Kerodon

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Lemma 4.3.7.8. Let $f: A \hookrightarrow B$ be a monomorphism of simplicial sets. Then the inclusion $f^{\triangleright }: A^{\triangleright } \hookrightarrow B^{\triangleright }$ is right anodyne, and the inclusion $A^{\triangleleft } \hookrightarrow B^{\triangleleft }$ is left anodyne.

Proof. We will prove the first assertion (the second follows by a similar argument). Let $T$ be the collection of all morphisms $f$ of simplicial sets for which $f^{\triangleright }$ is right anodyne. We wish to show that every monomorphism belongs to $T$. Since the collection $T$ is weakly saturated, it will suffice to show that every horn inclusion $f: \operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n$ belongs to $T$ (Proposition 1.4.5.13). In this case, we can identify $f^{\triangleright }$ with the with the horn inclusion $\Lambda ^{n+1}_{n+1} \hookrightarrow \Delta ^{n+1}$ (see Example 4.3.3.27). $\square$