Kerodon

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

Example 5.3.4.3. Let $n$ be a nonnegative integer and let $\operatorname{\mathcal{E}}$ denote the nerve of the partially ordered set $Q = \{ (i,j) \in [n] \times [n]: j \leq i \} $. Then there is a cocartesian fibration of $\infty $-categories $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^ n$, given on vertices by the formula $U(i,j) = i$. Let $\mathscr {F}: [n] \rightarrow \operatorname{Set_{\Delta }}$ denote the functor given by $\mathscr {F}(i) = \Delta ^ i$, so that vertices of the homotopy colimit can be identified with elements of $Q$. There is a unique morphism of simplicial sets $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ which is the identity at the level of vertices, which is a scaffold of the cocartesian fibration $U$. Moreover, $\lambda $ is monomorphism, and an $n$-simplex $(i_0, j_0) \leq (i_1, j_1) \leq \cdots \leq (i_ n, j_ n)$ belongs to the image of $\lambda $ if and only if $j_ n \leq i_0$. The case $n=3$ is depicted in the following diagram, where the image of $\lambda $ is indicated with solid arrows:

\[ \xymatrix@C =30pt@R=30pt{ & & & (3,3) \\ & & (2,2) \ar@ {-->}[ur] \ar [r] & (3,2) \ar [u] \\ & (1,1) \ar@ {-->}[ur] \ar [r] & (2,1) \ar [u] \ar [r] & (3,1) \ar [u] \\ (0,0) \ar@ {-->}[ur] \ar [r] & (1,0) \ar [r] \ar [u] & (2,0) \ar [u] \ar [r] & (3,0). \ar [u] } \]