2.1 Category theoretic preliminaries

We restrict our attention to categories that are small; i.e., where the objects and arrows of the category form a set. We assume that the reader is familiar with the basic axioms of category theory and the definitions of functors and natural transformations.Footnote ²

For our first equivalence relation, recall that categories $\mathcal {C}$ and $\mathcal {D}$ are isomorphic if there exist functors $F:\mathcal {C}\Rightarrow \mathcal {D}$ and $G:\mathcal {D}\Rightarrow \mathcal {C}$ such that $G\circ F=Id_{\mathcal {C}}$ and $F\circ G=Id_{\mathcal {D}}$ where $Id_{\mathcal {C}}$ and $Id_{\mathcal {D}}$ are the respective identity functors for $\mathcal {C}$ and $\mathcal {D}$ . In other words, F and G are such that:

• • for all objects A from $\mathcal {C}$ , $G(F(A))=A$ ;

• • for all arrows $h:A\to B$ from $\mathcal {C}$ , $G(F(h))=h$ ;

• • for all objects B from $\mathcal {D}$ , $F(G(A))=A$ ; and

• • for all arrows $i:C\to E$ from $\mathcal {D}$ , $F(G(i))=i$ .

Intuitively speaking, the functors take us forth and back returning us to exactly where we started. For our second equivalence relation, we first recall the following definition.

Definition 2.1. A natural transformation $\eta $ between functors $F,G:\mathcal {C}\Rightarrow \mathcal {D}$ is a natural isomorphism if for all objects A in $\mathcal {C}$

$$\begin{align*}\eta_{A}:F(A)\cong G(A). \end{align*}$$

Let us write $\eta _{\cdot }:F\cong _{nat}G$ to indicate this.

Let $Id_{\mathcal {C}}$ and $Id_{\mathcal {D}}$ be the identity functors for $\mathcal {C}$ and $\mathcal {D}$ respectively. Then recall that categories $\mathcal {C}$ and $\mathcal {D}$ are equivalent Footnote ³ if there are functors $F:\mathcal {C}\Rightarrow \mathcal {D}$ and $G:\mathcal {D}\Rightarrow \mathcal {C}$ such that $Id_{\mathcal {C}}$ and $G\circ F$ are naturally isomorphic and so are $Id_{\mathcal {D}}$ and $F\circ G$ . In other words, there is a natural transformation $\eta _{\cdot }$ from $Id_{\mathcal {C}}$ to $G\circ F$ such that for all objects A from  $\mathcal {C}$

$$\begin{align*}\eta_{A}:A\cong G(F(A)) \end{align*}$$

and a natural transformation $\nu _{\cdot }$ from $Id_{\mathcal {D}}$ to $F\circ G$ such that for all objects B from $\mathcal {D}$

$$\begin{align*}\nu_{D}:B\cong F(G(B)). \end{align*}$$

Or more compactly, $\eta _{\cdot }:Id_{\mathcal {C}}\cong _{nat}G\circ F$ and $\nu _{\cdot }:Id_{\mathcal {D}}\cong _{nat}F\circ G$ .

We see that equivalence is a natural weakening of isomorphism in that the functors take us back and forth to an object that is isomorphic to the one we started with but not necessarily identical to it. The fact that the isomorphisms $\eta _{\cdot }$ and $\nu _{\cdot }$ are natural ensures a kind of uniformity in the relationship between the isomorphism occurring at the object level. This provides our second equivalence relation. In contrast, our novel, third equivalence relation will abandon this uniformity requirement. We’ll describe this in a moment, but first we note a convenient alternative definition of equivalence between categories that will be useful later.

For our third equivalence relation, we move off the main menu. Informally speaking, my goal is to weaken categorical equivalence in such a way that we go back and forth to an isomorphic object, but we do not demand that the isomorphism is natural or uniform. I note—up front—that in most categorical settings this equivalence relation will be too weak to be of much significance. However, in the setting of categories based on theories it has a natural counterpart in relative interpretability. This is our reason for introducing it here.Footnote ⁵

To close the current section we observe that—in general—these equivalence relations are organized in a strict hierarchy:

2.2 Categories of theories

In the previous section, we described three equivalence relations between categories: isomorphism; equivalence; and objective equivalence. We now aim to apply these tools to makes comparisons between theories. As such, we define a natural category for the purposes of representing theories. To facilitate this, we assume for the rest of the paper that there is an inaccessible cardinal and that $\Omega $ is the least of them. This will allow us to treat our theory categories as small when their most obvious definition does not. Thus, unless otherwise stated our background theory for the rest of this paper will be $ZFC$ plus the assumption that an inaccessible cardinal exists.Footnote ⁶

Given a semantic outlook, the choice of objects for this category is—more or less—forced upon us, but with regard to arrows more options are available. For example one might also consider using: homomorphisms; embeddings or isomorphisms. Some helpful discussion around this is provided in [Reference Barrett and Halvorson4], however, for our purposes elementary embeddings provide a very natural fit with relative interpretability. We also note that at various times in the paper, we shall have cause to restrict our attention to various subcategories of $mod(T)$ as in some contexts it will be appropriate to consider particular kinds of models or embeddings. We now recall our equivalence relations as they are instantiated between theory categories.

2.2.1 Hierarchy?

It is then natural to ask whether the hierarchy described in Proposition 2.4 remains strict in context of theory categories. We now show that there are objectively equivalent theories that are not equivalent, but every pair of equivalent theories turns out to be isomorphic.

Proposition 2.7. There is a pair of theories that are objectively equivalent but not equivalent as categories.

Proof. Let T be the theory that says there are exactly two objects; and let S be the theory that says that there is exactly one object. To see that they are objectively equivalent, we describe functors $F:mod(T)\Leftrightarrow mod(S):G$ satisfying Definition 2.3. Since both T and S have exactly one isomorphism class, we may let the action of F on the objects of $\mathcal {C}$ be given by an arbitrary bijection between the models of T and S. Note that for any models $\mathcal {M}$ and $\mathcal {N}$ of S, there is exactly one elementary embedding between them. Thus, given $f:\mathcal {M}\to \mathcal {N}$ in $mod(T)$ , we let $F(f)$ be the unique arrow between $F(\mathcal {M})$ and $F(\mathcal {N})$ . In the other direction, we let $G(\mathcal {A})=F^{-1}(\mathcal {A})$ for models $\mathcal {A}$ of S. For arrows, we need to make some decisions. For any arrow $g:\mathcal {A}\to \mathcal {B}$ in $mod(S)$ , there will be two arrows between $G(\mathcal {A})$ and $G(\mathcal {B})$ in $mod(T)$ . To address this, we simply pick an arrow and its inverse between any two models of T. We then let $G(g)$ be that arrow; and for the arrow $g^{-1}:\mathcal {B}\to \mathcal {A}$ , we let $G(g^{-1})$ be its inverse. This makes sense since every elementary embedding between finite models is an isomorphism and thus has an inverse. It can then be seen that F and G are functors that witness an objective equivalence. We leave it to the reader to verify that $mod(T)$ and $mod(S)$ are not equivalent as categories.

We now show that equivalence between theory categories implies isomorphism. To see this, first observe that in any theory category $mod(T)$ the cardinality of any isomorphism class is $\Omega $ . More precisely, for any object $\mathcal {M}$ from $mod(T)$

$$\begin{align*}|[M]|=\Omega. \end{align*}$$

To see this, consider, for example, the category of models of Peano arithmetic, $mod(PA)$ . It is easy to see that for every ordinal $\alpha <\Omega $ , there will be a model of $PA$ that includes $\alpha $ in its domain. This means that following lemma suffices to prove this theorem.

Lemma 2.8. Suppose $\mathcal {C}$ and $\mathcal {D}$ are equivalent as categories and suppose that each of the isomorphism classes of these categories has the same cardinality; i.e., for all objects A from $\mathcal {C}$ and B from $\mathcal {D}$

$$\begin{align*}|[A]|=|[B]|. \end{align*}$$

Then $\mathcal {C}$ and $\mathcal {D}$ are isomorphic.

Proof. Suppose $F:\mathcal {C}\to \mathcal {D}$ is a functor witnessing the categorical equivalence; i.e., F is full, faithful and essentially surjective. We shall use F to define an isomorphism $H:\mathcal {C}\cong \mathcal {D}$ . To do this we shall pick a distinguished element from each of the isomorphism classes of $\mathcal {C}$ use the action of F on these elements to generate H’s action on the rest of  $\mathcal {C}$ .

We start by defining two endo-functors and natural transformations. First let $\cdot ^{*}:\mathcal {C}\to \mathcal {C}$ be an endo-functor that takes objects A from $\mathcal {C}$ and returns a distinguished element $A^{*}$ of $[A]$ such that for all objects $A_{0},A_{1}$ from $\mathcal {C}$ , $A_{0}^{*}=A_{1}^{*}$ whenever $A_{0}\cong A_{1}$ . Then let $\pi _{A}:A\cong A^{*}$ be an isomorphism such that $\pi _{A^{*}}=id_{A^{*}}$ .Footnote ⁹ For arrows $f:A\to B$ from $\mathcal {C}$ , let $f^{*}:A^{*}\to B^{*}$ be

$$\begin{align*}\pi_{B}\circ f\circ\pi_{A}^{-1}. \end{align*}$$

Note that $\pi _{\cdot }$ is a natural isomorphism from the identity functor on $\mathcal {C}$ to $\cdot ^{*}$ .

