Why the dual of some results are true while others are false?

Question

In mathematics, many results have their "dual" versions. In many cases, if a result is true, then its dual is true as well. However, there are some examples while the dual of a true statement is false. For example, consider the cube diagram in which all faces are homotopy-commutative squares,

the four vertical faces are homotopy pullback squares and the bottom square is a homotopy pushout square. Then the top square is a homotopy pushout square.

But the dual is false: If the four vertical faces are homotopy pushout squares and the bottom square is a homotopy pullback square. Then the top square may not be a homotopy pullback square.

Of course someone can show me an example disproving the dual. But I would like to ask the idea (philosophy?) here. Why the dual of a true statement is sometimes false?

Answer 1

Your first statement is too vague. Let's make it more specific: in category theory, we can dualize categorical statements by reversing the directions of all of the arrows. This always produces a second true statement, as long as you're careful to take the dual correctly.

Here's an example of taking the dual incorrectly: it's true that finite limits commute with filtered colimits in $\text{Set}$. It's not true that finite colimits commute with filtered limits in $\text{Set}$. But this isn't the dual statement: the dual statement is that finite colimits commute with filtered limits in $\text{Set}^{op}$ because I have to reverse the directions of all of the arrows.

One reason it's fun to prove facts about all categories is that dualizing them again gives me facts about all categories: here I am making crucial use of the fact that the opposite of a category is another category, and that all categories arise in this way. Similarly, the opposite of an abelian category is another abelian category, so I can play the same game there. But that fact above about $\text{Set}$ is just not a fact about a class of categories closed under taking opposites.

So let's return to your example.

1. The first problem is that you haven't reversed all of the arrows. Once you do, the role of the top and bottom faces swap (and they don't play symmetric roles): now it's the bottom face that maps up to the top face rather than the reverse.
2. The second problem (potentially; I don't know if this is actually a problem) is that you haven't specified what higher categories you're taking homotopy limits and colimits in. If you're only taking them in spaces, then you run into the same problem as above; the higher category of spaces is not equivalent to its opposite.