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

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 2

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 ?