Next we define a similar endo-functor $\cdot ^{\dagger }:\mathcal {D}\to \mathcal {D}$ that is tailored to match up with  $\cdot ^{*}$ . Note that since F establishes categorical equivalence it is essentially surjective and thus for any object E from $\mathcal {D}$ we may fix an object A from $\mathcal {C}$ such that $F(A)\cong E$ . So for all objects E from $\mathcal {D}$ , let $E^{\dagger }=F(A^{*})$ where A is some object from $\mathcal {C}$ such that $F(A)\cong E$ .Footnote ¹⁰ Then observe that

$$\begin{align*}E^{\dagger}=F(A^{*})\cong F(A)\cong E \end{align*}$$

so $E^{\dagger }\in [E]$ . For each object E from $\mathcal {D}$ , let $\rho _{E}:E\cong E^{\dagger }$ .Footnote ¹¹ For an arrow $g:E\to D$ from $\mathcal {D}$ , let $g^{\dagger }:E^{\dagger }\to D^{\dagger }$ be

$$\begin{align*}\rho_{D}\circ g\circ\rho_{E}^{-1}. \end{align*}$$

Note that $\rho _{\cdot }$ is a natural isomorphism between the identity functor on $\mathcal {D}$ and $\cdot ^{\dagger }$ .

Our goal now is to define H using $\cdot ^{*},\cdot ^{\dagger },\pi _{\cdot }$ and $\rho _{\cdot }$ . For all $A^{*}$ from $\mathcal {C}$ , let $H_{A^{*}}:[A^{*}]\to [F(A^{*})]$ be a bijection such that $H_{A^{*}}$ and F agree on $A^{*}$ ; i.e., $H_{A^{*}}(A^{*})=F(A^{*})$ . This makes sense given our assumptions about the cardinality of isomorphism classes in $\mathcal {C}$ and $\mathcal {D}$ . Now let $H:\mathcal {C}\to \mathcal {D}$ be defined as follows. For objects A from $\mathcal {C}$ , we let

$$\begin{align*}H(A)=H_{A^{*}}(A). \end{align*}$$

For arrows $f:A\to B$ from $\mathcal {C}$ , let $H(f):H(A)\to H(B)$ be

$$\begin{align*}\rho_{H(B)}^{-1}\circ F(f^{*})\circ\rho_{H(A)}. \end{align*}$$

To see that this makes sense, the following diagram is helpful.

Now we claim that H is a functor and that H witnesses that $\mathcal {C}$ and $\mathcal {D}$ are isomorphic as categories. We leave the reader to verify that H is a functor.

Claim. H establishes an isomorphism between $\mathcal {C}$ and $\mathcal {D}$ .

Proof. By design we know that H is a bijection on objects, so it suffices to show that H is full and faithful.

To see that H is full, suppose $g:H(A)\to H(B)$ is an arrow from $\mathcal {D}$ . It will suffice to show there is some $f:A\to B$ such that $H(f)=g$ . To see this first observe that $\rho _{H(B)}\circ g\circ \rho _{H(A)}^{-1}$ is an arrow from $H(A^{*})$ to $H(B^{*})$ . Then since $H(A^{*})=F(A^{*})$ and $H(B^{*})=F(B^{*})$ and F is full, we may fix some arrow $i:A^{*}\to B^{*}$ from $\mathcal {C}$ such that $F(i)=\rho _{H(B)}\circ g\circ \rho _{H(A)}^{-1}$ . We then let $f=\pi _{B}^{-1}\circ i\circ \pi _{A}$ and observe that this is an arrow from A to B. We then see chase the diagram observing that

$$ \begin{align*} H(f) & =\rho_{H(B)}^{-1}\circ F(f^{*})\circ\rho_{H(A)}\\ & =\rho_{H(B)}^{-1}\circ F(\pi_{B}\circ f\circ\pi_{A}^{-1})\circ\rho_{H(A)}\\ & =\rho_{H(B)}^{-1}\circ F(i)\circ\rho_{H(A)}\\ & =\rho_{H(B)}^{-1}\circ(\rho_{H(B)}\circ g\circ\rho_{H(A)}^{-1})\circ\rho_{H(A)}=g. \end{align*} $$

To see that H is faithful, suppose $f,g:A\to B$ are such that $H(f)=H(g)$ . We claim that $f=g$ . To see this first recall that

$$\begin{align*}H(f)=\rho_{H(B)}^{-1}\circ F(f^{*})\circ\rho_{H(A)}\ \ \&\ \ H(g)=\rho_{H(B)}^{-1}\circ F(g^{*})\circ\rho_{H(A)} \end{align*}$$

and so we see that $F(f^{*})=F(g^{*})$ . And since F is faithful, we see that $f^{*}=g^{*}$ . Then recall that

$$\begin{align*}f^{*}=\pi_{B}\circ f\circ\pi_{A}^{-1}\ \ \&\ \ g^{*}=\pi_{B}\circ g\circ\pi_{A}^{-1} \end{align*}$$

and so $f=g$ as required.

Our main claim then follows directly.

Corollary 2.9. If $mod(T)$ is equivalent to $mod(S)$ , then they are also isomorphic.

Thus, we see that when we restrict our attention to theory categories, the hierarchy we had in Proposition 2.4 is no longer strict. This may cause us to question the value separating them in our framework. I have a couple of things to say about this. First a small point: it is sometimes said in defense of categorical equivalence that it is superior to isomorphism in that it wipes away detail that is not pertinent when it comes to theory comparison. The corollary above tells us that, when it comes to theories, we are—in fact—concerned with isomorphism because it is the same thing as equivalence. Second and more substantively, we shall see in Section 3 that there are natural restrictions of this framework in which this hierarchy question becomes more difficult and interesting.

3.1 Back to category theory

We are now ready to compare the equivalence notions of category with those from relative interpretability. We start by observing that functions between models of one theory and another that are determined by interpretations give rise to functors.

Proof. t takes models of T and returns models of S. To obtain a genuine functor we need to also explain the action of t on arrows in the $mod(T)$ category. Given an elementary embedding $j:\mathcal {M}\to \mathcal {N}$ , we let $t(j)$ be the restriction of j to $t(M)$ . To see $t(j):t(\mathcal {M})\to t(\mathcal {N})$ is an elementary embedding consider a sequence $\bar {m}$ from $t(M)$ and a formula $\varphi (\bar {x})$ from $\mathcal {L}_{S}$ . Then we see that

$$ \begin{align*} t(\mathcal{M})\models\varphi(\bar{m}) & \Leftrightarrow\mathcal{M}\models t(\varphi)(\bar{m})\\ & \Leftrightarrow\mathcal{N}\models t(\varphi)(j\bar{m})\\ & \Leftrightarrow t(\mathcal{N})\models\varphi(j\bar{m}). \end{align*} $$

It is then easy to see that t preserves the identity arrow and composition.

It is worth nothing that this is a place where we make essential use of our requirement that arrows in a theory category are elementary embeddings.Footnote ²³

This tells us that interpretations determine functors and thus we can see the relative interpretations as special cases of functors between theory categories. However in general, many functors will not be determined by interpretations. For example any functor that takes a model and returns a model with greater cardinality cannot be determined by an interpretation. We address this by showing that the background framework for our interpretative equivalences can be seen as a very natural restriction of the framework we used for our equivalences between theory categories. To make this idea clearer, we shall first abstract away a formal framework in which the equivalences of Section 2 can be understood. First recall that a 2-category is a generalization of the ordinary notion of category obtained by adding an extra sort of arrows and governing their interaction with objects and ordinary arrows with new axioms.Footnote ²⁴ In this section, we shall call; ordinary arrows, 1-morphisms; and the new arrows, 2-morphisms. From our point of view the key point is that 2-morphisms can be seen as arrows between 1-morphisms, while 1-morphisms are arrows between objects. We can then define 1-isomorphisms and 2-isomorphisms in the obvious way. We shall say that two (small) categories are 1-equivalent, if there are functors going back and forth between them that are 2-isomorphic; in other words, they are categorically equivalent. The classic example of a 2-category is obtained by taking: every small category $\mathcal {C}$ as an object; functors F between categories as 1-morphisms; and natural transformations $\eta _{\cdot }$ between those functors as 2-morphisms.Footnote ²⁵ Let us call this $2\text {-}\mathbb {CAT}$ . This motivates the following definition.

Let’s call this the theory framework. It is then easy to see that isomorphism and categorical equivalence can be articulated using $\mathbb {TH}$ . More specifically, we see that:

With this in hand, we can now define a restriction of the theory framework that fits perfectly with our definitions from interpretability.

Let us call this the definable theory framework. We the observe that the equivalence relations given by interpretability fit very naturally here.

Proof. (1) and (3) are immediate. (2) Suppose that T and S are strictly bi-interpretable. Then let $t:mod(T)\to mod(S)$ and $s:mod(S)\to mod(T)$ be such that there exist formulae $\psi _{t}(x,y)\in \mathcal {L}_{T}$ and $\psi _{s}(x,y)\in \mathcal {L}_{S}$ defining isomorphisms over T and S respectively that witness the strict bi-interpretation. For a model $\mathcal {M}$ of T, let $\tau _{\mathcal {M}}:\mathcal {M}\cong s\circ t(\mathcal {M})$ be the isomorphism defined by $\psi _{t}(x,y)$ in $\mathcal {M}$ . Similarly, for models $\mathcal {N}$ of S, let $\sigma _{\mathcal {N}}:\mathcal {N}\cong t\circ s(\mathcal {N})$ . This satisfies the back and forth condition for bi-interpretation, it suffices to show that $\tau _{\cdot }$ and $\sigma _{\cdot }$ are natural transformations. We just prove this for $\tau _{\cdot }$ as the proof for $\sigma _{\cdot }$ is similar. Thus, it will suffice to show that the following diagram commutes.

To see this first observe that for $x\in s\circ t(M)$ , $j(x)=s\circ t(j)(x)$ , since $s\circ t(j)$ is just a restriction of j. Now suppose that $x\in M$ . Then we see that

$$\begin{align*}s\circ t(j)(\sigma_{\mathcal{M}}(x))=j(\sigma_{\mathcal{M}}(x))=\sigma_{\mathcal{N}}(j(x)) \end{align*}$$

as required.

I’d like to suggest that this result motivates a somewhat subtle conceptual turning of the tables. While relative interpretability has had a long history in mathematical logic, categorical equivalence relations between theory categories are relatively new on the scene. As such, it seems natural to think of categorical approaches as generalizations of the core notions from interpretability. The results above suggest a different story. The categorical equivalence relations can be seen as the prototype from which the interpretative picture can be derived by a natural restriction on the theory framework. We shall put this idea to work in Section 5, by considering a further natural restriction of the theory framework.

