Functorial first order theories interpretation

Question

Question will be a bit naive, so please, be kind.

Consider first order theories, $\Gamma, \Gamma'$ . Let $\mathcal{M}$ be the category of models for $\Gamma$ and $\mathcal{M}'$ be the category of models for $\Gamma'$, where morphism are arrows who respect the structure.

Let $U$ be a functor $U:\mathcal{M} \rightarrow \mathcal{M}'$. Does $U$ induces any kind of interpretation of $\Gamma$ in $\Gamma$'?

Answer 1

I believe that the answer in general has to be "no". This would be along the lines of the comments requiring additional restrictions on the functor $U$. Here is my attempted argument.

Let $\Gamma$ be the theory of abelian groups, let $\Gamma'$ be the theory of (arbitrary) groups, so then $\mathcal{M}$ is the category of abelian groups and $\mathcal{M}'$ is the category of (arbitrary) groups. We can then obviously define at least one functor $U: \mathcal{M} \to \mathcal{M}'$, namely the forgetful functor. But I don't think that there can be any interpretations of $\Gamma$ in $\Gamma'$ that preserve syntactic consequence, so no models (as theories) of $\Gamma$ in $\Gamma'$. (Obviously though there are models of $\Gamma'$ in $\Gamma$.)

Actually maybe that is a bad example, because it might actually just be suggesting that functors $U: \mathcal{M} \to \mathcal{M'}$ should correspond to models/theory morphisms/interpretations preserving syntactic consequence $\Gamma' \to \Gamma$ (i.e. as opposed to $\Gamma \to \Gamma'$).

So if nothing else, keep in mind that if there is a relationship, regardless of whether restrictions are required on $U$ in general or not, $U$ might actually be a contravariant functor, and not a covariant functor (as the question seems to have assumed). Maybe that relates to hyperdoctrines corresponding to contravariant functors (cf. this related question Hyperdoctrines and Contravariance ) -- I really don't know.

That having been said, I am also interested in the question of which restrictions needs to define on $U$ in general for this to be true.

Note that first-order theories can be interepreted themselves as individual categories, a "syntactic category" https://ncatlab.org/nlab/show/syntactic+category. I do not know for certain whether this is true, but this would seem to suggest that the "model categories" $\mathcal{M}$ and $\mathcal{M}'$ can in turn be considered "categories of categories", i.e. each object of $\mathcal{M}$/$\mathcal{M}'$ could be considered a category in some way, such that the morphisms between objects within $\mathcal{M}$ and $\mathcal{M}'$ can be considered functors.

(According to a subsection of the article on nLab, it seems that the category $\mathcal{M}$ of models of a theory $\Gamma$, at least when using classical logic/Boolean or De Morgan toposes, might correspond to a sheaf category, which is a subcategory of a presheaf category, which in turn is a functor category. But I don't really understand.)

Hence while you can in general define arbitrary kinds of morphisms between "functor categories", cf. a related question of mine "Alternatives" to Natural Transformations , in general it is desirable for the morphisms to be natural transformations. So when defining a functor $U: \mathcal{M} \to \mathcal{M}'$, ideally its components should probably correspond to a "natural transformation" in some way. But this would seem to be getting towards 2-category theory, which I won't pretend to understand.

The point I'm getting at is that the fact that theories $\Gamma$ and $\Gamma'$ themselves can be considered as having "(1-)categorical structure" (via interpretation as syntactical categories $C_{\Gamma}$, $C_{\Gamma'}$, suggests that $\mathcal{M}$/$\mathcal{M}'$ could be interpreted in an appropriate way such that they have "2-categorical structure" of some kind (not merely just "(1-)categorical structure"). Hence the "natural" (excuse the pun) place to begin looking for restrictions on the functor $U: \mathcal{M} \to \mathcal{M}'$ would be restrictions that ensure it respects that additional "2-categorical structure" as well. (I.e. whereas arbitrary functors only respect "(1-)categorical structure" by default.)

In particular it might be helpful if you can figure out how to interpret interpretations/models of one theory $\Gamma \to \Gamma'$ into another/inside of the other as functors between their corresponding syntactic categories $C_{\Gamma} \to C_{\Gamma'}$. Unfortunately I'm not sure whether this is possible and the nLab article on syntactic categories does not seem to mention this either way.