Both model theory and category theory formalize the idea of “structure” in some way. Is this purely different formal interpretations of an informal idea, or is there more of a formal relationship beyond this between these two subjects?
Ultimately, the connections between model theory and category theory are expressed by two points (which go in opposite directions):
- Model homomorphisms form categories in the natural way, where they become the morphisms. Thus, this aspect of model theory is generalized by category theory, in the same way that it is generalized from topology (continuous functions), algebra (algebraic structure homomorphisms), and many other subjects.
- However, model theory is unique compared to topology, algebra, and other subjects in the sense that it is the most general mathematical subject to study models of axioms expressible in particular logics. In fact, model theory encompasses any kind of axiomatically defined concept in this way, given a logic that is expressive enough. In particular, categories are defined by the axioms of category theory, which are amenable to model theory of a sufficiently expressive logic — driving the point further, categories are models of the category theory axioms.
Right now, the only logic that I feel familiar doing model theory in is first-order, but nevertheless the universal applicability of the subject stands.
On the subject of the expressivity of a logic, another concept I have read about recently is the idea that certain axioms in a “simpler logic” can nevertheless simulate more complicated or higher-order logics. For example, the ZFC axioms of set theory are actually first-order when we allow it to have an infinite set of axioms — the axiom schema is not actually an axiom on its own, but rather meant to represent a new entry in the set of axioms for each formula in the language of ZFC. However, ZFC can then simulate arbitrarily higher-order logics (as far as I understand.) This concept feels particularly trippy to me, but it is something I aim to learn more about once I study higher-order model theory.
Written January 2023.