3.1.1 Hierarchy?

As was the case with the categorical equivalence relations, it is natural to ask whether the equivalences given in Definition 3.5 are arranged in a strict hierarchy. I have only been able to provide partial answers to these questions. First we observe.

Proposition 3.11.

1. (1) If T and S are definitionally equivalent, then they are strictly bi-interpretable.

2. (2) If T and S are strictly bi-interpretable, then they are iso-congruent.

We now consider whether the top two positions collapse.

Problem 3.12. Is there a pair of theories that are strictly bi-interpretable but not definitionally equivalent?

We note first that there is a beautiful result from Visser and Friedman [Reference Visser and Friedman22] that could appear to establish that there is such a pair. However, it turns out that this is not the case since the interpretations used there are not identity-preserving. More specifically, they make use of a quotient interpretation.Footnote ²⁷ Nonetheless, the main theorem of that paper gives an indication of how rare such an example would be.

Theorem 3.13 [Reference Visser and Friedman22].

If T and S are sequentialFootnote ²⁸ theories that are strictly bi-interpretable, then T and S are definitionally equivalent.

The result is established using an internalization of the Cantor–Bernstein theorem. This tells us that if we are interested in comparing theories that can provide a foundation for mathematics, then they will certainly be sequential; and thus, whenever they are strictly bi-interpretable they are definitionally equivalent. This brings us to the lower two rungs on the ladder.

Problem 3.14. Is there a pair of theories that are iso-congruent but not strictly bi-interpretable?

The following provides a partial answer.

Proposition 3.15. If we restrict theory categories to countable models, there is a pair of theories that are iso-congruent but not strictly bi-interpretable.

Proof. Let D be the theory in the language $\mathcal {L}_{D}=\{<,d_{n}\}_{n\in \omega }$ which says that $<$ is a dense linear order without end points and that $d_{n}<d_{n+1}$ for all $n\in \omega $ . Let B be a theory in the language $\mathcal {L}_{B}=\{\prec ,b_{n}\}_{n\in \omega }$ that $\prec $ is dense linear order with no top point but with a bottom point that is $d_{0}$ and that $b_{n}<b_{n+1}$ for all $n\in \omega $ .Footnote ²⁹

First we show that iso-congruence. Let $t:mod(D)\to mod(B)$ discard everything below $d_{0}$ and preserve everything else. More precisely, let

$$ \begin{align*} \delta_{t}(x) & :=d_{0}<x\vee x=d_{0}\\ t(x\prec y) & :=x<y\\ t(d_{n}) & :=d_{n},\ \forall n\in\omega. \end{align*} $$

Let $s:mod(B)\to mod(D)$ discard the bottom element $b_{0}$ and then shift each $d_{n}$ up to the next remaining constant $b_{n+1}$ . More precisely, let

$$ \begin{align*} \delta_{s}(x) & :=x\neq b_{0}\\ \delta_{s}(x<y) & :=x\prec y\\ t(d_{n}) & :=b_{n+1},\ \forall n\in\omega. \end{align*} $$

Let $\mathcal {A}$ be a model of D. We claim that $s\circ t(\mathcal {A})\cong \mathcal {A}$ . To see this note that $s\circ t(\mathcal {A})$ is the submodel of $\mathcal {A}$ obtained by removing those elements $\leq d_{0}^{\mathcal {A}}$ . Observe $\mathcal {A}$ and $s\circ t(\mathcal {A})$ can be seen as infinite sequences of countable dense linear orders without endpoints demarcated by their interpretations of the $\langle d_{n}\rangle _{n\in \omega }$ sequence. Since these orders are countable, it can then be seen, via Cantor, that every such linear order on the sequence of $\mathcal {A}$ is isomorphic to every such linear order on the sequence in $s\circ t(\mathcal {A})$ . Thus $\mathcal {A}$ and $s\circ t(\mathcal {A})$ are clearly isomorphic. Similarly in the other direction, given a model $\mathcal {B}$ of B, we see that $t\circ s(\mathcal {B})$ is the submodel of $\mathcal {B}$ obtained by removing from the domain those elements below $b_{0}^{\mathcal {B}}$ and so it can be seen that $t\circ s(\mathcal {B})$ and $\mathcal {B}$ are isomorphic as required.

Remark Note, however, that these interpretations do not witness a strict bi-interpretation. This is because the isomorphism between $\mathcal {A}$ and $s\circ t(\mathcal {A})$ cannot be defined in $\mathcal {A}$ . To get some insight into this note that a formula $\sigma (x,y)$ representing such an isomorphism would need to be such that D entailed $\sigma (c_{n},c_{n+1})$ for every $n\in \omega $ .

To see that D and B are not strictly bi-interpretable, suppose toward a contradiction that they. Then fix $f:mod(D)\to mod(B)$ , $g:mod(B)\to mod(D)$ and formulae $\varphi _{D}(x,y)$ and $\varphi _{B}(x,y)$ from $\mathcal {L}_{D}$ and $\mathcal {L}_{B}$ respectively where:

• • $\mathcal {A}\cong g\circ f(\mathcal {A})$ for all models $\mathcal {A}$ of D; and

• • $\mathcal {B}\cong f\circ g(\mathcal {B})$ for all models $\mathcal {B}$ of B

and the relevant isomorphisms are defined by $\varphi _{D}$ and $\varphi _{B}$ respectively.

Now we observe that B and D are are too weak to define many functions.

Claim D cannot define any non-trivial isomorphisms.

Proof. Let $\mathcal {A}$ be a model of D. Then the only definable elements of $\mathcal {A}$ are those $d_{n}^{\mathcal {A}}$ for $n\in \omega $ . This means that the only bijections that can be defined in $\mathcal {A}$ are those that permute a finite some finite subsets of $\{d_{n}^{\mathcal {A}}\}_{n\in \omega }$ . But any bijection that gives such a finite permutation is not an isomorphism since it will break the required ordering on the $d_{n}$ sequence.

This means that if B and D are strictly bi-interpretable, then they are definitionally equivalent and so, in particular, D must be able to interpret B in such a way that the domain is preserved. Thus, the following claim suffices to establish the proposition.

Claim D cannot interpret B with a domain preserving translation.

Proof. See Appendix.

Note that proof illustrates a quite draconian limitation of interpretative techniques. We are blocked by what in this context seem like arbitrary restrictions on our ability to provide interpretations. This provides some motivation for the more general relations like categorical equivalence and Morita equivalence. We’d like to free ourselves of at least some of these bonds.

4.1 Multi-sorted languages

Our exposition here is very similar to Barrett and Halvorson [Reference Barrett and Halvorson4] and we’ll mainly aim to note divergences rather than provide a full exposition. A multi-sorted language $\mathcal {L}$ will consist of sorts, relations symbols and function symbols. We use $\sigma ,\tau ,\sigma _{0},\tau _{0},\sigma _{1},\tau _{1},\ldots $ for sorts. Relation symbols have an arity of the form $\sigma _{0}\times \cdots \times \sigma _{n}$ , where $\sigma _{0},\ldots ,\sigma _{n}$ are sorts from $\mathcal {L}$ . Function symbols f have an arity of the form $\sigma _{0}\times \cdots \times \sigma _{n}\to \sigma $ .Footnote ³⁰ For each sort $\sigma \in \mathcal {L}$ we shall have countably many variable symbols labeled with their sort $v_{0}^{\sigma },v_{1}^{\sigma },\ldots $ .Footnote ³¹ And we shall use $x^{\sigma },y^{\sigma }$ as metavariables of sort $\sigma $ .

A model $\mathcal {M}$ for $\mathcal {L}$ , will be a structure with a domain $M_{\sigma }$ for each sort $\sigma $ from $\mathcal {L}$ . Moreover for each relation symbol R from $\mathcal {L}$ with arity $\sigma _{0}\times \cdots \times \sigma _{n}$ , the interpretation of R in $\mathcal {M}$ , abbreviated $R^{\mathcal {M}}$ , will be a subset of $M_{\sigma _{0}}\times \cdots \times M_{\sigma _{n}}$ ; and for each function symbol with arity $\sigma _{0}\times \cdots \times \sigma _{n}\to \sigma $ , $f^{\mathcal {M}}$ will be a function from $M_{\sigma _{0}}\times \cdots \times M_{\sigma _{n}}$ to $M_{\sigma }$ . The terms and formulae of $\mathcal {L}$ can then be defined inductively in the obvious way as are the term denotation and satisfaction relations.Footnote ³² We’ll use $\varphi ,\psi ,\chi $ as metavariables for formulae and $s_{0},\ldots ,s_{n}$ as metavariables for terms.

4.2 Interpretation

We now define interpretation in the context of multi-sorted languages. This is a little more fiddly than Definition 3.1 since our work with Morita extensions pushes us to be very explicit in our treatment of function symbols. To facilitate this, we first translate formulae of a language $\mathcal {L}$ into a normal form that ensures that the only atomic formulae in which function symbols occur are of the form $x^{\sigma }=f(x_{0}^{\sigma _{0}},\ldots ,x_{n}^{\sigma _{n}})$ ; i.e., we can never have an atomic formula of the form $x^{\sigma }=f(g(x^{\sigma _{0}})$ . Let us call this function normal form. Once a formula is in this form, the usual translation process works smoothly. The following proposition establishes that nothing is lost if we just use formulae in function normal form.

We leave it to the reader to establish this, however, an example could be helpful to illustrate the effect. For simplicity, suppose $\mathcal {L}$ has just one sort and suppose $\varphi :=\exists x(R(x,f(x))\wedge g(x)=f(x))$ is a formula of $\mathcal {L}$ . Then we see that $\varphi $ is equivalence to the following formula in function normal form:

$$\begin{align*}\exists x(\forall y(y=f(x)\to Rxy)\wedge\forall z(z=g(x)\wedge y=f(x))). \end{align*}$$

Given the proposition above, let us assume without loss of generality for the remainder of this section that we only deal with formulae in function normal form. We can now provide our definition of an interpretation between multi-sorted languages.

