### Category Theory and Model Theory

#### Posted by David Corfield

The question of the relationship between category theory and model theory emerged in this thread. So I was interested to read some things David Kazhdan had to say about this relationship 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.

He continues,

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 model-theorists, 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. [Added: a description of the provenance of contemporary model theoretic work.] 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.

## Re: Category Theory and Model Theory

First I admit my ignorance:

a) I have only a very rough idea of model theory.

b) I still have essentially no good idea of what motivic integration is supposed to be

Still not, even though I was pointed to lecture notes back when discussion about what I called Integration without integration reminded people of motivic integration.

Since then, I have been thinking more about this idea of “integration without integration” in the sense of equivalence classes of differential forms (and, more generally, of $L_\infty$-algebra valued forms). I think it is a pretty cool thing, which, while known essentially to some people, hasn’t received much attention yet.

Even though off-topic in relation to the above entry, maybe I can mention a few details. I didn’t get the impression that this has any relation to motivic integration, but since I don’t know what motivic integration is, that doesn’t mean anything. So if anyone does see a relation to motivic integration, let us know.

All right. So this is what “integration without integration” in my sense is about:

Say you have a smooth 1-form $A \in \Omega^1([0,1])$ on the standard interval and wish to determin its integral $\int_0^1 A$. Let me assume that $A$ vanishes in a neighbourhood of the boundary of the interval.

Here is a funny way to determine the integral:

DefinitionDefine an equivalence relation $A \sim A'$ on 1-forms on the interval by saying that two 1-forms are in the same class if there is aclosed1-form $\hat A \in \Omega^1_{closed}(D^2)$ on the disk which restricts to $A$ on the upper semi-circle of $S^1 = \partial D^2$ and to $A'$ on the lower semicircle.Proposition.There is a bijection between equivalence classes $\Omega^1([0,1])/\sim$ and the real numbers $\Omega^1([0,1])/\sim \simeq \mathbb{R} \,.$ This bijection is realized on representatives by the integration map $\int_{[0,1]} : \Omega^1([0,1]) \to \mathbb{R} \,.$In words, this says that we can integrate a 1-form without integrating it by passing to the equivalence class of all those 1-forms which would yield the same integral had we decided to compute it.

Proof. Of course by Stokes theorem any two 1-forms on the intervakl which are connected by a closed 1-form on the disk have the same integral. Conversely, to see that for every two 1-forms $A$ and $A'$ with the same integral there is a closed 1-form on the disk interpolating between them, construct that closed 1-form explicitly as follows:

Attach the two intervals end-by end and identify $[0,2]$ with endpoints identified with $S^1 = \partial D^2$. This gives a single 1-form $\tilde A = \tilde A_\sigma d\sigma \in \Omega^1(S^1)$ with the property that $\int_{S^1} \tilde A = 0 \,.$ The task is to extend this to a closed 1-form $\hat A$ on the disk.

Choose polar coordinates $(0 \leq \sigma \lt 2\pi,0 \lt r \leq 1)$ away from the origin and choose a smoothing function $f : [0,1] \to [0,1]$, i.e an orientation preserving diffeomorphism constant in a neighbourhood of the boundary of the interval.

Then setting $\hat A := f(r) \tilde A_\sigma(\sigma) \, d \sigma + f'(r) (\int_0^\sigma \tilde A_\sigma(s)\,d s) \, d r$ does the job. This is well defined precisely because $\int_0^{2\pi} A_\sigma(s) = 0$.

Maybe you are not convinced yet that this is more than a weird collection of trivialities. It becomes a bit more interessting when we change perspective as follows:

There is a non-concrete generalized smooth space (as described in convenient category of smooth spaces) with the funny name $S(CE(u(1)))$ which is the

classifying spacefor closed 1-forms: for every manifold $X$ we have a bijection between smooth flat 1-forms on $X$ and smooth maps from $X$ into this classifying space $Maps(X,S(CE(u(1)))) \simeq \Omega^1_{closed}(X) \,.$The above computation shows that

Proposition.The fundamental group of $S(CE(u(1)))$ is $\mathbb{R}$.Even more: writing $\mathbf{B}\mathbb{R}$ for the one-object groupoid version of the group $\mathbb{R}$ we have:

The concretization of the fundamental groupoid of $S(CE(u(1)))$ is $\mathbf{B}\mathbb{R}$:

$\Pi_1(S(CE(u(1)))) = \mathbf{B}\mathbb{R} \,.$

To see this, one has to notice that there is a

family versionof the above yoga:Let $U$ be any Euclidean space (the statment also works with $U$ any manifold). Let now $A$ be any closed 1-form on the interval times $U$ $A \in \Omega^1_{closed}([0,1]\times U) \,,$ i.e. a $U$-family of closed 1-forms. For $\{u^1\}$ a global coordinate system of $U$ we can decompose this as $A := A_\sigma d\sigma + \sum_i A_{u^i} d u^i \,.$

Then we have the following generalization:

Definition.For $A,A' \in \Omega^1_{flat}([0,1]\times U)$ define an equivalence relation which says that $A \sim A'$ precisely if there is a flat 1-form $\hat A \in \Omega^1(D^2\times U)$ on the disk times $U$ which interpolates between $A$ and $A'$ in the above sense.Proposition:Now the equivalence classes are real-valued smooth functions on $U$:$\Omega^1_{closed}([0,1] \times U)/\sim \simeq C^\infty(U,\mathbb{R}) \,.$ The equivalence is realized on representatives by the integral $\int_{[0,1]}$ at fixed $u \in U$

$\int_{[0,1]} (-)|_{u} : \Omega^1_{flat}([0,1]\times U) \to C^\infty(U,\mathbb{R}) \,.$

Proof:

Again it is clear that if there is an interpolating 1-form, then the integrals agree. Conversely, if the integrals agree proceed as above, build a 1-form $\tilde A \in \Omega^1_{closed}(S^1 \times U)$ to be decomposed as $\tilde A = \tilde A_\sigma d\sigma + \sum_i \tilde A_{u^i} \, d u^i \,.$

Then set as before $\hat A = f(r) (\tilde A_\sigma d\sigma + \sum_i \tilde A_{u^i}\, d u^i) + f'(r)(\int_0^\sigma \tilde A_\sigma \, d\sigma) dr \,.$

To see that this is indeed closed differentiate under the integral sign and use the flatness of $\tilde A$.

There are analogous stories to be told for “integration without integration” of higher forms. That amounts to computing the concrete fundamental $n$-groupoid

$\Pi_n(S(CE(b^{n-1}u(1))))$

of the non-concrete generalized smooth classifying space of closed $n$-forms.