TITLE: The most general context of Mather's Cube Theorems
QUESTION [12 upvotes]: 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.

REPLY [17 votes]: Let $\mathcal{X}$ be an $\infty$-category (i.e., a homotopy theory) which admits small homotopy colimits, a set of small generators, and has the property that homotopy colimits in $\mathcal{X}$ commute with homotopy pullback. Then $\mathcal{X}$ satisfies the Mather cube theorem if and only if $\mathcal{X}$ is an $\infty$-topos: that is, it can be described as a left exact localization of an $\infty$-category of presheaves of spaces. (I learned this from Charles Rezk). Such homotopy theories are studied extensively in my book "Higher Topos Theory" (see in particular Proposition 6.1.3.10 and the remark which follows it).