Take the tour ×  Quite simply, I'd like to know what is the
broadest or most natural context in which either (or both) of
Mather's cube theorems hold. If you like, this may mean any of
• What properties of $Top$ or $Top^*$ are essential to the
proofs?   • (where) are model/homotopical categories verifying
Mather's theorems studied as such in the literature?   • Are
there more examples known verifying Mather's theorems?  I ask
because Mather's proof strikes me as fairly gritty and seems to
rely on explicit cellular constructions.  For reference, the
cube theorems concern a cubical diagram whose faces commute up
to homotopy in a coherent way, and assert    1. If one pair of
opposite faces are homotopy push-outs and the two remaining
faces adjecent the source vertex are homotopy pull-backs, then
the final two faces are also homotopy pull-backs   2. If two
pairs of opposite faces are homotopy pull-backs, and the
remaining face adjacent the target vertex is a homotopy push-
out, then the remaining face is a homo