Back in 1969, F. William Lawvere (in his Dialectica paper ‘Adjointness in Foundations’) remarked on what he called “the familiar Galois connection between sets of axioms and classes of models”.
But even if it has long been familiar to some category theorists and theoretical computer scientists, the idea that Lawvere is referring to here seems not to have been picked up that widely. Certainly, I’ve not been able to find a neat stand-alone presentation which is accessible e.g. to philosophers doing a first course in model theory. So I’m trying to put together some notes to fill the gap. (If anyone can point me to helpful aids to thinking about these things that I might have missed in googling around, then I’d be very pleased to hear about them.)
Don’t expect novel fireworks, though! The name of the game is to get at routine points about theories and their models by a slightly unfamiliar route (and I’m still trying to work out just what points fall out of this approach in a natural way). However, approaching familiar territory from an unfamiliar angle can be illuminating. So I think it is probably going to be worth the effort.