Definition 4.2. Let T and S be theories in multi-sorted languages $\mathcal {L}_{T}$ and $\mathcal {L}_{S}$ respectively. We say that a function t is a translation from the sentences of $\mathcal {L}_{S}$ to those of $\mathcal {L}_{T}$ if: for all sorts $\sigma \in \mathcal {L}_{S}$ , $t(\sigma )$ is a sort in $\mathcal {L}_{T}$ ; and

• • for all sorts $\sigma \in \mathcal {L}_{S}$ , there is a formula $\delta _{t,\sigma }$ of $\mathcal {L}_{T}$ with at most $v_{0}^{t(\sigma )}$ free,

• • for all relation symbols R of $\mathcal {L}_{T}$ of arity $\sigma _{0}\times \cdots \times \sigma _{n}$ , there is a formula $t(R)$ with at most the variables $v_{0}^{t(\sigma _{0})},\ldots ,v_{n}^{t(\sigma _{n})}$ free, and

• • for all function symbols g of $\mathcal {L}_{T}$ of arity $\sigma _{0}\times \cdots \times \sigma _{n}\to \sigma _{n+1}$ , there is a formula $t(g)$ with at most the variables $v_{0}^{t(\sigma _{0})},\ldots ,v_{n+1}^{t(\sigma _{n+1})}$ free,

such that for formulae $\varphi $ of $\mathcal {L}_{S}$ (in function normal form), $t(\varphi )$ is defined recursively as follows:

• • if $\varphi :=Rx_{0}^{\sigma _{0}}\ldots x_{n}^{\sigma _{n}}$ where $R\in \mathcal {L}_{S}$ is a relation symbol of arity $\sigma _{0}\times \cdots \times \sigma _{n}$ , then

  $$\begin{align*}t(\varphi)=t(R)(x_{0}^{t(\sigma_{0})},\ldots,x_{n}^{t(\sigma_{n})}), \end{align*}$$

• • if $\varphi :=(x_{n+1}^{\sigma _{n+1}}=g(x_{0}^{\sigma _{0}},\ldots ,x_{n}^{\sigma _{n}}))$ where $g\in \mathcal {L}_{S}$ is a function symbol of arity $\sigma _{0}\times \cdots \times \sigma _{n}\to \sigma $ , then

  $$\begin{align*}t(\varphi)=t(g)(x_{0}^{t(\sigma_{0})},\ldots,x_{n+1}^{t(\sigma_{n+1})}), \end{align*}$$
  where $t(g)$ is a formula of $\mathcal {L}_{T}$ such that
  $$\begin{align*}T\vdash\forall x_{0}^{t(\sigma_{0})}\ldots \forall x_{n}^{t(\sigma_{n})}\exists!x^{\sigma_{n+1}}\ t(g)(x_{0}^{t(\sigma_{0})},\ldots,x_{n+1}^{t(\sigma_{n+1})}), \end{align*}$$

• • $t(\neg \varphi )=\neg t(\varphi )$ ;

• • $t(\varphi \wedge \psi )=t(\varphi )\wedge t(\psi )$ ; and

• • $t(\forall x^{\sigma }\,\varphi )=\forall x^{t(\sigma )}(\delta _{t,\sigma }(x^{t(\sigma )})\to t(\varphi ),$

where $t(R)(x_{0}^{\sigma _{0}},\ldots ,x_{n}^{\sigma _{n}})$ is the result of replacing each $v_{i}^{t(\sigma _{i})}$ with $x_{i}^{t(\sigma _{i})}$ while changing bound variables when required to avoid clashes, and analogous remarks apply to $t(g)(x_{0}^{t(\sigma _{0})},\ldots ,x_{n+1}^{t(\sigma _{n+1})})$ and $\delta _{t,\sigma }(x^{t(\sigma )})$ .

We say that T interprets S via a translation t, if for all sentences $\varphi \in \mathcal {L}_{S}$ we have

$$\begin{align*}S\vdash\varphi\ \Rightarrow\ T\vdash t(\varphi). \end{align*}$$

Then as with ordinary interpretation, we see that an interpretation gives rise to a function from models of T to models of S. The following example shows how a single-sorted theory $ZFC$ can be used to interpret a multi-sorted theory.

The following theorem generalizes our earlier Theorem 3.2 linking interpretations to mod-functors.

Theorem 4.4. If T interprets S via $t:\mathcal {L}_{S}\to \mathcal {L}_{T}$ , then t determines a function

$$\begin{align*}t^{*}:mod(T)\to mod(S) \end{align*}$$

such that for all models $\mathcal {M}$ of T, formulae $\varphi (x_{0}^{\sigma _{0}},\ldots ,x_{n}^{\sigma _{n}})$ , we have

$$\begin{align*}\mathcal{M}\models t(\varphi)(m_{0},\ldots,m_{n})\ \Leftrightarrow\ t^{*}(\mathcal{M})\models\varphi(m_{0},\ldots,m_{n}) \end{align*}$$

when for all $i\leq n$ , $\mathcal {M}\models \delta _{t,\sigma _{i}}(m_{i})$ .

We omit the proof, which is similar to that of Theorem 3.2. As above, we shall just write $t^{*}$ instead of t below. We can now define analogues of the interpretative equivalences described in Section 3. Everything works much the same, however, we shall take a little care to explain how isomorphisms and elementary equivalences work in this setting.

Definition 4.5. Let us say that $f:\mathcal {M}\to \mathcal {N}$ is a homomorphism if $f:\prod _{\sigma \in \mathcal {L}}(M_{\sigma }\to N_{\sigma })$ is such that: for all relation symbols R of arity $\sigma _{0}\times \cdots \times \sigma _{n}$ and $m_{0},\ldots ,m_{n}$ from $M_{\sigma _{0}},\ldots ,M_{\sigma _{n}}$ respectively,Footnote ³³

$$\begin{align*}\mathcal{M}\models Rm_{0}\ldots m_{n}\ \Rightarrow\ \mathcal{N}\models Rf_{\sigma_{0}}(m_{0})\ldots f_{\sigma_{n}}(m_{n}), \end{align*}$$

and for all function symbols g of arity $\sigma _{0}\times \cdots \times \sigma _{n}\to \sigma $ and $m_{0},\ldots ,m_{n}$ from $M_{\sigma _{0}},\ldots ,M_{\sigma _{n}}$ respectively,

$$\begin{align*}g^{\mathcal{M}}(m_{0},\ldots,m_{n})=g^{\mathcal{N}}(f_{\sigma_{0}}(m_{0}),\ldots,f_{\sigma_{n}}(m_{n})). \end{align*}$$

We say that $f:\mathcal {M}\to \mathcal {N}$ is an embedding if f is a homomorphism and for all relation symbols (including identity) R of arity $\sigma _{0}\times \cdots \times \sigma _{n}$ and $m_{0},\ldots ,m_{n}$ from $M_{\sigma _{0}},\ldots ,M_{\sigma _{n}}$ respectively,

$$\begin{align*}\mathcal{M}\models Rm_{0}\ldots m_{n}\ \Leftrightarrow\ \mathcal{N}\models Rf_{\sigma_{0}}(m_{0})\ldots f_{\sigma_{n}}(m_{n}). \end{align*}$$

We say that $f:\mathcal {M}\to \mathcal {N}$ is an isomorphism if there is some homomorphism $g:\mathcal {N}\to \mathcal {M}$ such that

$$\begin{align*}g\circ f=id_{\mathcal{M}}\text{ and }f\circ g=id_{\mathcal{N}}. \end{align*}$$

We say that $f:\mathcal {M}\to \mathcal {N}$ is an elementary embedding if for all formulae $\varphi (x_{\sigma _{0}},\ldots ,x_{\sigma _{n}})$ of $\mathcal {L}$ and $m_{0},\ldots ,m_{n}$ from $M_{\sigma _{0}},\ldots ,M_{\sigma _{n}}$ respectively,

$$\begin{align*}\mathcal{M}\models\varphi(m_{0}\ldots m_{n})\ \Leftrightarrow\ \mathcal{N}\models\varphi(f_{\sigma_{0}}(m_{0})\ldots f_{\sigma_{n}}(m_{n})). \end{align*}$$

For a theory T in a multi-sorted language $\mathcal {L}$ , we now let $mod_{mult}(T)$ be the category whose objects are models of T and whose arrows are elementary embeddings between them. The following proposition establishes that interpretations also determine functors in the multi-sorted setting. The proof is similar to that of Lemma 3.6.

Then in preparation for the more general notion of strict bi-interpretability, we define what it means for an isomorphism to be definable over some theory.

Definition 4.7. Suppose $\mathcal {L}$ is a multi-sorted language with sorts $\{\sigma _{i}\}_{i\in I}$ . Let T be an $\mathcal {L}$ -theory. We say that $\{\psi _{i}(x,y)\}_{i\in I}$ defines a function over T if for all models $\mathcal {N}$ and $\mathcal {M}$ of T where $\mathcal {N}$ is a submodel of $\mathcal {M}$ there is some $f:\mathcal {M}\cong \mathcal {N}$ such that for all $i\in I$ and all $m_{0},m_{1}\in M_{\sigma _{i}}$

$$\begin{align*}f_{\sigma_{i}}(m_{0})=m_{1}\ \Leftrightarrow\ \mathcal{M}\models\psi_{i}(m_{0},m_{1}). \end{align*}$$

Finally, we are can define the restricted framework suitable for multi-sorted interpretation.

We may then define the standard equivalence relations in much same way as we did above. This time we take the category theoretic equivalences as the prototype from which the ordinary definitions are obtained.

