I recently read about Galois connections, and that they show up in a lot of different places in mathematics. Given there apparent ubiquity, I thought they might have a rich theory. However, when asking other people in my department, I got varying answers as to why they were not particularly interesting and/or were not worth studying. It is hard to tell if this is due to a bias or whether there could be an objective argument made as to why mathematicians invest a lot of effort in characterizing and classifying one class of objects such as groups or topological spaces and not another.
I am interested in answers that either address the particular theory of Galois connections (and possibly arguing that they are in fact worth studying) or attempt to give a general criteria for what makes a good mathematical theory.