$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 4.8.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq 0$ be an integer. The following conditions are equivalent:
- $(1)$
For every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the set $\pi _{n}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), f)$ consists of a single element.
- $(2)$
Every diagram $\operatorname{\partial \Delta }^{n+2} \rightarrow \operatorname{\mathcal{C}}$ can be extended to an $(n+2)$-simplex of $\operatorname{\mathcal{C}}$.
Proof.
By virtue of Proposition 4.8.2.13, we can replace $\operatorname{\mathcal{C}}$ by $\operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}})$ and thereby reduce to the case where the $\infty $-category $\operatorname{\mathcal{C}}$ is $(n+2)$-coskeletal. In this case, the $\infty $-category $\operatorname{\mathcal{C}}$ is locally $n$-truncated (Proposition 4.8.2.8), and satisfies condition $(1)$ if and only if it is locally $(n-1)$-truncated. Applying Corollary 4.8.3.9, we see that $(1)$ is equivalent to the following:
- $(1')$
The tautological map $\operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ is a trivial Kan fibration.
The equivalence of $(1')$ and $(2)$ now follows from Corollary 3.5.4.24.
$\square$