The n-Category Café

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:

Definition Define 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 a closed 1-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 space for 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 version of 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.

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

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.

OK, it looks like I’ll be the first “working model theorist” to take the bait and respond to this post…

Short answer: I’m fascinated by all the recent applications of my field to “mainstream math” (Hrushovski’s proof of the Mordell-Lang Conjecture, connections with complex analytic geometry and with motivic integration, etc. etc.), but as more of a “pure model theorist” myself, I feel like I understand very little of it and I’m not sure what to make of it all. And I think this is typical of most researchers who currently call themselves “model theorists.” Research in model theory seems to be going on in many different directions at once right now, and few people would claim to have a good understanding of all the various subfields.

I started out in model theory being very interested in categories of models (under the natural notion of embedding – “elementary embeddings”). In many ways this is orthogonal to the research program that Kazhdan and Macintyre seem to be indicating – they’re advocating (as I understand) an emphasis on categories of definable sets. To give an example from math, in the case of algebraically closed fields, the category of models is the category of all such fields (with the usual field embeddings), and the category of definable sets is (equivalent to) the category of Zariski-constructible sets over a prime field (with all the Zariski-constructible functions as your arrows).

My impression is that the logicians and the category theorists have come up with two very different programs for generalizing classical mathematics. And if you ask, “What are the essentially properties of mathematical object X?”, then mathematical logic and category theory often seem to suggest two totally different answers. (Caveat: but there are connections between the two, e.g. in topos theory.) Even in 2008, model theorists tend to be very “ensembliste” and have a more classical view of mathematical structures than, it seems, your typical modern number theorist or algebraist.

As for Makkai’s work on categorical model theory, it sounds very interesting, and I once briefly made an effort to read some of his papers, but they were too daunting for me. Category-theoretic methods have not permeated mainstream model theory (yet?).

I guess there are two main flavors of models - saturated and “other”. It’s true that model theorists break the link between formalism and substance, but they often re-establish it at a different level, by working in a large saturated model of a complete theory. I’d say that’s the first non-obvious hummock, because these saturated models are not standard mathematical objects. When model theorists study the real numbers, for example, they’ll usually work in a huge real-closed field (with infinitesimals, infinite elements, etc.).

To the next hummock: recently a very productive idea, if you have an incomplete theory, is to look at its “model companion” (when this exists). The motivating example is to start with the theory of fields, then the model companion is the theory of algebraically closed fields. (You can express this in terms of the categories of models of the two theories - but I don’t think there’s a deep point of contact there.)

A great example is to begin with fields equipped with a differential map, and the model companion is differentially closed fields, that is differential fields where all possible differential polynomials have solutions… but there is no natural example of such a thing! You can play the same game with fields with an automorphism, to get fields with a generic automorphism. Zilber tackled fields with an exponential map to get his “pseudoexponentiation”. And so on. These are some of the sorts of objects that model theorists have discovered and are working on, which are finding applications in number theory and elsewhere, which may be why the subject is becoming fashionable.

Another current theme is to extend stable methods to unstable theories. Stable theories are where this business of “forking” comes in: basically where you have an independence relation which behaves like algebraic independence in an algebraically closed field. With this you can build up powerful machinery mirroring algebraic geometry. This is great… but most theories aren’t stable. So people have been looking for broader classes of structures where you can do similar things. Examples are simple theories (e.g pseudofinite fields) and metastable theories (e.g algebraically closed valued fields). There are various others.

Everything I’ve said so far fits into classical “point-set” model theory (sometimes moving slightly beyond first-order logic), but this is still pretty rich, and includes Morley and Zilber’s work, and Hrushovski’s proof of Mordell-Lang. I don’t know a huge amount about the category-theory/model-theory boundary. I don’t think a framework has been fully worked out. Certainly if it’s going to develop, you’re looking in the right places: also Prest’s done a lot of work on the model theory of modules; and motivically, see especially Denef and Loeser. From another angle, not related to motivic matters, there’s an intriguing paper by Hrushovski on definable groupoids.

(Postscript - here I am rambling on about model theory, and I’ve just realised you’re a co-blogger of Alexandre Borovik elsewhere!)

Thanks very much for these comments. I’ll enjoy reading them when back from holiday.

I would love to hear more about this. My impression, which could easily be wrong, is that some subjects admit a model-theoretic approach and a ‘mainstream’ approach. For example, differential algebraic geometry can be approached (I think) using differentially closed fields and differentially definable sets but also using varieties with a fixed vector field, jet spaces, and so on.

It would be great to read an exposition that compares the two approaches in specific cases and in general, how concepts can be translated back and forth, how one point of view might be better or worse for looking at certain things, etc. I have the impression that some people can translate easily between the two points of view, but not many.

