Mather's cube theorem for the category of topological spaces says that given a homotopy-commutative cube:

If one pair of opposite faces are homotopy pushouts and the two remaining faces adjecent the source vertex are homotopy pullbacks, then the final two faces are also homotopy pullbacks.

What is the geometric intuition behind this theorem?

firstcube theorem, correct? It seems some authors say “Mather’s cube theorem” to reference thesecondcube theorem. $\endgroup$ – Santana Afton Nov 2 '19 at 9:22