Our second meeting of the Model Theory Reading Group, and Nathan Bowler — a category theory PhD student — gave a terrific overview talk taking us through the rest of the second chapter of Hodges. Moreover, he briefly introduced us to Martin Hyland’s idea that it is illuminating to think of the relation between the set of all theories and the power set of the set of all structures, together with the ‘semantics’ function from theories to their set of models, and the ‘syntax’ function back from a set of structures to the smallest theory true of just that set as forming a Galois connection. A fascinating glimpse (resolution: I must get more on top of this sort of category-theoretic stuff).
Despite occasional light touches, Hodges’s second chapter is pretty relentless stuff (a few more vivid examples of various notions, and neat illustrations of various distinctions would surely have been very welcome). So Nathan’s talk turned a session that promised to be very hard going into an enjoyable and illuminating occasion.
1 thought on “Theories, models and Galois Connections”
For the n-category café take on this topic, try Todd Trimble’s post and follow up.