Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 4.6.7.26. Let $\operatorname{\mathcal{C}}$ be a simplicial category containing a pair of objects $X$ and $Y$, and assume that the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. Then the simplicial set $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X,Y)$ is also an $\infty $-category.

Proof. Let $i: A \hookrightarrow B$ be an inner anodyne morphism of simplicial sets. We wish to show that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d]^{i} & \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X,Y) \ar [d] \\ B \ar [r] \ar@ {-->}[ur] & \Delta ^0 } \]

admits a solution. By virtue of Corollary 4.6.7.17, we can rephrase this as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Phi (A) \ar [r] \ar [d]^{\Phi (i)} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \ar [d] \\ \Phi (B) \ar [r] \ar@ {-->}[ur] & \Delta ^0. } \]

Note that $\Phi (i)$ is a monomorphism (Remark 4.6.7.15) and a categorical equivalence (Corollary 4.6.7.25), so the desired result follows from Lemma 4.5.5.2. $\square$