Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 3.4.2.6. Suppose we are given a commutative diagram of simplicial sets

3.46
\begin{equation} \label{diagram:homotopy-pushout-square1} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ C \ar [r] & D, } \end{gathered} \end{equation}

where $f$ is a monomorphism. Then (3.46) is a homotopy pushout square if and only if the induced map $C \coprod _{A} B \rightarrow D$ is a weak homotopy equivalence.

Proof. For every Kan complex $X$, we obtain a commutative diagram of simplicial sets

3.47
\begin{equation} \label{diagram:pushout-lemma2} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,X) & \operatorname{Fun}(B,X) \ar [l]_{u} \\ \operatorname{Fun}(C,X) \ar [u] & \operatorname{Fun}(D,X), \ar [l] \ar [u] } \end{gathered} \end{equation}

where $u$ is a Kan fibration (Corollary 3.1.3.3). It follows that the diagram (3.47) is a homotopy pullback square if and only if the induced map

\[ \operatorname{Fun}(D, X) \rightarrow \operatorname{Fun}(C,X) \times _{ \operatorname{Fun}(A,X)} \operatorname{Fun}(B,X) \simeq \operatorname{Fun}( C \coprod _{A} B, X) \]

is a weak homotopy equivalence (Example 3.4.1.5). Consequently, the diagram (3.46) is a homotopy pushout square if and only if, for every Kan complex $X$, the induced map $\operatorname{Fun}(D, X) \rightarrow \operatorname{Fun}( C \coprod _{A} B, X)$ is a homotopy equivalence of Kan complexes. By virtue of Corollary 3.1.7.5), this is equivalent to the requirement that the morphism $C \coprod _{A} B \rightarrow D$ is a weak homotopy equivalence. $\square$