Kerodon

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

Remark 4.6.1.20. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, and let $e: q(X) \rightarrow q(Y) )$ be a morphism in $\operatorname{\mathcal{D}}$. By construction, we have a pullback diagram of simplicial sets

4.53
\begin{equation} \begin{gathered}\label{equation:homotopy-pullback-square-of-morphism-spaces} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e} \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [d] \\ \{ e\} \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( q(X), q(Y) ). } \end{gathered} \end{equation}

It follows from Proposition 4.6.1.19 that the vertical maps in this diagram are Kan fibrations, so that (4.53) is also a homotopy pullback square. Stated more informally, we have a homotopy fiber sequence

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( q(X), q(Y) ). \]