Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 1.4.7.7 (The Spine of a Simplex). Let $n \geq 0$ and let $\Delta ^{n}$ be the standard $n$-simplex (Construction 1.1.2.1). We let $\operatorname{Spine}[n]$ denote the simplicial subset of $\Delta ^{n}$ whose $k$-simplices are monotone maps $\sigma : [k] \rightarrow [n]$ satisfying $\sigma (k) \leq \sigma (0) + 1$. We will refer to $\operatorname{Spine}[n]$ as the spine of the simplex $\Delta ^{n}$. More informally, it is comprised of all vertices of $\Delta ^{n}$, together with those edges which join adjacent vertices. The spine $\operatorname{Spine}[n]$ is a simplicial set of dimension $\leq 1$, which we can identify with the directed graph $G$ depicted in the diagram

\[ \xymatrix { 0 \ar [r] & 1 \ar [r] & 2 \ar [r] & \cdots \ar [r] & n. } \]

Under this identification, the map $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ corresponds to the inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ (see Example 1.2.6.2). Invoking Proposition 1.4.7.3 and Theorem 1.4.7.1, we obtain the following:

$(a)$

The inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne.

$(b)$

For any $\infty $-category $\operatorname{\mathcal{C}}$, the restriction map $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{Spine}[n], \operatorname{\mathcal{C}})$ is a trivial Kan fibration.