Kerodon

Proof. We first show that $(a_ n)$ implies $(b_ n)$. Fix a vertex $x \in X$ and an $(n-1)$-simplex $\sigma : \Delta ^{n-1} \rightarrow X$ such that $\sigma |_{ \operatorname{\partial \Delta }^{n-1}}$ is the constant map taking the value $x$. Amalgamating $\sigma $ with the constant map $\Lambda ^{n}_{n} \rightarrow X$, we obtain a morphism $\tau : \operatorname{\partial \Delta }^{n} \rightarrow X$. If condition $(1)$ is satisfied, then $\tau $ can be extended to an $n$-simplex of $X$. Theorem 3.2.2.10 then guarantees that the (pointed) homotopy class $[\sigma ]$ is the identity element of the group $\pi _{n-1}(X,x)$.

We now show that $(b_ n)$ implies $(a_ n)$. Let $\tau : \operatorname{\partial \Delta }^{n} \rightarrow X$ be any morphism of simplicial sets. Using Lemma 3.2.4.11, we see that the restriction $\tau |_{ \Lambda ^{n}_{n} }$ is nullhomotopic. Applying the homotopy lifting property (Remark 3.1.5.3), we conclude that $\tau $ is homotopic to a morphism $\tau ': \operatorname{\partial \Delta }^{n} \rightarrow X$ for which $\tau '|_{ \Lambda ^{n}_{n} }$ is the constant map taking the value $x$, for some vertex $x \in X$. In particular, $\tau '$ is constant when restricted to the $(n-2)$-skeleton of $\Delta ^{n}$. If the homotopy group $\pi _{n-1}(X,x)$ is trivial, then Theorem 3.2.2.10 guarantees that $\tau '$ can be extended to an $n$-simplex of $X$. Applying Remark 3.1.5.3 again, we conclude that $\tau $ can also be extended to an $n$-simplex of $X$. $\square$