About Galois connections

A Galois connection is a pair of complementary maps with certain properties between a pair of complementary sets.

Galois connections provide a mathematical basis for abstract interpretation, and they are possibly easiest to understand from examples given in that context.

We can think of our pair of sets as being a set of concrete values (such as {0, 1, 2, ... }) and a set of abstract values (such as {int, even, odd, power-of-two, prime, ... }).

We then think of our pair of maps as being an abstractifying map which takes elements of the concrete value set and sends them to elements of the abstract value set, and a concretizing map which takes elements of the abstract value set and sends them to elements of the concrete value set.

This pair of maps is a Galois connection when, if some abstract value (such as even) maps to some set of concrete values (such as {2, 4, 6, 8, ... }), then any concrete value in that set (such as 8) also maps to some (possibly specialized) instance of the abstract value (such as power-of-two).

That's not quite the whole story, but it seems like the most intuitive version to me.

Now there's nothing saying that the sets have to be "abstract" or "concrete" or that the maps have to be "abstractifying" or "concretizing" -- those are just from this one application in abstract interpretation. On the other hand, the structure inherent in the Galois connection does lend itself to relationships like this.