Example 7.5.6.5 (Projectively Cofibrant Squares). A commutative diagram of simplicial sets
7.58
\begin{equation} \begin{gathered}\label{equation:projectively-cofibrant-squares} \xymatrix@C =50pt@R=50pt{ A \ar [r]^-{f_0} \ar [d]^{f_1} & A_0 \ar [d]^{f_1} \\ A_1 \ar [r]^-{f_0} & A_{01} } \end{gathered} \end{equation}
is projectively cofibrant (when regarded as a functor $[1] \times [1] \rightarrow \operatorname{Set_{\Delta }}$) if and only if the morphisms
\[ f_0: A \rightarrow A_0 \quad \quad f_1: A \rightarrow A_1 \quad \quad (f'_1, f'_0): A_0 {\coprod }_{A} A_1 \rightarrow A_{01} \]
are monomorphisms of simplicial sets. Equivalently, (7.58) is projectively cofibrant if it is a pullback square consisting of monomorphisms.