5.1 Morita extension

The framework reviewed in the previous section allows us to compare theories articulated in multi-sorted languages. However, there are many occasions where we might want to compare, say, a single-sorted theory with a multi-sorted theory but the tools above do not—in general—allow for this.Footnote ³⁴ For example, we might want to compare the axiomatization of category theory with sorts for arrows and objects with its axiomatization using only arrows. In this section, we describe a framework developed by Barrett and Halvorson [Reference Barrett and Halvorson4] for exactly this purpose. We have two main goals in this section. First, we aim to generalize their approach so that we not only get a new notion of equivalence, but further a new notion of interpretation: Morita interpretation. Second, we use this to show that there is another natural restriction of $\mathbb {TH}$ that gives us Morita equivalence and more. I’m going to depart a little from the presentation of Barrett and Halvorson [Reference Barrett and Halvorson4], although most of the deviations are minor. The definition below describes a way of taking a theory T articulated in some language $\mathcal {L}$ and adding a new sort $\sigma $ corresponding to one of four common type constructions.

• • (Product) There exist sorts $\sigma _{0},\sigma _{1}\in \mathcal {L}$ , $\mathcal {L}^{*}=\mathcal {L}\cup \{\sigma ,\pi _{0},\pi _{1}\}$ where $\pi _{0}:\sigma \to \sigma _{0}$ and $\pi _{1}:\sigma \to \sigma _{1}$ ; and $T^{*}$ is T extended by

  $$\begin{align*}\forall x_{0}^{\sigma_{0}}\forall x_{1}^{\sigma_{1}}\exists!y^{\sigma}(\pi_{0}(x_{0}^{\sigma_{0}})=y^{\sigma}\wedge\pi_{1}(x_{1}^{\sigma_{1}})=y^{\sigma}). \end{align*}$$

• • (Co-product) There exist sorts $\sigma _{0},\sigma _{1}\in \mathcal {L}$ , $\mathcal {L}^{*}=\mathcal {L}\cup \{\sigma ,\pi _{0},\pi _{1}\}$ where $\pi _{0}:\sigma _{0}\to \sigma $ and $\pi _{1}:\sigma _{1}\to \sigma $ ; and $T^{*}$ is T extended by

  $$ \begin{align*} & \forall y^{\sigma}(\exists x_{0}^{\sigma_{0}}\ \pi_{0}(x_{0}^{\sigma_{0}})=y^{\sigma}\vee\exists x_{1}^{\sigma_{1}}\ \pi_{1}(x_{1}^{\sigma_{1}})=y^{\sigma})\wedge\\ & \forall x_{0}^{\sigma_{0}}\forall x_{1}^{\sigma_{1}}(\pi_{0}(x_{0}^{\sigma_{0}})\neq\pi_{1}(x_{1}^{\sigma_{1}})). \end{align*} $$

• • (Subsort) There exists a sort $\sigma _{0}\in \mathcal {L}$ and a formula $\varphi (x^{\sigma _{0}})$ from $\mathcal {L}$ ; $\mathcal {L}^{*}=\mathcal {L}\cup \{\sigma ,\pi \}$ where $\pi :\sigma \to \sigma _{0}$ ; and $T^{*}$ is T extended by

  $$ \begin{align*} & \forall x^{\sigma_{0}}(\varphi(x^{\sigma_{0}})\leftrightarrow\exists z^{\sigma}\ \pi(z^{\sigma})=x^{\sigma_{0}}))\wedge\\ & \forall x_{0}^{\sigma},x_{1}^{\sigma}(\pi(x_{0}^{\sigma})=\pi(x_{1}^{\sigma})\to x_{0}^{\sigma}=x_{1}^{\sigma}). \end{align*} $$

• • (Quotient) There exists sort $\sigma _{0}$ and a formula $\varphi (x^{\sigma _{0}},y^{\sigma _{0}})$ in $\mathcal {L}$ and T proves $\varphi (x^{\sigma _{0}},y^{\sigma _{0}})$ represents an equivalence relation; $\mathcal {L}^{*}=\mathcal {L}\cup \{\sigma ,\pi \}$ where $\pi :\sigma _{0}\to \sigma $ ; and $T^{*}$ is T extended by

  $$ \begin{align*} & \forall x_{0}^{\sigma_{0}},x_{1}^{\sigma_{0}}(\pi(x_{0}^{\sigma_{0}})=\pi(x_{1}^{\sigma_{0}})\leftrightarrow\varphi(x_{0}^{\sigma_{0}},x_{1}^{\sigma_{0}}))\wedge\\ & \forall y^{\sigma}\exists x^{\sigma_{0}}(\pi(x^{\sigma_{0}})=y^{\sigma}). \end{align*} $$

Let $T^{+}$ be a theory in a language $\mathcal {L}^{+}$ extending $\mathcal {L}^{*}$ with possibly new relation and function symbols. We say that $T^{+}$ is a mixed Morita successor of T if $T^{+}$ interprets $T^{*}$ in the sense of Definition 4.2.Footnote ³⁵

Thus, we are allowed to add new sorts corresponding to products, co-products, subsorts and quotients. For example, if I am working in the $PA$ in the language of arithmetic, I could add a subsort corresponding to the prime numbers. The distinction between pure and mixed Morita successors is introduced for a technical reason. Barrett and Halvorson [Reference Barrett and Halvorson4] define a Morita extension to be what we have called a mixed Morita successor. However, we shall do things a little differently by only using pure Morita successors and then using a multi-sorted interpretation at the end. We demonstrate below that this makes no substantive difference. However, the approach taken here makes for an easier comparison with the categorical approach above. We shall generally omit the “pure” and call pure Morita successors, “Morita successors,” unless confusion could arise.

We then note that when $T^{*}$ is a Morita expansion of T any model of T can be expanded to obtain a model of $T^{*}$ that is unique up to isomorphism.

From this it follows that $T^{*}$ is model-theoretic conservative extension of T. Thus, we might say that whatever is added by a Morita expansion comes at a relatively low cost.

5.2 Morita interpretation and equivalence(s)

Using Morita expansions, we can now describe a notion of Morita interpretation that is implicit in Barrett and Halvorson [Reference Barrett and Halvorson4]. The main work of this section will be establishing that these interpretations can be understood as giving rise to functors in $\mathbb {TH}_{mult-def}$ .

Observe that unlike ordinary relative interpretation, Morita interpretation has two components. First we make a Morita expansion, then we make the (usually multi-sorted) interpretation. We’d like now to define something like a mod-functor for Morita interpretation, however, the initial Morita expansion poses a problem: since we are adding new domains, there will be many different ways to populate them. Thus a Morita interpretation does not determine a particular functor but rather a family of them.

Note that $t^{\dagger }$ takes models of T while $t^{*}$ takes models of $T^{+}$ . The idea here is that we restrict our attention to functors $t^{\dagger }$ that yield models that could have been obtained by taking a pure Morita extension and then applying the mod-functor $t^{*}$ . Notice, however, that while we have a definition of compatible functors, we are yet to show that any such functors exist. To establish this, we first define what we call the canonical Morita expansion of a model $\mathcal {M}$ of T. The basic idea is to use the obvious operations on the original sort domains to define the new sort domains.

We now aim to prove that functors compatible with interpretations exist. To do this, we first recall a technical theorem from [Reference Barrett and Halvorson4] that will be used further during this paper. In order for this theorem to make sense, we first need to explain the notion of a code. A detailed discussion of this is provided after Example 4.5 in [Reference Barrett and Halvorson4], but we’ll content ourselves here with a brief overview. Suppose that $T^{+}\subseteq \mathcal {L}_{T^{+}}$ is a Morita successor of $T\subseteq \mathcal {L}_{T}$ and that $\sigma $ is the new sort that was added. Despite being a new sort, we can still understand the behavior of objects of sort $\sigma $ using objects from sorts from the original language $\mathcal {L}_{T}$ . For example, if $\sigma $ is formed as a product of sorts $\tau _{0}$ and $\tau _{1}$ , using projection functions $\pi _{0}$ and $\pi _{1}$ , then we see that any $x^{\sigma }$ is determined by a unique pair $z_{0}^{\tau _{0}}$ and $z_{0}^{\tau _{1}}$ . More precisely, we see that

$$\begin{align*}\pi_{0}(x^{\sigma})=z_{0}^{\tau_{0}}\wedge\pi_{1}(x^{\sigma})=z_{1}^{\tau_{1}} \end{align*}$$

and we call such a formula a code for variables of sort $\sigma $ . A code provides a kind of bridge between the old sorts of $\mathcal {L}_{T}$ and the new sorts of $\mathcal {L}_{T^{+}}.$ Similarly, a code for each of the other three sort types can also be defined. The following theorem then shows how the behavior of the new sorts can be understood from the perspective of the original language.

Theorem 5.7 [Reference Barrett and Halvorson4].

