Kerodon

Example 3.1.2.5. For $0 \leq i \leq n$, the inclusion map $\{ i\} \hookrightarrow \Delta ^ n$ is anodyne. To prove this, let $\operatorname{Spine}[n]$ denote the spine of the $n$-simplex, so that the inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne (Example 1.5.7.7) and therefore anodyne (Example 3.1.2.4). It will therefore suffice to show that the inclusion $\{ i\} \hookrightarrow \operatorname{Spine}[n]$ is anodyne, which is clear (it can be written as a composition of pushouts of the inclusions $\{ 0\} \hookrightarrow \Delta ^1$ and $\{ 1\} \hookrightarrow \Delta ^1$).