Corollary 4.5.4.15 (01FL)

Corollary 4.5.4.15. Let $i: A \rightarrow B$ and $i': A' \rightarrow B'$ be morphisms of simplicial sets. Assume that $i$ is a monomorphism and that either $i$ or $i'$ is a categorical equivalence. Then the induced map

\[ (A \times B') \coprod _{ (A \times A') } (B \times A') \rightarrow B \times B' \]

is a categorical equivalence.

Proof. By virtue of Proposition 4.5.4.11, it will suffice to show that the diagram

4.39

\begin{equation} \begin{gathered}\label{diagram:categorical-pushout-corollary} \xymatrix@R =50pt@C=50pt{ A \times A' \ar [r] \ar [d] & B \times A' \ar [d] \\ A \times B' \ar [r] & B \times B' } \end{gathered} \end{equation}

is a categorical pushout square. This follows from the criterion of Proposition 4.5.4.10: if $i$ is a categorical equivalence, then the horizontal maps in the diagram (4.39) are categorical equivalences (Remark 4.5.3.7). Similarly, if $i'$ is a categorical equivalence, then the vertical maps in the diagram (4.39) are categorical equivalences. $\square$