Suppose $T^{+}\subseteq \mathcal {L}_{T^{+}}$ is a Morita successor of $T\subseteq \mathcal {L}_{T}$ . Suppose $\varphi (x_{0}^{\sigma },\ldots ,x_{m}^{\sigma },y_{0}^{\sigma _{0}},\ldots ,y_{n}^{\sigma _{n}})$ where $\sigma \in \mathcal {L}_{T^{+}}\backslash \mathcal {L}_{T}$ and $\sigma _{0},\ldots ,\sigma _{m}\in \mathcal {L}_{T}$ . Suppose that for $i\leq m$ , $\xi _{i}(x^{\sigma },z_{0}^{\tau _{0}},z_{1}^{\tau _{1}})$ is a code for variables of sort $\sigma $ where $\tau _{0}$ and $\tau _{1}$ are sorts from $\mathcal {L}_{T}$ . Then there is a formula $\varphi ^{*}(z_{0,0}^{\tau _{0}},z_{0,1}^{\tau _{1}},\ldots ,z_{m}^{\tau _{0}},z_{m,1}^{\tau _{1}},y_{0}^{\sigma _{0}},\ldots ,y_{n}^{\sigma _{n}})$ of $\mathcal {L}_{T}$ such that $T^{+}$ proves that for all $x_{0}^{\sigma },\ldots ,x_{m}^{\sigma },y_{0}^{\sigma _{0}},\ldots ,y_{n}^{\sigma _{n}},z_{0,0}^{\tau _{0}},z_{0,1}^{\tau _{1}},\ldots ,z_{m}^{\tau _{0}},z_{m,1}^{\tau _{1}}$ if $\bigwedge _{i\leq m}\xi _{i}(x_{i}^{\sigma },z_{i,o}^{\tau _{0}},z_{i,1}^{\tau _{1}})$ , then

$$\begin{align*}\varphi(x_{0}^{\sigma},\ldots,x_{m}^{\sigma},y_{0}^{\sigma_{0}},\ldots,y_{n}^{\sigma_{n}})\leftrightarrow\varphi^{*}(z_{0,0}^{\tau_{0}},z_{0,1}^{\tau_{1}},\ldots,z_{m}^{\tau_{0}},z_{m,1}^{\tau_{1}},y_{0}^{\sigma_{0}},\ldots,y_{n}^{\sigma_{n}}). \end{align*}$$

Informally speaking, we are using the code formulae $\xi _{i}$ to form a bridge between the formula $\varphi $ of $\mathcal {L}_{T^{+}}$ and an equivalent formula $\varphi ^{*}$ of the original language $\mathcal {L}_{T}$ .Footnote ³⁸ A little more formally, we are replacing all the variables of sort $\sigma $ with pairs of variables of sort $\tau _{0}$ and $\tau _{1}$ and then modifying the formula $\varphi $ to accommodate this change. Given the appropriate coding formulae, we end up with a formula $\varphi ^{*}$ from the original language, $\mathcal {L}_{T}$ , that is equivalent to $\varphi $ according to $T^{+}$ . The upshot of this is that we see that T is capable of simulating what happens in $T^{+}$ . Thus, it might be argued that the expansion is—in some sense—harmless. We then use this result in the following lemma to establish the existence of compatible functors for Morita expansions.

Proof. First we define $j^{+}:\mathcal {M}^{+}\to \mathcal {N}^{+}$ where $j_{\sigma }^{+}=j_{\sigma }$ for all $\sigma $ from $\mathcal {L}$ and then we show that $j^{+}$ is an elementary embedding. If $\sigma $ is a sort from $\mathcal {L}$ we let $j_{\sigma }^{+}=j_{\sigma }$ . So suppose $\sigma $ is the new sort from $\mathcal {L}_{T^{+}}\backslash \mathcal {L}_{T}$ defined from sorts $\sigma _{0},\sigma _{1}$ from $\mathcal {L}_{T}$ . We we define $j_{\sigma }^{+}$ depending on how it was formed:

• • if $\sigma $ is a product formed from $\sigma _{0},\sigma _{1}$ , and $\langle m_{0},m_{1}\rangle \in M_{\sigma }^{+}$

  $$\begin{align*}j_{\sigma}^{+}(\langle m_{0},m_{1}\rangle)=\langle j_{\sigma_{0}}^{+}(m_{0}),j_{\sigma_{1}}^{+}(m_{1})\rangle; \end{align*}$$

• • if $\sigma $ is a co-product sort formed from $\sigma _{0},\sigma _{1}$ and $\langle m,i\rangle \in M_{\sigma }^{+}$

  $$\begin{align*}j_{\sigma}^{+}(\langle m,i\rangle)=\begin{cases} \langle j_{\sigma_{0}}^{+}(m),i\rangle, & \text{if }i=0,\\ \langle j_{\sigma_{1}}^{+}(m),i\rangle, & \text{if }i=1; \end{cases} \end{align*}$$

• • if $\sigma $ is a subsort of $\sigma _{0}$ formed using $\varphi (x^{\sigma _{0}})$ and $m\in M_{\sigma }^{+}$

  $$\begin{align*}j_{\sigma}^{+}(m)=j_{\sigma_{0}}^{+}(m);\text{ and} \end{align*}$$

• • if $\sigma $ is a quotient on $\sigma _{0}$ formed using $\varphi (x^{\sigma _{0}},y^{\sigma _{0}})$ , let

  $$\begin{align*}j_{\sigma}^{+}([m]_{\varphi})=[j_{\sigma_{0}}^{+}(m)]_{\varphi}. \end{align*}$$

We then claim that $j^{+}$ gives an elementary embedding. Let $\varphi (x_{0}^{\sigma },\ldots ,x_{m}^{\sigma },y_{0}^{\sigma _{0}},\ldots ,y_{k}^{\sigma _{k}})$ be a formula of $\mathcal {L}_{T^{+}}$ and $m_{0},\ldots ,m_{n},a_{0},\ldots ,a_{k}$ where $m_{0},\ldots ,m_{n}$ are from $M_{\sigma }$ and $a_{0},\ldots ,a_{k}$ are from $M_{\sigma _{0}},\ldots ,M_{\sigma _{k}}$ respectively. We claim that

$$\begin{align*}\mathcal{M}^{+}\models\varphi(m_{0},\ldots,m_{n},a_{0},\ldots,a_{k}) \Leftrightarrow \mathcal{N}^{+}\models\varphi(j_{\sigma}^{+}(m_{0}),\ldots j_{\sigma}^{+}(m_{n}),j_{\sigma_{0}}^{+}(a_{0}),\ldots,j_{\sigma_{k}}^{+}(a_{k})). \end{align*}$$

We’ll just do the case when $\sigma $ is a co-product sort formed from sorts $\tau _{0},\tau _{1}$ from $\mathcal {L}_{T}$ . Then $M_{\sigma }=(M_{\tau _{0}}\times \{0\})\cup (M_{\tau _{1}}\times \{1\})$ . For each $i\leq n$ , let $\xi _{i}(x_{i}^{\sigma },z_{i,0}^{\tau _{0}},z_{i,1}^{\tau _{1}})$ be: the formula $\pi _{0}(z_{i,0}^{\tau _{0}})=x_{i}^{\sigma }$ if $m_{i}=\langle m_{i}^{*},0\rangle $ for some $m_{i}^{*}\in M_{\tau _{0}}$ ; and $\pi _{1}(z_{i,1}^{\tau _{1}})=x_{i}^{\sigma }$ otherwise. Each of these formulae $\xi _{i}$ is a code for the sort $\sigma $ . We then see that for all $i\leq n$ , we may fix $c_{i,0}\in M_{\tau _{0}}$ and $c_{i,1}\in M_{\tau _{1}}$ such that $\mathcal {M}^{+}\models \xi _{i}(m_{i},c_{i,0},c_{i,1})$ . Thus we see that

$$\begin{align*}\mathcal{M}^{+}\models\bigwedge_{i\leq n}\xi_{i}(m_{i},c_{i,0},c_{i,1}). \end{align*}$$

Moreover, using the definition of $j^{+}$ and it can be seen that $\mathcal {N}^{+}\models \bigwedge _{i\leq n}\xi _{i}(j_{\sigma }^{^{+}}(m_{i}), j_{\tau _{0}}(c_{i,0}),j_{\tau _{1}}(c_{i,1}))$ .

By the previous lemma we may fix a formula $\varphi ^{*}(z_{0,0}^{\tau _{0}},z_{0,1}^{\tau _{1}},\ldots ,z_{n,0}^{\tau _{0}},z_{n,1}^{\tau _{1}},y_{0}^{\sigma _{0}},\ldots ,y_{k}^{\sigma _{k}})$ such that $T^{+}$ proves that for all $x_{0}^{\sigma },\ldots ,x_{m}^{\sigma }$ and $z_{0,0}^{\tau _{0}},z_{0,1}^{\tau _{1}},\ldots ,z_{n,0}^{\tau _{0}},z_{n,1}^{\tau _{1}},y_{0}^{\sigma _{0}},\ldots ,y_{k}^{\sigma _{k}}$ if $\bigwedge _{i\leq n}\xi _{i}(x_{i}^{\sigma },z_{i,0}^{\tau _{0}},z_{i,1}^{\tau _{1}})$ then

$$\begin{align*}\varphi(x_{0}^{\sigma},\ldots,x_{m}^{\sigma},y_{0}^{\sigma_{0}},\ldots,y_{k}^{\sigma_{k}})\leftrightarrow\varphi^{*}(z_{0,0}^{\tau_{0}},z_{0,1}^{\tau_{1}},\ldots,z_{n,0}^{\tau_{0}},z_{n,1}^{\tau_{1}},y_{0}^{\sigma_{0}},\ldots,y_{k}^{\sigma_{k}}). \end{align*}$$

Finally, we put this together to see that

$$ \begin{align*} & \mathcal{M}^{+}\models\varphi(m_{0},\ldots,m_{n},a_{0},\ldots,a_{k})\\ \Leftrightarrow & \mathcal{M}\models\varphi^{*}(c_{0,0},c_{0,1},\ldots,c_{n,0},c_{n,1},a_{0},\ldots,a_{k})\\ \Leftrightarrow & \mathcal{N}\models\varphi^{*}(j_{\tau_{0}}(c_{0,0}),j_{\tau_{1}}(c_{0,1}),\ldots,j_{\tau_{0}}(c_{n,0}),j_{\tau_{1}}(c_{n,1}),j_{\sigma_{0}}(a_{0}),\ldots,j_{\sigma_{k}}(a_{k})\\ \Leftrightarrow & \mathcal{N}^{+}\models\varphi(j_{\sigma}^{+}(m_{0}),\ldots j_{\sigma}^{+}(m_{n}),j_{\sigma_{0}}^{+}(a_{0}),\ldots,j_{\sigma_{k}}^{+}(a_{k})) \end{align*} $$

as required.

It is easy to see that Lemma 4.6 gives us the successor case in a proof by induction establish the result above holds for Morita expansions more generally. Finally, we can use this to show that Morita interpretations are compatible with functors between theory categories.

This puts us in position, analogous to that in Section 3, to define a framework in which Morita interpretation can be understood as another natural restriction of the theory framework based in category theory.

