What (if anything) can we infer about a first order theory from its category of models?

Question

Does the structure of the category of models of a given first order theory where the arrows are embeddings tell us anything interesting about the theory itself?

Two models $M$ and $M'$ are isomorphic if and only if there exists an embedding $f : M \to M'$ whose inverse $f^{(-1)}$ is an embedding from $M'$ to $M$. Also, embeddings compose. Thus, we have a category.

As a trivial observation, we can rehash the Löwenheim–Skolem theorem in categorical language, if we take some effort to remove the finite models ahead of time.

Picking a satisfiable theory $T$ with no finite models produces a category $C$. If we take $C$ and produce a new category $D$ that identifies all the arrows in every homset in $C$, we get a category that is isomorphic to the category of infinite cardinals, where the homset $\text{Hom}(\kappa, \lambda)$ is a singleton if and only if $\kappa \le \lambda$ and $\text{Hom}(\kappa, \lambda)$ is empty otherwise.

The above statement is a consequence of the Löwenheim–Skolem theorem; it might be equivalent, though, if "cardinal gaps" are always detectable by the structure of the category.

Answer 1

We can say a lot about the first-order theory by just looking at its category of models. Let's use elementary embeddings instead of just embeddings. They are the 'right' arrows to take because they respect first-order logic. You could also work with just embeddings to get some sort of positive logic, which is also interesting. But that is harder and you asked about first-order logic.

So just to remind you: for structures $M$ and $N$ we call a function $f: M \to N$ an elementary embedding if for every first-order formula $\phi(\bar{x})$ and every tuple $\bar{a} \in M$ we have $$ M \models \phi(\bar{a}) \quad \Longleftrightarrow \quad N \models \phi(f(\bar{a})). $$ For a first-order theory $T$ let us denote by $\mathbf{Mod}(T)$ the category of models of $T$ with elementary embeddings.

Indeed, we can already detect the Löwenheim-Skolem theorem. For this we have the language of accessible categories. Roughly this works as follows: we can give a purely category-theoretic definition of what a $\lambda$-presentable object is (see here, although there they call it "compact" instead of "presentable"). In $\mathbf{Mod}(T)$, for any $\lambda > |T|$, we have that an object $M$ is $\lambda$-presentable iff $|M| < \lambda$. So this gives us a notion of size. Then we can formulate the Löwenheim-Skolem theorem roughly as "every object is a colimit of small (i.e. $< |T|^+$) objects".

One easy thing that we can see is whether or not $T$ is complete. Namely, $T$ is complete exactly when $\textbf{Mod}(T)$ has the Joint Embedding Property (JEP). JEP means that whenever we have objects $M_1$ and $M_2$ that then there is an object $N$ and arrows $M_1 \to N \leftarrow M_2$. It should be a nice exercise to prove that this is equivalent to being complete.

Another thing that we can easily see from $\mathbf{Mod}(T)$ is whether or not the theory is $\kappa$-categorical for any $\kappa$. This is again done using the notion of presentability. We can detect the 'size' of an object purely category-theoretically and then we can just ask if all the objects of that 'size' are isomorphic.

Just to contrast, let's look at something that $\textbf{Mod}(T)$ cannot detect. It cannot detect whether or not $T$ has quantifier elimination. Indeed, for any theory $T$ we can add a relation symbol $R_\phi(\bar{x})$ for every formula in the language of $T$ and then let $T'$ extend $T$ by saying $\forall \bar{x}(\phi(\bar{x}) \leftrightarrow R_\phi(\bar{x}))$ (we call $T'$ the Morleyisation of $T$). Then $T'$ has quantifier elimination, but $\textbf{Mod}(T)$ and $\textbf{Mod}(T')$ are clearly equivalent categories.

Finally I should say something about classification theory. This is a very deep subject where the aim is to classify theories by how well-behaved they are. The class of stable theories is the most well-known here and the most studied, but there are also more general classes like simple theories. We can detect whether or not a theory is stable or simple (or neither) from its category of models! I think that we can do much more, but this is all very recent research...

Answer 2

There is a very interesting result by Campion, Cousins, and Ye in the context of elementary embeddings. In particular, the Lascar group of a first order theory $T$ is isomorphic to the fundamental group of the classifying space of $Mod(T)$ (where, $Mod(T)$ is the category of models of $T$ with elementary embeddings).

Here is a link to the paper.