Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$

Corollary 5.1.3.8. Let $q: Y \rightarrow S$ be a morphism of simplicial sets, and suppose we are given a pushout diagram

$\xymatrix@R =50pt@C=50pt{ A \ar [r]^-{i} \ar [d] & A' \ar [d] \\ B \ar [r] & B' }$

in the slice category $(\operatorname{Set_{\Delta }})_{/S}$. If $i$ is a monomorphism and $q$ is either a left or right fibration, then the induced diagram

5.4
$$\begin{gathered}\label{equation:mapping-space-homotopy-pullback} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{/S}(A', Y) & \operatorname{Fun}_{/S}(A, Y) \ar [l] \\ \operatorname{Fun}_{/S}(B',Y) \ar [u] & \operatorname{Fun}_{/S}(B, Y) \ar [u] \ar [l] } \end{gathered}$$

is a homotopy pullback square of Kan complexes.

Proof. Diagram (5.4) is automatically a pullback square (Remark 5.1.3.7). If $q$ is a right fibration and $i$ is a monomorphism, then the horizontal maps in (5.4) are Kan fibrations (Proposition 5.1.3.5). Invoking Example 3.4.1.5, we deduce that $(\ref{equation:mapping-space-homotopy-pullback})$ is also a homotopy pullback square. $\square$