In his classic Dialectica paper `Adjointness in foundations’ (1969), F. William Lawvere writes of `the familiar Galois connection between sets of axioms and classes of models, for a fixed [signature]’.
But even if long familiar folklore to category theorists, the idea doesn’t in fact seem to be that widely known. The ideas however are pretty enough, elementary enough, and illuminating enough to be worth rehearsing briskly in an accessible stand-alone form. Here’s my attempt.
I wrote these notes a while ago (inspired by an improptu talk by Nathan Bowler), as part of a longer planned piece (hence the perhaps overkill first chapter). The longer piece is now on the back burner. But this excerpt is reasonably polished and might be useful. (The previous version has now been improved/corrected in the light of Laurant Méhats’ comments. Further corrections or suggestions for improvement gratefully received as usual.)