Remark 7.6.6.3 (03HK)

Remark 7.6.6.3. Let $\operatorname{Spine}[ \operatorname{\mathbf{Z}}_{\geq 0} ]$ denote the simplicial subset of $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$ whose $k$-simplices are sequences of nonnegative integers $(n_0, n_1, \cdots , n_ k )$ satisfying $n_0 \leq n_1 \leq \cdots \leq n_ k \leq n_0 + 1$. Then $\operatorname{Spine}[ \operatorname{\mathbf{Z}}_{\geq 0} ]$ is a $1$-dimensional simplicial set, which corresponds (under the equivalence of Proposition 1.1.6.9) to the directed graph $G$ indicated in the diagram

\[ 0 \rightarrow 1 \rightarrow 2 \rightarrow 3 \rightarrow \cdots . \]

Moreover, the linearly ordered set $(\operatorname{\mathbf{Z}}_{\geq 0}, \leq )$ can be identified with the path category $\operatorname{Path}[G]$ of Construction 1.3.7.1. It follows that the inclusion map $\operatorname{Spine}[ \operatorname{\mathbf{Z}}_{\geq 0} ] \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$ is inner anodyne (Proposition 1.5.7.3).

In particular, for any $\infty $-category $\operatorname{\mathcal{C}}$, the restriction map

\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{Spine}[ \operatorname{\mathbf{Z}}_{\geq 0} ], \operatorname{\mathcal{C}}) \]

is a trivial Kan fibration (Theorem 1.5.7.1). Stated more informally, every sequence of composable morphisms

\[ X(0) \xrightarrow { f_0 } X(1) \xrightarrow { f_1} X(2) \xrightarrow { f_2} X(3) \xrightarrow { f_3} X(4) \rightarrow \cdots \]

admits an essentially unique extension to a functor $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{C}}$.