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
- 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
- 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 homotopy push-out.