \]

where the (filtered) colimit is taken over the collection of all finite subsets $S \subseteq [0,1]$ containing $0$ and $1$.