Given a theory describing some class of mathematical objects (groups, rings, orders, ...) we can construct the sentences one might consider about these, model the symbols as being about an actual set with actual functions, relations, etc, and talk about how manipulating sentences to get proofs relates to truth in the models. Thanks to category theory, we can replace all of this with a lot of abstract nonsense! And who doesn't like abstract nonsense?
In this talk we'll introduce some category theory and construct a category representing syntax and provability in a given theory. We'll discuss how this allows us to talk about models as a special case of structure preserving maps. As in the classical case, we get an adjunction (generalized Galois connection) between syntax and semantics. We'll also use this to engage in some shenanigans! Given a non-commutative algebraic structure, we'll pretend its commutative, even recovering classical results about commutative structures in the non-commutative case. We'll talk about how the real numbers become more subtle and interesting, with distinct characterizations giving different structures. We'll even model a theory that describes surjections of N onto R. (Don't worry, the surjections of N onto R aren't real and they can't hurt you.)