$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.2.3.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\overline{f}: A^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a diagram, where $A$ is a weakly contractible simplicial set. The following conditions are equivalent:
- $(1)$
The diagram $\overline{f}$ carries each edge of $A^{\triangleleft }$ to an isomorphism in $\operatorname{\mathcal{C}}$.
- $(2)$
The restriction $f = \overline{f}|_{A}$ carries each edge of $A$ to an isomorphism in $\operatorname{\mathcal{C}}$, and $\overline{f}$ is a limit diagram.
Proof.
Without loss of generality, we may assume that $f$ carries each edge of $A$ to an isomorphism in $\operatorname{\mathcal{C}}$. Under this assumption, we can restate $(1)$ and $(2)$ as follows:
- $(1')$
For every vertex $a \in A$, the edge
\[ \Delta ^1 \simeq \{ a\} ^{\triangleleft } \hookrightarrow A^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}} \]
is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.
- $(2')$
The morphism $\overline{f}$ is a limit diaram.
Using Corollary 3.1.7.2, we can choose an anodyne morphism $i: A \hookrightarrow B$, where $B$ is a Kan complex. Note that $f$ can be regarded as a morphism from $A$ to the core $\operatorname{\mathcal{C}}^{\simeq }$, which is also a Kan complex (Corollary 4.4.3.11). We can therefore extend $f$ to a morphism of Kan complexes $g: B \rightarrow \operatorname{\mathcal{C}}^{\simeq }$. Moreover, the morphism $i$ is left cofinal (Proposition 7.2.1.5) and therefore left anodyne (Proposition 7.2.1.3). It follows that the induced map $B {\coprod }_{A} A^{\triangleleft } \hookrightarrow B^{\triangleleft }$ is inner anodyne (Example 4.3.6.5), so that we can choose a functor $\overline{g}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{g}|_{B} = g$ and $\overline{g}|_{A^{\triangleleft }} = \overline{f}$.
It follows from Corollary 7.2.2.3 that $\overline{f}$ is a limit diagram if and only if $\overline{g}$ is a limit diagram. Since $A$ is weakly contractible, the Kan complex $B$ is contractible. In particular, every vertex $a \in A$ can be regarded as a final object of $B$. The equivalence of $(1')$ and $(2')$ now follows from Corollary 7.2.2.6.
$\square$