## Model Theory and Category Theory July 29, 2008

Posted by dcorfield in Uncategorized.

David Kazhdan has some interesting things to say about model theory, and in particular its relationship to category theory, in his Lecture notes in Motivic Integration.

In spite of it successes, the Model theory did not enter into a “tool box” of mathematicians and even many of mathematicians working on “Motivic integrations” are content to use the results of logicians without understanding the details of the proofs.

I don’t know any mathematician who did not start as a logician and for whom it was “easy and natural” to learn the Model theory. Often the experience of learning of the Model theory is similar to the one of learning of Physics: for a [short] while everything is so simple and so easily reformulated in familiar terms that “there is nothing to learn” but suddenly one find himself in a place when Model theoreticians “jump from a tussock to a hummock” while we mathematicians don’t see where to “put a foot” and are at a complete loss.

So we have two questions:

• a) Why is the Model theory so useful in different areas of Mathematics?
• b) Why is it so difficult for mathematicians to learn it ?

But really these two questions are almost the same – it is difficult to learn the Model theory since it appeals to different intuition. But exactly this new outlook leads to the successes of the Model theory. One difficulty facing one who is trying to learn Model theory is disappearance of the “natural” distinction between the formalism and the substance. For example the fundamental existence theorem says that the syntactic analysis of a theory [the existence or non-existence of a contradiction] is equivalent to the semantic analysis of a theory [the existence or non-existence of a model].

The other novelty is related to a very general phenomena. A mathematical object never comes in a pure form but always on a definite background. Finding a new way of constructions usually lead to substantial achievements.

For example, a differential manifold is “something” which is locally like a ball. But we almost never construct a differential manifold $X$ by gluing it from balls. For a long time the usual way to construct a differential manifold $X$ was to realize it at a subvariety of a simple manifold $M$ [a sphere, a projective space etc.].

A substantial progress in topology in the last 20 years comes from a “simple observation” due to physicists one can realize a differential manifold $X$ as quotient of an “infinite-dimensional submanifold” $Y$ of a “simple” infinite-dimensional manifold $M$. For example Donaldson’s works on the invariants of differential 4-manifolds are based on the consideration of the moduli space of self-dual connections which is the quotient of the “infinite-dimensional submanifold” of self-dual connections by the gauge group.

This tension between an abstract definition and a concrete construction is addressed in both the Category theory and the Model theory. The Category theory is directed to a removal of the importance of a concrete construction. It provides a language to compare different concrete construction and in addition provides a very new way to construct objects as “representable functors” which allows to construct objects internally. This construction is based on the Yoneda’s lemma which I consider to be most important result of the Category theory.

On the other hand, the Model theory is concentrated on gap between an abstract definition and a concrete construction. Let $T$ be a complete theory. On the first glance one should not distinguish between different models of $T$, since all the results which are true in one model of $T$ are true in any other model. One of main observations of the Model theory says that our decision to ignore the existence of differences between models is too hasty. Different models of complete theories are of different flavors and support different intuitions. So an attack on a problem often starts which a choice of an appropriate model. Such an approach lead to many non-trivial techniques for constructions of models which all are based on the compactness theorem which is almost the same as the fundamental existence theorem.

On the other hand the novelty creates difficulties for an outsider who is trying to reformulate the concepts in familiar terms and to ignore the differences between models.

So there’s a mathematician looking to model theory. Now for a model theorist reaching out to mathematics. Here’s Angus MacIntyre in Model theory: Geometrical and set-theoretic aspects and prospects, The Bulletin of Symbolic Logic Volume 9, 2003, pp. 197–212.

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 affine 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 than anything yet known in “set­-theoretic model theory”.

and

Tarski’s set-­theoretic foundational formulations are still favoured by the majority of model­theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable (to the geometers), this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly.

I think I’m right in saying that the geometric model theory MacIntyre promotes in this article is the kind of model theory which concerns Kazhdan in his notes. If the latter’s right, even if geometric model theory is closer to mainstream maths, it’s still hard to take on the model theoretic mind-set.

