Kerodon

Remark 1.2.4.5. Let $0 \leq i \leq n$ be integers with $n > 0$. Then the horn $\Lambda ^{n}_{i}$ is connected. If $n = 1$ or $n=2$, this follows by inspection (see Examples 1.2.4.3 and 1.2.4.4). For $n \geq 3$, the inclusion map $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ is bijective on simplices of dimension $\leq 1$, so the desired result follows from Proposition 1.2.1.22 (together with the connectedness of the standard simplex $\Delta ^{n}$; see Example 1.2.1.7).