Kerodon

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

Remark 5.3.2.14. Let $n$ be a nonnegative integer, and suppose we are given a diagram of simplicial sets

\[ X(0) \rightarrow X(1) \rightarrow X(2) \rightarrow \cdots \rightarrow X(n). \]

For each integer $0 \leq i \leq n$, let $\Delta ^ n_{\geq i}$ denote the nerve of the linearly ordered set $\{ i < i+1 < \cdots < n \} $, which we regard as a simplicial subset of $\Delta ^ n$. Applying Remark 5.3.2.12 repeatedly, we can identify the mapping simplex $ \underset { \longrightarrow }{\mathrm{holim}}( X(0) \rightarrow \cdots \rightarrow X(n) )$ with the iterated pushout

\[ (\Delta ^ n \times X(0)) \coprod _{ ( \Delta ^{n}_{\geq 1} \times X(0)) } (\Delta ^{n}_{\geq 1} \times X(1) ) \coprod _{ ( \Delta ^{n}_{\geq 2} \times X(1))} \cdots \coprod _{ (\{ n\} \times X(n-1))} (\{ n\} \times X(n)). \]