Category theory and model theory as "natural" counterparts

Question

I am aware of the profound discussion of the relationship between category theory and model theory (e.g. at The n-Category Café) but do wonder why category theory (CT) is not opposed to model theory (MT) from the very beginnings and firstly on a rather superficial level and why it's mostly set theory to play the "natural" counterpart to category theory (e.g. as a foundation of mathematics).

This is just a loose list of superficial analogies (to be taken with at least two grains of salt):

1. Theories in MT define classes of structures just as categories do in CT: theories describe structures "from the inside", categories describe structures "from the outside".

2. The relation of "equal up to isomorphism" (between structures/objects) plays a dominant role both in MT and CT.

3. There are related notions of equivalence of theories (bi-interpretability) and of categories (equivalence of categories). (Thanks to John Goodrick, who clarified this for me.)

4. Both CT and MT are strongly related to universal algebra:

   MT = universal algebra + logic (Chang/Keisler),

   CT = a language to further abstract away from the standard notions of universal algebra (Tarlecki)

5. CT and MT both seem to need set theory to provide concrete models (of theories and categories, resp.).

6. CT and MT can sometimes do without standard set models and provide typical "self-models":

   CT has "hom-set-models" (→ Yoneda)

   MT has "term-models" (→ Henkin).

7. David Kazdhan's questions concerning MT:

   a) Why is the Model theory so useful in different areas of Mathematics?

   b) Why is it so difficult for mathematicians to learn it ?

   apply equally well to CT. And also his preliminary answer does:

   One difficultly facing one who is trying to learn Model theory is disappearance of the ”natural” distinction between the formalism and the substance.

8. First-order theories with an infinite model give rise to arbitrarily large models, their class of models thus - being a proper one - corresponds to a large category.

9. The name of the important model-theoretic concept "categoricity" is striking. [Addendum: "Category theory provides a notion of 'unique specification’ that is related to categoricity in an interesting way, which remains to be clarified." (Steven Awodey in Completeness and Categoricity, Part II: Twentieth-Century Metalogic to Twenty-First-Century Semantics, p. 91)]

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The following questions arise naturally:

Question #1: Why are these - admittedly vague - analogies so seldomy discussed in introductory textbooks on both MT and CT (presuming some basic knowledge of the respective other theory)? Even if these analogies are misleading, it would be of help to know the reasons-why early.

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Question #2: Which concepts can be translated more or less directly from CT to MT and vice versa? Is there a translation scheme?

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Question #3: What are the specific strengths and weaknesses of CT and MT, compared to each other?

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Question #4: Can the levels of abstraction of MT and CT be compared?

Answer 1

You are comparing apples and organges. Model theory should be compared with categorical logic, not category theory. Conversely, category theory should be compared with algebra, not model theory.

Model theory is the study of set-theoretic models of theories expressed in first-order classical logic. As such it is a particular branch of categorical logic, which is the study of models of theories, without insistence on set theory, first order, or classical reasoning.

Answer 2

I find this a difficult question to answer, but let me try for your Q1. It could be that some people don't feel comfortable promoting vague analogies, or indeed spending time discussing them. Signal-noise ratio, to be blunt. In particular, your point 9 is not really the sort of thing we want to spend time belabouring. Point 7 does not say much about either model theory or category theory; the fact I can't eat rocks or wood says little about the common material composition of either. Point 5 is again an observation that both lions and tables have legs.

There is, I think common ground between ideas from model theory and categorical frameworks; but this is something where the devil is in the detail and not in the blue sky. 'Tis very like a whale, one might say.

In my Philistine opinion, of course.

Answer 3

I'm not an expert in model theory anyway I'll try to answer your questions.

From what I get your problem come from the fact that both model theory and category theory are related with the study of stuctured objects and morphisms between them. There are categories which aren't at all build up from structured objects and morphisms stucture-preserving, for instance monoids, groups and posets are categories too, and seeing this objects as categories is useful for some applications. Model theory instead deal exactly with models of a theory which are exactly stuctured objects and the stucture preserving morphisms, so it deals with categories of models of given theories (to be exact if I'm not mistaking, model theory also deal with theories' morphisms and derived morphisms between theories' models, but also this can be seen in terms objects and morphism).

After this not too short introduction let's try to answer your questions:

Answer #1: I suppose that the textbook you are referring to were written in time when the deep connection between model theory and category theory weren't well known. Try to take a look to book about categorical logic.

