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?



Geometric intuition for Mather's cube theorem
I'm not sure what counts as an intuitive explanation, but this is sort of how I think about it.


Say that $B=B\_1\cup B\_2$ and $B\_0=B\_1\cap B\_2$. This is the second pushout square.


Now let $E\_1$ be a bundle over $B\_1$ and let $E\_2$ be a bundle over $B\_2$, and suppose that the restriction to $B\_0$ is the same for both bundles -- call it $E\_0$. These are the two given pullback squares. Now let $E$ be the union of $E\_1$ and $E\_2$ along $E\_0$. This is the other pushout square. $E$ should be a bundle over $B$ whose restriction to $B\_1$ (resp. $B\_2$) is the bundle $E\_1$ (resp. $E\_2$). 


