Is there a relationship between model theory and category theory? - MathOverflow

Is there a relationship between model theory and category theory?

2010-01-16T08:29:34Z

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

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)?

Answer by Charles Stewart for Is there a relationship between model theory and category theory?

2010-01-16T14:14:54Z

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 anway, to talk down his achievement).

Blackburn &al (2001, pp40-41) 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.

Ref: Blackburn, de Rijke, & Venema (2001) Modal Logic, CUP.

Answer by Mike Shulman for Is there a relationship between model theory and category theory?

2010-01-16T15:06:21Z

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!)

Answer by François G. Dorais for Is there a relationship between model theory and category theory?

2010-01-16T15:16:45Z

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).

Answer by Tom Leinster for Is there a relationship between model theory and category theory?

2010-01-17T02:57:49Z

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.

Answer by David Corfield for Is there a relationship between model theory and category theory?

2010-01-18T11:55:55Z

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