$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Elem{Elem}$Background/Motivation: Inspired by an interesting question by Joel, I’ve been wondering about the relationship between two very natural ways to define the category of “all models of $T$” where $T$ is a first-order theory.
Let us assume that $T$ is a complete theory with infinite models. Then on the one hand we can define the category $\Mod(T)$ whose objects are all the models of $T$ and whose morphisms are all homomorphisms in the sense of model theory – i.e. functions $\varphi\colon M \rightarrow N$ such that
For any $n$-ary relation $R$ in the language of $T$, if $M \models R^M(a_1, \ldots, a_n)$, then $N \models R^N(\varphi(a_1), \ldots, \varphi(a_n))$;
and
$\varphi(f^M(a_1, \ldots, a_n)) = f^N(\varphi(a_1), \ldots, \varphi(a_n))$ for any $n$-ary function symbol $f$ in the language of $T$.
Also, one can define another category $\Elem(T)$ whose objects are also all the models of $T$, but whose morphisms are only the elementary embeddings, that is, functions which preserve the truth of all first-order formulas. As a model theorist, I’m more used to thinking about the category $\Elem(T)$, and this latter category arises naturally if one cares about which sets are definable but one does not particularly care about which sets are definable by positive quantifier-free formulas.
Question: What sorts of category-theoretic properties automatically transfer from $\Mod(T)$ to $\Elem(T)$, or from $\Elem(T)$ to $\Mod(T)$?
To be clear, by a “category-theoretic property” I mean something that is preserved by an equivalence of categories.
Another related question is:
Question: Suppose we have a set of category-theoretic properties which we know characterize all the categories $C$ which are equivalent to $\Mod(T)$ for some $T$ [or to $\Elem(T)$ for some $T$]. Can we use this to characterize the categories which are equivalent to $\Elem(T)$ [respectively, $\Mod(T)$] for some $T$?
Here are a couple of basic facts I know, which may or may not be useful here. First of all, every category $\Elem(T)$ is equivalent to a category $\Mod(T')$ for some other theory $T'$ – namely, the “Morleyization” $T'$ of $T$, where we expand the language by adding new predicates for every definable set (and iterating $\omega$ times), thereby forcing $T'$ to have quantifier elimination. However, it is certainly not true that every category $\Mod(T)$ is equivalent to a category of the form $\Elem(T')$ – for instance, $\Mod(T)$ might not have colimits of $\omega$-directed chains, but $\Elem(T)$ always will (by Tarski’s elementary chain theorem).
Addendum: As Joel pointed out, there is a third possible notion of “morphism” for this category: the “strong homomorphisms” $\varphi\colon M \rightarrow N$ which commute with the interpretations of function symbols and have the property that for any $n$-ary relation $R$ in the language of $T$, $$M \models R^M(a_1, \ldots, a_n) \iff N \models R^N(\varphi(a_1), \ldots, \varphi(a_n)).$$
I’d also be interested to learn about any relationships between the category of models with strong homomorphisms and the other two categories above.
Best Answer
Well, that's not an answer to your question but perhaps it is still a comment you might useful. I happend to have spent some time thinking about how to define a category of models of a theory, and it appeared to me that a reasonable way is to consider the category of families of models of a theory. There are not much indications that this is a good way, though, but still, you can define something like a model category structure on this category, and, perhaps more importantly, this model category structure does give rise to an interesting set-theoretic invariant (the covering number) in the (model theoretically degenerate) case of the theory of equality (and is an actually model category). I am not sure whether you find this helpful, though, and it is definitely not an answer to your question. Actually, I've written a text about this, sort of a report on whatever little I understand now...