TITLE: The interplay between certain aspects of interpretability, model theory and category theory
QUESTION [14 upvotes]: 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: 

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?
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?
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".

REPLY [4 votes]: Regarding the first question: 
(a) Theories with equivalent categories of models are not necessarily bi-interpretable.  One can come up with simple examples in propositional logic (first-order logic with only zero-place relation symbols).  For example, let $T_1$ be the empty theory in the language $\{ p_0,p_1,\dots \}$, and let $T_2$ be the theory with axioms $\{ p_0 \vdash p_i \}$ for all $i$.  Both $Mod(T_1)$ and $Mod(T_2)$ are discrete categories with $2^{\aleph _0}$ elements, hence they are equivalent.
(b) Bi-interpetable theories do have equivalent categories of models.  This result follows from the fact that an interpretation $F:T_1\to T_2$ gives rise to a functor $F^*:Mod(T_2)\to Mod(T_1)$; and if $T_1$ and $T_2$ are bi-interpretable, then $F$ has a "pseudo-inverse" $G$ which gives rise to a inverse $G^*$ to $F^*$.  
The previous paragraph was a bit sketchy. A more elegant argument can be given using categorical logic: the theory $T_i$ gives rise to a syntactic category $C_{T_i}$.  If $T_1$ and $T_2$ are bi-interpretable, then the pretopos completion of $C_{T_1}$ is equivalent to the pretopos completion of $C_{T_2}$.  It then follows that the category $LF(C_{T_1},Sets)$ of logical functors from $C_{T_1}$ into the category of sets is equivalent to the category $LF(C_{T_2},Sets)$ of logical functors from $C_{T_2}$ into the category of sets.  But $LF(C_{T_i},Sets)$ is equivalent to $Mod(T_i)$.   [These ideas are all contained in the latter half of Makkai and Reyes, First order categorical logic.]
Note: I am taking "bi-interpretable" as synonymous with "definitional equivalence."  Note that mutual interpretability does not imply bi-interpretability.  See Andreka et al., "Mutual definability does not imply definitional equivalence, a simple example."  http://www.math-inst.hu/pub/algebraic-logic/amn-definote.pdf