And then we may define our standard equivalences in this restricted framework.

Once again, we see that the category theoretic framework provides a clean housing of a notion of interpretation. However, we must note that Barrett and Halvorson [Reference Barrett and Halvorson4] define a different notion, Morita equivalence, which raises the question: where does their notion Morita equivalence fit above? The answer is that it is the same as definitional Morita equivalence and we shall now prove this. As mentioned above, the key issue is whether we use pure or mixed Morita successors. Let us recall the definition of what we shall call standard Morita equivalence.

It is easy to see that T and S are definitionally Morita equivalent if T and S have pure Morita expansions $T^{*}$ and $S^{*}$ such that they are definitionally equivalent. One might thus worry that by using a succession of mixed Morita expansions rather than pure ones, we may be able to interpret more and thus obtain more standard Morita equivalences than definitional Morita equivalences. The following lemma show that this worry is misplaced.

Proof. We proceed by induction the Morita successors of T that culminate in $T^{*}$ . We suppose for our induction hypothesis that we have a sequence $T=T_{0}^{*},T_{1}^{*},\ldots ,T_{n}^{*}$ of mixed Morita successors of T in languages $\mathcal {L}=\mathcal {L}_{0}^{*},\ldots ,\mathcal {L}_{n}^{*}$ ; and for each $i\leq n$ , there is a Morita expansion $T_{i}^{+}$ in $\mathcal {L}_{i}^{+}$ such that an interpretation of $T_{i}^{+}$ that is $T_{i}^{*}$ . Let $T_{n+1}^{*}$ be a mixed Morita expansion of $T_{n}^{*}$ in the language $\mathcal {L}_{n+1}^{*}$ . We claim that there is a Morita expansion of $T_{n}^{+}$ such that $T_{n+1}^{*}$ can be obtained by interpretation. Thus, it suffices to show that for the new sort $\sigma $ added to $\mathcal {L}_{n+1}^{*}$ , this sort can also added as a Morita expansion of $T_{n}^{+}$ . First, we note that $\mathcal {L}_{n}^{*}$ and $\mathcal {L}_{n}^{+}$ must have the same sorts. Thus, it is easy to see that any product or coproduct sort added to $\mathcal {L}_{n}^{*}$ could also be added to $\mathcal {L}_{n}^{+}$ .

Things are a little more difficult with subsorts and quotient sorts since they are defined using a formula of $\mathcal {L}_{n}^{*}$ that may contain more relation and function symbols than $\mathcal {L}_{n}^{+}$ . We focus on the quotient sort case as it is more complex. We suppose that $\sigma $ is a quotient sort formed from the sort $\sigma _{0}$ from $\mathcal {L}_{n}^{*}$ formed using the formula $\varphi (x^{\sigma _{0}},y^{\sigma _{0}})$ from $\mathcal {L}_{n}^{*}$ . It then suffices to show that there is a formula $\varphi ^{*}(x^{\sigma _{0}},y^{\sigma _{0}})$ such that

$$\begin{align*}T_{n}^{*}\vdash\forall x^{\sigma_{0}}\forall y^{\sigma_{0}}(\varphi(x^{\sigma_{0}},y^{\sigma_{0}})\leftrightarrow\varphi^{*}(x^{\sigma_{0}},y^{\sigma_{0}})). \end{align*}$$

This follows directly from Theorem 5.7. Thus, $T_{n}^{+}$ is also able to define $\sigma $ and so we may form $T_{n+1}^{+}$ by adding the quotient sort $\sigma $ using the formula $\varphi ^{*}(x^{\sigma _{0}},y^{\sigma _{0}})$ . Then any new relation symbols involving $\sigma $ can be procured by interpretation as required.

Putting this together, we see that whenever we have a series of mixed Morita interpretations witnessing standard Morita equivalence, those interpretations can be recovered using a pure Morita expansion following by an interpretation. Thus, we get the following.

Thus, we now see that the equivalence relation of [Reference Barrett and Halvorson4] fits very comfortably into the framework provided by this paper. Let us now look at some examples of definitional Morita equivalence.

5.2.2 A more interesting example

We now employ this framework on an example from Visser and Friedman [Reference Visser and Friedman22] that we mentioned above. In the context of interpretations that are not identity preserving (and thus, not that of this paper), this example was used to provide a pair of theories that were (non-strictly) bi-interpretable but not definitionally equivalent. In the context of this paper, it makes for a pleasing example of Morita equivalence.

Let $AS$ be the theory in the language $\mathcal {L}_{AS}$ with a single sort $ob$ and a two place relation symbol $\in $ . Let it have the following two axioms:

• • $\exists x\forall y\ y\notin x$ ; and

• • $\forall x\forall y\exists z\forall w(w\in z\leftrightarrow w\in x\vee w=y)$ .

Let $ACF$ be the theory in the language $\mathcal {L}_{ACF}$ with two sorts $ob$ and $cl$ where we use lower case variables for the $ob$ sort and upper case variables for the $cl$ sort. Let the non-logical vocabulary consist of: a relation symbol $\varepsilon $ of arity $ob\times cl$ ; and a function symbol F of sort $ob\to cl$ . Let it be axiomatized by the following axioms:

• • $\exists X\forall y\ \neg y\varepsilon X$ ;

• • $\forall X\forall y\exists Z\forall w(w\varepsilon Z\leftrightarrow w\varepsilon X\vee w=y)$ ;

• • $\forall X\forall Y(\forall z(z\varepsilon X\leftrightarrow z\varepsilon Y)\to X=Y)$ ;

• • $\forall X\exists y\ F(y)=X$ .

Proposition 5.16. $AS$ is Morita equivalent to $ACF$ .

Proof. We let $AS^{+}$ be a Morita expansion of $AS$ with a quotient sort on $ob$ given by the formula $\varphi (x,y):=\forall z(z\in x\leftrightarrow z\in y)$ . Call this new sort $\sigma $ and let $\pi :ob\to \sigma $ be the associated function. Use upper case letters for $\sigma $ -variables.

It suffices to show that $AS^{+}$ is definitionally equivalent to $ACF$ . Let $t:\mathcal {L}_{ACF}\to \mathcal {L}_{AS^{+}}$ be such that $t(ob)=ob$ and $t(cl)=\sigma $ . Let

$$ \begin{align*} t(x=y) & :=x=y\\ t(X=Y) & :=X=Y\\ t(x\varepsilon Y) & :=\exists z(\pi(z)=Y\wedge x\in z)\\ t(F(x)=Y) & :=\pi(x)=y. \end{align*} $$

Let $s:\mathcal {L}_{AS^{+}}\to \mathcal {L}_{ACF}$ be such that $s(ob)=ob$ and $s(\sigma )=cl$ . Let

$$ \begin{align*} s(x=y) & :=x=y\\ s(X=Y) & :=X=Y\\ s(x\in y) & :=x\varepsilon F(y)\\ s(\pi(x)=Y) & :=F(x)=Y. \end{align*} $$

It’s easy to see that this gives a definitional equivalence.

6.1 Open questions

We now explore the failure of reversals in the table a little further by addressing the following open questions posed by Barrett and Halvorson [Reference Barrett and Halvorson4].Footnote ³⁹

These problems focus on whether the differences between Morita equivalence and definitional equivalence collapse when we consider theories that are sufficiently strong. In other words, they ask whether the relationships along the top of the table reverse. We answer these two questions below.

Problem 6.5(2)

We answer this question negatively by providing a theory that is Morita equivalent to $ZF$ but is not definitionally equivalent to $ZF$ . Let $ZF_{twin}$ be $ZF$ with the axiom of extensionality replaced by axiom stating that for any set there is exactly one other set with the same members. More specifically,

$$\begin{align*}\forall x\exists!x^{*}(x\neq x^{*}\wedge\forall w(w\in x\leftrightarrow w\in x^{*})). \end{align*}$$

Thus, we have a strange modification of $ZF$ in which every set has a twin. Note that $ZF_{twin}$ is unable to define an element of any of its models since there will always be a twin. For this reason, we also demand that our axiomatization of $ZF$ includes the axiom of collection instead of replacement. In $ZF$ this makes no difference but in contexts where extensionality fails it is important since genuine class functions are difficult to find.Footnote ⁴⁰

Theorem 6.6.

1. (1) $ZF_{twin}$ is Morita equivalent to $ZF$ .

2. (2) $ZF_{twin}$ is not definitionally equivalent to $ZF$ .

Proof. (1) To Morita interpret $ZF_{twin}$ in $ZF$ add the subsort with domain X defined by transfinite recursion such that

$$ \begin{align*} X_{0}=\ & \{\langle0,\emptyset\rangle,\langle1,\emptyset\rangle\}\\ X_{\alpha+1}=\ & \{\langle0,y\rangle\ |\ y\in\mathcal{P}(X_{\alpha})\}\cup\\ & \{\langle1,y\rangle\ |\ y\in\mathcal{P}(X_{\alpha})\}\cup X_{\alpha}\\ X_{\lambda}=\ & \bigcup_{\alpha<\lambda}X_{\alpha}\text{ for limit }\lambda \end{align*} $$

and $X=\bigcup _{\alpha \in Ord}X_{\alpha }$ . We then define a relation $\in _{X}$ on this sort which is such that for $\langle i,x\rangle ,\langle j,y\rangle \in X$

$$\begin{align*}\langle i,x\rangle\in_{X}\langle j,y\rangle\ \Leftrightarrow\ x\in y. \end{align*}$$

The idea is to build in the twins by tagging sets with the natural numbers $0$ and $1$ . This is clearly a model of $ZF_{twin}$ . Call this Morita expansion $ZF^{+}$ . To Morita interpret $ZF$ in $ZF_{twin}$ we define a quotient sort following [Reference Scott19]. In $ZF_{twin}$ , let us say that a set x is extensional if whenever y and z have the same members, then $y\in x$ iff $y\in z$ . We $\varphi _{1}(x)$ say that there is a transitive set Y such that $x\in Y$ and every element of Y is extensional. Then let $\varphi _{0}(x,y)$ say that x and y have the same members. It can then be seen that the quotient sort given by $\varphi _{0}$ over $\varphi _{1}$ gives a model of $ZF$ . Call the expansion $ZF_{twin}^{+}$ . One can then see that $ZF_{twin}^{+}$ and $ZF^{+}$ are definitionally equivalent and so $ZF_{twin}$ and $ZF$ are Morita equivalent.

