$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 3.5.4.24. Let $X$ be a simplicial set and let $n \geq -2$ be an integer. The following conditions are equivalent:
- $(1)$
The comparison map $f: \operatorname{cosk}_{n+2}(X) \rightarrow \operatorname{cosk}_{n}^{\circ }(X)$ is a trivial Kan fibration.
- $(2)$
Every morphism $\operatorname{\partial \Delta }^{n+2} \rightarrow X$ can be extended to an $(n+2)$-simplex of $X$.
- $(3)$
The weak $(n+1)$-coskeleton $\operatorname{cosk}_{n+1}^{\circ }(X)$ coincides with $\operatorname{cosk}_{n+1}(X)$.
Proof.
The equivalence of $(2)$ and $(3)$ follows from Remark 3.5.4.21. The implication $(3) \Rightarrow (1)$ follows from the observation that $f$ factors as a composition
\[ \operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}}) \twoheadrightarrow \operatorname{cosk}_{n+1}^{\circ }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{cosk}_{n+1}(\operatorname{\mathcal{C}}) \twoheadrightarrow \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}), \]
where the outer maps are trivial Kan fibrations (Proposition 3.5.4.22). We will complete the proof by showing that $(2)$ implies $(3)$. Suppose we are given a morphism $\sigma _0: \operatorname{\partial \Delta }^{n+2} \rightarrow \operatorname{\mathcal{C}}$. Since $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ is $(n+1)$-coskeletal we can extend $F \circ \sigma _0$ to an $(n+2)$-simplex $\overline{\sigma }$ of $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$. If condition $(2)$ is satisfied, then the lifting problem
\[ \xymatrix { \operatorname{\partial \Delta }^{n+2} \ar [r]^{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{n+2} \ar@ {-->}[ur] \ar [r]^{ \overline{\sigma } } & \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}) } \]
admits a solution; in particular, $\sigma _0$ can be extended to an $(n+2)$-simplex of $\operatorname{\mathcal{C}}$.
$\square$