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

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.

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...

• Thank you very much for the thorough answer. What does $|T|^+$ mean? Jan 5 at 0:54
• @GregoryNisbet $|T|$ is the cardinality of the set of formulas in the language of $T$, up to equivalence modulo $T$. So usually you would have $|T| = |L| + \aleph_0$, where $L$ is the language of $T$ ('usually' because you could do something like adding a lot of distinct constant symbols and declaring the all to be equal in $T$). Of course, $|T|^+$ is just the next cardinal after $|T|$. Jan 5 at 11:05

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.