Definition 7.7.5.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ satisfies the first Mather cube theorem if, for every cubical diagram
7.88
\begin{equation} \label{equation:mather-cube-general} \begin{gathered} \xymatrix@R =50pt@C=50pt{\overline{A} \ar [rr] \ar [dd] \ar [dr] & & \overline{B} \ar [dd] \ar [dr] & \\ & \overline{C} \ar [rr] \ar [dd] & & \overline{D} \ar [dd] \\ A \ar [rr] \ar [dr] & & B \ar [dr] & \\ & C \ar [rr] & & D } \end{gathered}\end{equation}
having the property that the back and left faces
\[ \xymatrix@R =50pt@C=50pt{ \overline{A} \ar [r] \ar [d] & \overline{B} \ar [d] & \overline{A} \ar [r] \ar [d] & \overline{C} \ar [d] \\ A \ar [r] & B & A \ar [r] & C } \]
are pullback squares and the top and bottom faces
\[ \xymatrix@R =50pt@C=50pt{ \overline{A} \ar [r] \ar [d] & \overline{B} \ar [d] & A \ar [r] \ar [d] & B \ar [d] \\ \overline{C} \ar [r] & \overline{D} & C \ar [r] & D } \]
are pushout squares, then the front and right faces
\[ \xymatrix@R =50pt@C=50pt{ \overline{C} \ar [r] \ar [d] & \overline{D} \ar [d] & \overline{B} \ar [r] \ar [d] & \overline{D} \ar [d] \\ C \ar [r] & D & B \ar [r] & D } \]
are also pullback squares.
We say that $\operatorname{\mathcal{C}}$ satisfies the second Mather cube theorem if, for every cubical diagram (7.88) where the the front, back, left, and right faces are pullback squares and the bottom face is a pushout square, the top face is also a pushout square.