Example 3.2.4.20. Let $n$ be a positive integer. For $0 \leq i \leq n$, the horn $\Lambda ^{n}_{i}$ is weakly contractible. This follows from Remark 3.2.4.18, since the inclusion map $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ is a weak homotopy equivalence (Proposition 3.1.6.14) and the simplex $\Delta ^ n$ is contractible (Example 3.2.4.2).
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