Every math undergraduate loves their torus, with the 18 triangles needed to compute homology. With quantum cubes, we automatically label axes discretely with powers of a letter.
So when does a square include a 2-cell? Look at the bottom left square.
A category theorist will immediately see that this is a limit diagram for the product of X and Y. But the square defined by the generator of H1 does not yield a nice square, although it does commute.