Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 5.2.6.7. Let $\overrightarrow {X}$ be a diagram of simplicial sets

\[ X(0) \xrightarrow {F(1)} X(1) \xrightarrow { F(2) } X(2) \xrightarrow {F(3)} \cdots \xrightarrow {F(n)} X(n), \]

and suppose that the simplicial set $X(0)$ is empty. It follows from Remark 5.2.6.4 that the fiber $\{ 0\} \times _{ \Delta ^ n } M( \overrightarrow {X} )$ is also empty, so the inclusion map

\[ \operatorname{N}_{\bullet }( \{ 1 < 2 < \cdots < n \} ) \times _{ \Delta ^ n } M( \overrightarrow {X} ) \hookrightarrow M( \overrightarrow {X} ) \]

is bijective. If $n > 0$, then we can use Remark 5.2.6.5 to identify $M( \overrightarrow {X} )$ with the mapping simplex of the truncated diagram $X(1) \rightarrow X(2) \rightarrow X(3) \rightarrow \cdots \rightarrow X(n)$.