Kerodon

Proof of Proposition 4.8.3.12. Let $\operatorname{\mathcal{C}}$ be an integer and let $m > 0$ be an integer. It follows immediately from the definitions that, if the restriction map

is injective, then $\operatorname{\mathcal{C}}$ is minimal in dimension $m$. We claim that, if this condition is satisfied, then $\theta _{m+1}$ is surjective: that is, every morphism $\tau _0: \operatorname{\partial \Delta }^{m+1} \rightarrow \operatorname{\mathcal{C}}$ can be extended to an $(m+1)$-simplex of $\operatorname{\mathcal{C}}$. Fix an integer $0 < i < m+1$. Our assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category guarantees that we can choose an $(m+1)$-simplex $\tau $ of $\operatorname{\mathcal{C}}$ satisfying $\tau |_{ \Lambda ^{m+1}_{i} } = \tau _0 |_{ \Lambda ^{m+1}_{i} }$. In particular, $\tau $ and $\tau _0$ have the same restriction to the $(m-1)$-skeleton of $\Delta ^{m+1}$. Invoking the injectivity of $\theta _{m}$, we conclude that $\tau |_{ \operatorname{\partial \Delta }^{m+1} } = \tau _0$.

We now prove the converse. Assume that $\operatorname{\mathcal{C}}$ is minimal in dimension $m$ and that $\theta _{m+1}$ is surjective; we wish to show that $\theta _{m}$ is injective. Let $\sigma _0$ and $\sigma _1$ be $m$-simplices of $\operatorname{\mathcal{C}}$ which have the same restriction to $\operatorname{\partial \Delta }^{m}$; we wish to show that $\sigma _0 = \sigma _1$. Let

be the filtration of Lemma 3.1.2.12, so that $X(0) = (\Delta ^1 \times \operatorname{\partial \Delta }^ m) \cup ( \{ 1\} \times \Delta ^ m)$ and the inclusion map $X(i) \hookrightarrow X(i+1)$ is inner anodyne for $0 \leq i < m$. There is a unique morphism of simplicial sets $h_0: X(0) \rightarrow \operatorname{\mathcal{C}}$ such that $h_0 |_{ \{ 1\} \times \Delta ^ m }$ coincides with $\sigma _1$, and $h_0|_{ \Delta ^1 \times \operatorname{\partial \Delta }^{m} }$ factors through the projection map $\Delta ^1 \times \operatorname{\partial \Delta }^{m} \twoheadrightarrow \operatorname{\partial \Delta }^ m$. Since $\operatorname{\mathcal{C}}$ is an $\infty $-category, we can extend $h_0$ to a diagram $h_{m}: X(m) \rightarrow \operatorname{\mathcal{C}}$. Invoking the surjectivity of $\theta _{m+1}$, we see that $h_{m}$ can be extended to a morphism $h: \Delta ^1 \times \Delta ^ m \rightarrow \operatorname{\mathcal{C}}$ satisfying $h|_{ \{ 0\} \times \Delta ^ m } = \sigma _0$. By construction, $h$ is an isomorphism from $\sigma _0$ to $\sigma _1$ in the $\infty $-category $\operatorname{Fun}( \Delta ^ m, \operatorname{\mathcal{C}})$ whose image in $\operatorname{Fun}( \operatorname{\partial \Delta }^ m, \operatorname{\mathcal{C}})$ is an identity morphism. Since $\operatorname{\mathcal{C}}$ is minimal in dimension $m$, it follows that $\sigma _0 = \sigma _1$. $\square$