Kerodon

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

Lemma 3.1.2.8. For every pair of integers $0 < i \leq n$, the horn inclusion $f_0: \Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ is a retract of the inclusion map $f: (\Delta ^1 \times \Lambda ^ n_ i) \coprod _{ \{ 1\} \times \Lambda ^{n}_ i} ( \{ 1\} \times \Delta ^ n) \hookrightarrow \Delta ^1 \times \Delta ^ n$.

Proof. Let $A_{}$ denote the simplicial subset of $\Delta ^1 \times \Delta ^ n$ given by the union of $\Delta ^1 \times \Lambda ^ n_ i$ with $\{ 1 \} \times \Delta ^ n$. To prove Lemma 3.1.2.8, it will suffice to show that there exists a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times \Lambda ^{n}_{i} \ar [r] \ar [d]^{f_0} & A_{} \ar [r] \ar [d]^{f} & \Lambda ^{n}_{i} \ar [d]^{f_0} \\ \{ 0\} \times \Delta ^ n \ar [r] & \Delta ^1 \times \Delta ^ n \ar [r]^-{r} & \Delta ^ n } \]

where the left horizontal maps are given by inclusion and the horizontal compositions are the identity maps. To achieve this, it suffices to choose $r$ to be given on vertices by the map of partially ordered sets

\[ r: [1] \times [n] \rightarrow [n] \quad \quad r(j,k) = \begin{cases} i & \text{ if $j=1$ and $k \leq i$ } \\ k & \text{ otherwise. } \end{cases} \]
$\square$