Answer #2: As I said above categories can be viewed as models of a particular (first order) theory, by the way this is not really useful because of the size issues I mentioned above. By the way category theory via notions of categories (with enough structure), functors (preserving the said structure) and natural transformations offer a new way to define the notion of theory, model and model transformation. In this way it become possible to study the notion of model of a theory in any category, where classical model theory become simply the study the theory of models in $\mathbf{Set}$, the category of sets and functions.

Answer #3:I don't know if there's any satisfactory answer to this question, mostly because as I said category theory and model theory are really different theories which aims to study different objects (the first one deal with theories and models, the second with categories, functors and natural transformations, but also other objects if we consider higher category theory as category theory). Maybe it could be more interesting studying the relation between classical (i.e. set theoretic) model theory and categorical model theory, but I don't know enough to talk about this.

Answer #4:If by level of abstraction you mean if one can be consider as a special case of the other I guess the answer is yes and no: you can build a first order theory of categories, functors natural transformation but from another point of view model theory can be completely rephrased in categorical term. Seeing from this point of view the question seems to me very similar to the chicken or the egg causality dilemma, and I don't think it's really useful this point of view, I would never consider group theory just as the study of the models of the theory of groups. :)

I hope this helps.

Answer 4

(Sorry for bumping this to the top. I should have checked the date it was posted and just simply let it lie.)

Really, the question makes no sense to me. A few points, in no particular order:

1. We may consider models of a classical first-order theory $T$ in any Boolean category. So there's really no sense in contrasting model theory with category theory; the former studies models of first-order theories in $\mathbf{Set}$, while the latter gives you the tools to study first-order models in much more general contexts. I don't see the contrast; as others have said, these are apples and oranges.

2. If we're going to study a theory $T$, then morphisms between models of $T$ should be assumed to preserve everything that can be expressed in the language of $T$. For example, if $T$ is an algebraic theory, we should consider homomorphisms between $T$-models. If $T$ is a first-order theory, then we should consider elementary embeddings between $T$-models. In fact (although I admit to not having much knowledge in this area), I think that category theory usually chooses the morphisms for you, automatically. For example, if $L$ is an Lawvere theory and $X,Y : L \rightarrow \mathbf{C}$ are models of $L$ in a finite product category $\mathbf{C}$, then a homomorphism $\varphi : X \rightarrow Y$ is just a natural transformation $X \Rightarrow Y$. We don't really get much choice in the matter!

3. Let me expand on the above point a little. If $T$ is a first-order theory in the language of $\{\in\}$ and $X$ and $Y$ are $T$-models, then there is nothing natural about functions $f : X \rightarrow Y$ satisfying $x \in y \rightarrow f(x) \in f(y)$. Remember, $T$ is best viewed as a first-order theory, not as some particular presentation for that theory! The fact that we decided to axiomatize $T$ using the signature $\{\in\},$ rather than $\{\subseteq,x \mapsto \{x\}\}$ or something else entirely, is irrelevant, and in any reasonable definition of "first-order theory", the theory $T$ would not "remember" how it was defined. So we really do have to preserve all the structure that $T$ can express; we don't get a choice in the matter.

4. Let $\mathsf{AlgGrp}$ denote the algebraic theory of groups. Let $\mathsf{1stGrp}$ denote the 1st-order theory freely generated by $\mathsf{AlgGrp}$, whatever that means. Write $\mathbf{AlgGrp}$ and $\mathbf{1stGrp}$ for the corresponding categories of models in $\mathbf{Set}$. Then there is a forgetful functor $\mathbf{1stGrp} \rightarrow \mathbf{AlgGrp}.$ It is faithful and surjective on objects. But it is not full.

5. On the other hand, if $T$ is an algebraic theory and $F(T)$ denotes the coherent theory freely generated by $T$, then (someone correct me if I'm wrong here) the corresponding categories of models in $\mathbf{Set}$ should be equivalent.

6. Category theory isn't inherently restricted to studying well-behaved categories, in much the same way as order-theory isn't inherently restricted to studying lattice-orders. Its harder if your category is missing lots of limits and colimits, etc. - just like its harder to understand a poset thats missing most of its meets and joins - but it can be done. The fact that categories like $\mathbf{1stGrp}$ whose morphisms are elementary embeddings lack most limits and colimits doesn't make them impossible to study. Although of course, the kinds of results you can expect to prove are different. Your theorems will end up "feeling" much more model-theoretic than algebraic, but you can still use category theory to prove them.

7. Let me just reiterate that category theory is bigger than algebra. True, its most successful application domain is (currently) in algebra, but as it matures, it will begin to be used more-and-more frequently to study categories of models of much more expressive theories than algebraic theories.

Answer 5

Question #3: What are the specific strengths and weaknesses of CT and MT, compared to each other?

CT probably copes better with objects of infinite nature ?