Kerodon

Example 4.3.3.28. Let $n \geq 0$, and let $\Delta ^{n}$ denote the standard $n$-simplex. Using Example 4.3.3.27, we see that there is a unique isomorphism of simplicial sets $(\Delta ^{n})^{\triangleright } \simeq \Delta ^{n+1}$, which carries each vertex $i \in \{ 0, 1, \ldots , n \} $ to itself and the cone point of $( \Delta ^{n} )^{\triangleright }$ to the final vertex $n+1$. This isomorphism carries the simplicial subset $(\operatorname{\partial \Delta }^{n})^{\triangleright } \subseteq ( \Delta ^{n} )^{\triangleright }$ to the horn $\Lambda ^{n+1}_{n+1} \subseteq \Delta ^{n+1}$. Similarly, the left cone $( \operatorname{\partial \Delta }^{n} )^{\triangleleft }$ is isomorphic to the horn $\Lambda ^{n+1}_{0}$.