I just came back to this post looking for something else, and this time I clicked through the links (one of which, to my pleasant surprise, was to an expository paper of Hrushovski’s). I’m nearly completely ignorant of all this, but it looks like Macintyre and Kazhdan were making contradictory points.

Macintyre writes, “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.” And “I stress that it is the technology (here very advanced) of definability theory that matters, not at all any special model or set-theoretic construction.” And “In none of the above examples are specific models of much interest. What is fertile is to understand the definability theory. One uses geometric model theory to pass to applications in geometry.”

On the other hand, we have Kazhdan: “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.”

While I hardly even know what they’re talking about, it’s Macintyre’s point of view that smells right to me.

On the categories mailing list, Jonathan Chiche is asking for references to further category theoretic treatments of model theory.

Adding to what we have in this thread, so far we have Colin McLarty:

…there are some pointed remarks, about a tendency to avoid explicitly using categorical tools that are in effect already being used, in Pillay’s part of

S. Buss et al. The Prospects for Mathematical Logic in the Twenty-First Century

Unsystematic uses occur throughout the literature, often without using the word “category.” I would mention especially

Lou van den Dries, Lou Tame Topology and O-minimal Structures

D.Haskell et al. (eds) Model Theory, Algebra, and Geometry

and from Phil Scott

There will be an upcoming workshop June 19-20 at the CRM (Centre de Recherches Mathematiques) at the University of Montreal, dedicated to Michael Makkai’s 70th birthday, which is exactly on this theme. Its local organizers are Robert Seely and me, along with two model theorists (Bradd Hart and Tommy Kucera), who were students of Makkai.

More on model theory and category theory – Model theory and the Tannakian formalism by Moshe Kamensky:

The aim of this paper is to exhibit the analogy and relationship between two seemingly unrelated theories. On the one hand, the Tannakian formalism, giving a duality theory between affine group schemes (or, more generally, gerbs) and a certain type of categories with additional structure, the Tannakian categories. On the other hand, a general notion of internality in model theory, valid for an arbitrary first order theory, that gives rise to a definable Galois group. The analogy is made precise by deriving (a weak version of) the fundamental theorem of the Tannakian duality (3.7) using the model theoretic internality.

Perhaps not so surprising given the list of entries on the page reconstruction theorem includes Tannakian theorems and Lawvere’s reconstruction theorem.

Chapters of the book – Models, Logics, and Higher-Dimensional Categories: A Tribute to the Work of Mihaly Makkai – should help mutual understanding between model theorists and category theorists. For example, Victor Harnik says of part of his chapter – Model Theory vs. Categorical Logic: Two Approaches to Pretopos Completion (a.k.a. $T^{eq}$) – that it can be seen as “a model-theorist’s view of the category-theoretic thinking”.

Continuing the addition of things to think about if we ever get round to reconsidering the relationship between category theory and model theory, here’s Terry Tao commenting on the announcement of a proof of the ABC conjecture by Mochizuki:

I have always been fond of the idea that model-theoretic connections between objects (e.g. relating two objects by comparing the sentences that they satisfy) are at least as important in mathematics as the more traditional category-theoretic connections (where morphisms are the fundamental connective tissue between objects) or topological connections (where the objects are gathered into some common topological space or metric space in order to compare them). A good example is the Ax-Grothendieck theorem, in which a result that is easy to prove in the positive characteristic setting can then be transferred to the characteristic zero setting by a model-theoretic connection, even though there is no immediately obvious morphism, functor, or natural transformation from positive characteristic to zero characteristic, nor is there an immediately obvious topological sense in which the zero characteristic setting is a limit of the positive characteristic one. (This is not to say that there the connection between positive characteristic and zero characteristic that is relevant for Ax-Grothendieck can’t be thought of in categorical or topological terms if one really wanted to view it that way – it probably can – but that the model-theoretic way of looking at it seems much more natural, in my opinion.)

Scheme theory, in my mind, does an excellent job of capturing the categorical and topological ways of connecting objects in algebraic or arithmetic geometry, but only engages in the model-theoretic connections in a rather restricted fashion. (An ideal in a commutative ring can be thought of model-theoretically as the set of all identities that can be deduced from a set of generator identities from the laws of commutative algebra (i.e. high school algebra), and so schemes capture the model theory of this algebra well; but there isn’t an obvious mechanism in place in scheme theory to capture the model theory of more sophisticated theories that might also be relevant in arithmetic geometry.)

So that would make an interesting question of whether there’s a good category theoretic approach here. Or might moving to quasicategories help? I see someone at MO is wondering something similar.