# Is there a relationship between model theory and category theory?

According to Chang and Keisler's "Model Theory", Model Theory = Universal Algebra + Logic. Model theory generalized Universal Algebra in the sense that we allow relations while in Universal Algebra we only allow functions.

Also, we know that Category Theory generalized Universal Algebra. From wikipedia:

Blockquote Given a list of operations and axioms in universal algebra, the corresponding algebras and homomorphisms are the objects and morphisms of a category. Category theory applies to many situations where universal algebra does not, extending the reach of the theorems. Conversely, many theorems that hold in universal algebra do not generalise all the way to category theory.

So this suggest there might be some overlapping between Model Theory and Category Theory. I hope some one can elaborate about the relation (if there is)?

• "Conversely, many theorems that hold in universal algebra do not generalise all the way to category theory" - well, it depends on what it's meant by "category theory", but isn't everything that happens in universal algebra also expressible in the language of category theory? May 2, 2016 at 10:31

Between model theory and category theory broadly conceived: not anything really compelling, because a category, on its own, does not stand as an interpretation for anything.

Between model theory and categorical logic, however: yes, I think the overlap is large.

A spot of history: the man most deserving, in my opinion, of being called the father of model theory is Alfred Tarski, who came from a Polish school of logic that, I understand, was very much within the algebraic school. His model theory was more in the vein of a reworking of the Polish-style algebraic logic (this is not, in anyway, to talk down his achievement).

Blackburn et al (2001, pp 40-41) talk of a might-have-been for the Jónsson-Tarski representation theorem:

...while modal algebras were useful tools, they seemed of little help in guiding logical intuitions. The [theorem] should have swept this apparent shortcoming away for good, for in essence they showed how to represent modal algebras as the structures we now call models! In fact, they did a lot more that this. Their representation technique is essentially a model building technique, hence their work gave the technical tools needed to prove the completeness result that dominated [work on modal logic before Kripke].

They go on to present a nice anecdote showing how Tarski did not seem to think this algebraic approach provided a semantics for modal logic, even after Kripke stressed how important it was to Kripke semantics. It seems that sometimes algebraic logic and model theory are more similar than they appear.

Like model theory, categorical logic can seem to be a special way of doing algebraic logic. And with some theories, model theory and algebraic logic sometimes seem to differ only in trivialities; with categorical logic I am more hesitant in making sweeping judgements, but it sometimes feels that way to me too.

• That's an interesting take on Tarski. Can you expand on the rôle of the Jónsson-Tarski Theorem? I think my view is obstructed since I only think of cylindrical algebras trough Tarski's lens... Jan 16, 2010 at 15:01
• @François: Expand on what? The role it plays in algebraic modal logic? It is analogous to Stone's representation theorem: where that theorem tells us that the Lindenbaum algebras (LAs) are subalgebras of set algebras like, Jónsson-Tarski is an analogy where set algebras are replaced by ultrafilter extensions. The analogy isn't perfect: the distance from representation theorem to completeness theorem is a bit longer in modal logic than it is in propositional logic. Is that the sort of thing you were after? Jan 17, 2010 at 1:53
• @Charles: I had the impression there was more to it from what you said. I guess it's just the length of the road that prevented Tarski from seeing that far ahead. I'll check the details in Blackburn & al when I have a chance. Jan 17, 2010 at 2:45

Categorical logic is definitely one place to look. Although in my very limited experience model theory seems to be mostly interested in models built out of sets, while categorical logic is usually primarily interested in models in more general categories.

Another place to look is the theory of locally presentable and accessible categories, which are the categories of models (in Set) of a large class of theories. A number of model-theoretic ideas and techniques come up in their study, and the converse might be true too. The standard books are:

• Adámek and Rosicky, Locally presentable and accessible categories
• Makkai and Paré, Accessible categories: the foundations of categorical model theory

(Makkai and Paré certainly thought there was a relationship!)

Another link than the one explained by Charles Stewart is the relation with accessible categories. This was proposed by Michael Makkai and Bob Paré as a category theoretic foundation for model theory (Accessible categories: the foundations of categorial model theory, Contemporary Mathematics 104, AMS, 1989). I found this approach particularly compelling.

The basic idea is to think of the of models of a complete theory as forming a category with elementary embeddings as morphisms. The fact that this is an accessible category is basically the Löwenheim-Skolem Theorem. I really like the fact that this view is not limited by first-order logic. For example, it applies to infinitary logics and Abstract Elementary Classes just as well.

Another connection comes though classifying topoi (see Mac Lane & Moerdijk, Sheaves in Geometry and Logic, Chapter X). There are also strong ties with Abstract Stone Duality (I'm still trying to catch up there, so I can't say much more).

• Ah, Mike Shulman scooped me! Maybe somebody can expand on the link with ASD? Jan 16, 2010 at 15:19

There certainly is. Last year there was a largeish conference in Durham, New Directions in the Model Theory of Fields, which had the connections between model theory and category theory as its "second theme".

Perhaps the talk most relevant to your question was that of Martin Hyland, Categorical Model Theory. You can see a video on the website, but unfortunately it seems to start part-way through the talk. Anyway, he started by saying that everything he was going to explain was known in 1982, which perhaps was a reference to Makkai-Pare (as mentioned by Mike Shulman and F.G. Dorais) and that era.

A distinguished, but non-categorical, logician who seems to strongly support categorical model theory is Angus Macintyre. Here's his introduction to 'Model theory: geometrical and set-theoretic aspects and prospects', Bulletin of Symbolic Logic 9 (2003):

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older "sets of points in affined or projective space" no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and "transfer of structure" is incomparably more flexible and powerful then anything yet known in "set-theoretic model theory".

It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called "Definability Theory" in the near future.

We had a chat about this topic over here, prompted by remarks by David Kazhdan.

• This is an interesting discussion (+1). But I consider myself a counterexample to one of Kazhdan's theses: I am a mathematician that did not start out (nor end up, thus far!) in mathematical logic, but I learned about model theory in my final year of grad school (a fellow grad student friend of mine told me a bit about it, and then I read up on it for the 2003 Arizona Winter School) and have had an interest in it ever since. I think it is quite easy to pick up model theory -- at least a nontrivial amount of it -- compared to many subfields of my own discipline, e.g. p-adic Hodge theory. Jan 18, 2010 at 13:31

The theory of Abstract Elementary Classes, which were introduced by Shelah as an abstract axiomatization of elementary classes was recently connected with Accessible categories.

• Beke, Rosicky. Abstract elementary classes and accessible categories.
• M. Lieberman's PhD Thesis.

This connection was better understood since the recent join of Sebastian Vasey.

• Rosicky Lieberman Vasey. Universal abstract elementary classes and locally multipresentable categories
• Rosicky Lieberman Vasey. Internal sizes in μ-abstract elementary classes

• Rosicky Lieberman Vasey. Forking independence from the categorical point of view

The theory of institutions is based on the category theory and formalizes the notion of logics. The model theory based on the theory of institutions is described in Institution-independent Model Theory by Razvan Diaconescu, 2008.

"A rather classical viewpoint is formulated in : Model theory = logic + universal algebra. A rather different and more radical perspective which reflects the success of model theoretic methods in some areas of classical mathematics is given in : Model theory = algebraic geometry - fields. From a formal specification viewpoint, in a similar tone, one may say that Model theory = logical semantics - specification. ... formal specification theory requires a much more abstract view on model theory than the conventional one. The institution theory of Goguen and Burstall [30, 75] arose out of this necessity. Institutions. The theory of institutions is a categorical abstract model theory which formalizes the intuitive notion of a logical system, including syntax, semantics, and the satisfaction relation between them."