MacIntyre ends with some Prospects, including

There are various hints in the literature as to categorical foundations for model­-theory [21]. The type spaces seem fundamental [28], the models much less so. Now is perhaps the time to give new foundations, with the flexibility of those of algebraic geometry. It now seems to me natural to have distinguished quantifiers for various particularly significant kinds of morphism (proper, étale, flat, finite, etc), thus giving more suggestive quantifier­-eliminations. The traditional emphasis on logical generality generally obscures geometrically significant features [19].

and

I sense that we should be a bit bolder by now. There are many issues of uniformity associated with the Weil Cohomology Theories, and major definability issues relating to Grothendieck’s Standard Conjectures. Model theory (of Henselian fields) has made useful contact with motivic considerations, including Kontsevich’s motivic integration [6]. Maybe it has something useful to say about “algebraic geometry over the one element field” [25], ultimately a question in definability theory.

[21] is Lawvere’s Quantifiers and Sheaves, Actes Congrès. Intern. Math. 1970, pp. 329-334. I wonder what model theorists think of Makkai and Paré (1989), Accessible categories: The foundation of Categorical Model Theory, Contemporary Mathematics, AMS.

1. dcorfield - July 31, 2008

More on the relationship between category theory and model theory from Ravi Rajani and Mike Prest Model-theoretic imaginaries and coherent sheaves, 2008:

Model theory has evolved in two sharply different directions. One is set-based, centred around pure model theory and applications to various mathematical structures: here even the language of category theory is only beginning to be heard. In contrast is the sort of model theory which is set in rather general category-theoretic, or topos-theoretic, contexts and which often looks to non-classical logics or computer science for its inspirations and applications. Our results sit in the rather sparsely populated territory between these and our hope is that this paper will help to bridge the gap between these rather different kinds of model theory. Our paper is directed mainly to set-based model-theorists in that we show how finitely presented and coherent functors arise through the imaginaries construction. This opens a door to the use of functorial techniques in model theory. Use of such techniques has proved to be enormously effective in the model theory of additive structures and we see no reason why this will not extend to the model theory of more general structures. The distances between these different sorts of model theory should not, however, be underestimated. As we ourselves found, it is quite possible to prove a result and then discover that it, or something very close to it, exists already in the literature but in a form which, without the benefit of hindsight, looks completely different.

What we do here is show the equivalence of categories of imaginaries (of various kinds) with categories of “small” (finitely generated, finitely presented, coherent) functors. We do this first for certain locally finitely presented categories and then, by localising, for much more general “definable categories” (categories of models of coherent theories). Then we discuss the corresponding notion of interpretation.

Some of our results may be derived from the results and proofs in [19], [17], [8] but our proofs, indeed our whole approach, is very different, being rooted in set-based model theory and the development of the model theory of additive categories. The emphases also are different: here we present the results as equivalences between categories of certain functors and certain imaginaries; in the category-based literature the final form of the results is usually a “conceptual completeness” theorem (see [16], [17], [18]) which might be expressed as an equivalence of 2-categories (see [8] for a number of examples).

Shelah seems to be the originator of imaginaries:

Imaginaries belong to set-based model theory. Indeed, thinking of them as forming a category was a novel step which was taken by Herzog…

but

…given a coherent theory $T$ in a first-order language $L$, there is an associated category of “positive existential imaginaries” – a functorial version of Shelah’s imaginaries – which can be defined in purely categorical terms as a certain category of coherent sheaves.

2. Seamus - August 11, 2008

Model theory and Category theory are also both relevant to structuralist views in the philosophy of maths. Anachronistic as it is, one can discern model-theoretic and structuralist motivation in David Hilbert’s Foundations of Geometry, for example.

3. Morgan - November 24, 2008

I would Like to know, if is there any form of this theory in every domain ? by this, I mean an abstraction in order to operate over concept? linking and compose to uncerstand and evolve? Most of this ideas come from the monoid theory used in computer science.