I have some questions about the interplay of interpretability, model theory and category theory. Since I had difficulties in finding literature or other helpful information about this topic, it would be great if somebody of you can help me. 


For a first-order theory $T$, let $Mod(T)$ denote the category of all models of $T$ where the arrows are the homomorphisms (or - if you want so - take the elementary embeddings). Let $I\_T$ denote the spectrum of $T$. That is, $I\_T$ is a function which takes some cardinal $\kappa$ as argument and which outputs the number of non-isomorphic models of $T$ having cardinality $\kappa$. I will also refer to the syntactical notion of interpretability (i.e. "theory $T\_1$ interprets theory $T\_2$") and mutual interpretability ("$T\_1$ interprets $T\_2$ and vice versa"). For what follows, assume that $T\_1$ and $T\_2$ are arbitrary first-order theories. Here are the three questions for which I would like to know whether there are any answers: 


1. Are $Mod(T\_1)$ and $Mod(T\_2)$ equivalent (in the category theoretical sense) whenever $T\_1$ and $T\_2$ are mutually interpretable? Does the converse holds?
2. Are $Mod(T\_1)$ and $Mod(T\_2)$ equivalent (in the category theoretical sense) whenever $I\_{T\_1}=I\_{T\_2}$? Does the converse holds?
3. Does $I\_{T\_1}=I\_{T\_2}$ holds whenever $T\_1$ and $T\_2$ are mutually interpretable? Does the converse holds?


I will be grateful for any comments, answers and also for references to literature (papers, books,...) in which these questions or similar ones are discussed. Thank you!


Note that I do not expect that these questions are "profound conjectures" or something like that. Maybe that among the experts, their answers are already known and even "trivial". I was just thinking about some model theoretical topics and in this context, the three questions from above started to occupy me. Since I am not able to find any literature concerned with similar questions, I just would like to know whether there are some people in this forum who know more than I do and could share their knowledge with me.


**Edit**: Following some remarks of some of the commentators of this post, the above questions may also be interesting if we replace "mutual interpretable" with "bi-interpretable". 



The interplay between certain aspects of interpretability, model theory and category theory
Here is a negative answer to question 2 and the converse of question 3.


Let $T\_1$ be the theory of the integers under successor $\langle\mathbb{Z},S\rangle$. This theory asserts that $S$ is bijective and has no cycles of any finite length. That theory is complete, since all models of size $\aleph\_1$ are isomorphic, consisting of $\aleph\_1$ many $\mathbb{Z}$-chains, and more generally there is only one model of uncountable size $\kappa$ for any uncountable $\kappa$. But there are countably infinite many countable models up to isomorphism, since every model consists of some countable number of $\mathbb{Z}$-chains. So $I\_{T\_1}$ is the function that says there are $\aleph\_0$ many countably infinite models, but only one model each in any uncountable cardinality. 


Let $T\_2$ be the theory of infinitely many distinct constants $c\_n$. There are countably infinitely many countable models of this theory, depending on the number of objects in the model that are not the interpretation of any $c\_n$, but there is only one model up to isomorphism in any uncountable cardinality, since any model of size $\kappa$ has the distinct $c\_n$'s and then $\kappa$ many additional points. 


Thus, both of these theories have the same number of models in any cardinality: no finite models, $\aleph\_0$ many countable models, one model in each uncountable cardinality. So $I\_{T\_1}=I\_{T\_2}$. 


But $T\_1$ is not interpretable in $T\_2$, since if it were, the interpretation would involve only finitely many constants, but one can define only finitely many points in a model of the reduct of $T\_2$ to those finitely many constants plus one parameter, whereas in any model of $T\_1$, we can define infinitely many points relative to any parameter.
So this is a counterexample to the converse of question 3. 


Note also that $T\_1$ has no rigid models, since every $\mathbb{Z}$-chain admits translations, but $T\_2$ has two rigid models, namely, the model with only the constants, and the model with the constants plus one more point. (Also, $T\_2$ has models with only finitely many automorphisms.) These features show that $\text{Mod}(T\_1)$ and $\text{Mod}(T\_2)$ are not equivalent as categories, so this provides a negative instance to question 2.