(2) We claim that $ZF_{twin}$ cannot even interpret $ZF$ . Suppose toward a contradiction that there is an interpretation $t:\mathcal {L}_{\in }\to \mathcal {L}_{\in }$ such that for all $\varphi \in \mathcal {L}_{\in }$

$$\begin{align*}ZF\vdash\varphi\Rightarrow ZF{}_{twin}\vdash t(\varphi). \end{align*}$$

Now consider the sentence $\psi $ that says there is a unique set that has no members; i.e., $\exists !x\forall y\ y\notin x$ . t must translate this to a sentence of the form

$$\begin{align*}\exists!x(\delta_{t}(x)\wedge\forall y\ \neg t(\in)(x,y)). \end{align*}$$

But this would mean that $ZF_{twin}$ was able to define an empty set which is impossible.

We also note that this example can be generalized to hold for $PA$ and not just $ZF$ . To see this we first recall a standard result. Let $ZF_{fin}$ be $ZF$ with the axiom of infinity removed and its negation added. Moreover, suppose we use the axiom of set schema of set induction rather than Foundation.Footnote ⁴¹

Fact 6.7. $PA$ and $ZF_{fin}$ are definitionally equivalent.Footnote ⁴²

It is then easy to see that the proof above made no use of the axiom of infinity, thus there is a theory $ZF_{fin,twin}$ that is Morita equivalent but not definitionally equivalent with $ZF_{fin}$ . Fact 6.7, then entails that $PA$ is also Morita equivalent but not definitionally equivalent with $ZFC_{fin,twin}$ .

Problem 6.5(3)

We answer the question affirmatively by using a natural condition—Morita completeness—suggested by Barrett and Halvorson [Reference Barrett and Halvorson4]—that entails that definitional equivalence follows from Morita equivalence. The idea here is that some theories have sufficient expressive strength that they can replicate the effects of Morita expansion by defining objects internally without the need to add new sorts. We shall say that such a theory is Morita complete. The following definition is intended to capture this idea.

Definition 6.8. Let us say that a theory T in a single sorted language $\mathcal {L}_{T}$ is Morita complete if each of the following hold:

1. (1) For all $\varphi _{0}(x),\varphi _{1}(x)\in \mathcal {L}_{T}$ there exist formulae, $\varphi _{0\times 1}(x),\pi _{0}(x,y)$ and $\pi _{1}(x,y)$ of $\mathcal {L}_{T}$ such that T proves $\pi _{0}$ and $\pi _{1}$ represent functions and T proves

   $$\begin{align*}\forall x\forall y(\varphi_{0}(x)\wedge\varphi_{1}(y)\to\exists!z(\varphi_{0\times1}(z)\wedge\pi_{0}(z,x)\wedge\pi_{1}(z,y)). \end{align*}$$

2. (2) For all $\varphi _{0}(x),\varphi _{1}(x)\in \mathcal {L}_{T}$ there exist formulae, $\varphi _{0+1}(x),\sigma _{0}(x,y)$ and $\sigma _{1}(x,y)$ of $\mathcal {L}_{T}$ such that T proves $\sigma _{0}$ and $\sigma _{1}$ represent functions and

   $$ \begin{align*} T\vdash & \forall z(\varphi_{0+1}(z)\to\exists x((\varphi_{0}(x)\wedge\sigma_{0}(x,z))\vee(\varphi_{1}(x)\wedge\sigma_{1}(x,z)))\wedge\\ & \forall x\forall y\forall z(\varphi_{0}(x)\wedge\varphi_{1}(y)\to\neg(\sigma_{0}(x,z)\wedge\sigma_{1}(y,z))). \end{align*} $$

3. (3) For all $\varphi _{0}(x)$ and $\varphi _{1}(x,y)$ from $\mathcal {L}_{T}$ if T proves that $\varphi _{1}$ is an equivalence relation on $\varphi _{0}$ , then there exist formulae $\varphi _{0/1}(x)$ and $\rho (x,y)$ such that T proves that $\rho $ represents a function and

   $$ \begin{align*} T\vdash & \forall x\forall y\forall z(\varphi_{0}(x)\wedge\varphi_{0}(y)\to(\rho(x,z)\wedge\rho(y,z)\leftrightarrow\varphi_{1}(x,y)))\wedge\\ & \forall y(\varphi_{0}(y)\to\exists!x(\varphi_{0/1}(x)\wedge\rho(y,x))). \end{align*} $$

Note that we did not include a clause for subsorts above. This is because any theory can replicate the effect of subsorts. Then we observe that $ZF$ is an example of a Morita complete theory.

Proposition 6.9. $ZF$ is Morita complete.

Proof. It will suffice to show that (1)–(3) of Definition 6.8 hold. (1) Suppose we have formulae $\varphi _{0}(x)$ and $\varphi _{1}(x)$ from the language of set theory. We let $\varphi _{0\times 1}(x)$ be the formula $\exists y,z(\varphi _{0}(y)\wedge \varphi _{1}(z)\wedge x=\langle x,y\rangle $ . Then let $\pi _{0}(x,y)$ say that there is some z such that $x=\langle y,z\rangle $ , and let $\pi _{1}(x,z)$ say that there is some y such that $x=\langle y,z\rangle $ .

(2) Suppose $\varphi _{0}(x)$ and $\varphi _{1}(x)$ are formulae of set theory. Let $\varphi _{0+1}(x)$ say that there is some $z,i$ such that $x=\langle z,i\rangle $ and either: $i=0$ and $\varphi _{0}(z)$ ; or $i=1$ and $\varphi _{1}(z)$ . Let $\sigma _{0}(x,y)$ say that $y=\langle x,0\rangle $ and let $\sigma _{1}(x,y)$ say that $y=\langle x,1\rangle $ .

(3) Suppose that $ZF$ proves that $\varphi _{1}(x,y)$ describes an equivalent relation on $\varphi _{0}(x)$ . Let $\rho (x,y)$ say that y is the set of those z of least rank such that $\varphi _{1}(x,z)$ .Footnote ⁴³ Let $\varphi _{0/1}(x)$ say that there is some z such that $\varphi _{0}(z)$ and $\rho (z,y)$ .

To establish the result, we prove the following lemma, which is essentially a special case of Lemma 5.7 of [Reference McEldowney18].

Lemma 6.10. If T is Morita complete in a single sorted language $\mathcal {L}_{T}$ and $T^{+}$ is a Morita successor of T, then $T^{+}$ and T are strictly bi-interpretable.

Proof. Given that this is essentially a special case of a more general result, we just do the case for products. Suppose $T^{+}$ is formed by adding a product sort $\sigma $ based on subsorts $\sigma _{0}$ and $\sigma _{1}$ from $\mathcal {L}_{T}$ defined by formulae $\varphi _{0}(x)$ and $\varphi _{1}(x)$ respectively. Let $\rho _{0}:\sigma \to \sigma _{0}$ and $\rho _{1}:\sigma \to \sigma _{1}$ be the associated projection functions. Then since T is Morita complete we may fix formulae $\varphi _{0\times 1}(x)$ , $\pi _{0}(x,y)$ and $\pi _{1}(x,y)$ from $\mathcal {L}_{T}$ such that T proves that $\pi _{0}$ and $\pi _{1}$ represent functions and that

$$\begin{align*}\forall x\forall y(\varphi_{0}(x)\wedge\varphi_{1}(y)\to\exists!z(\varphi_{0\times1}(z)\wedge\pi_{0}(z,x)\wedge\pi_{1}(z,y)). \end{align*}$$

We then define an interpretation $t:\mathcal {L}_{T^{+}}\to \mathcal {L}_{T}$ such that $t(\sigma )$ is sent to the only sort in $\mathcal {L}$ and $\delta _{t,\sigma }(x)$ is $\varphi _{0\times 1}(x)$ . We then let $t(\rho _{0})=\pi _{0}$ and $t(\rho _{1})=\pi _{1}$ . In the other direction we let $t^{+}:\mathcal {L}_{T}\to \mathcal {L}_{T^{+}}$ be the trivial interpretation.

Given a model $\mathcal {M}$ of T, it is easy to see that $t^{+}\circ t(\mathcal {M})=\mathcal {M}$ . Essentially, t just adds the new sort and then $t^{+}$ discards it. From the other direction, suppose $\mathcal {N}^{+}$ is a model of $T^{+}$ . We define an isomorphism between $\mathcal {N}^{+}$ and $t\circ t^{+}(\mathcal {M})$ with a formula $\psi (x,y)$ of $\mathcal {L}_{T^{+}}$ as follows. For the single sort of $\mathcal {L}_{T}$ , we just use the identity. For the new product sort, we let $\psi (x,y)$ be

$$\begin{align*}\exists x_{0}\exists x_{1}(\rho_{0}(x)=x_{0}\wedge\rho_{1}(x)=x_{1}\wedge\pi_{0}(y,x_{0})\wedge\pi_{1}(y,x_{1})). \end{align*}$$

It can then be seen that this defines an isomorphism.

Theorem 6.11. If T and S are sequential Morita-complete theories that are Morita equivalent, then T and S are definitionally equivalent.

Proof. Suppose T and S are Morita equivalent, Morita complete theories. Suppose that $T^{*}$ and $S^{*}$ are the Morita expansions witnessing this. Given that T and S are Morita complete we see by induction on Lemma 6.10 that T and $T^{*}$ are strictly bi-interpretable and so are S and $S^{*}$ . Thus since $S^{*}$ are and $T^{*}$ are definitionally equivalent, we see that T and S are strictly bi-interpretable. So by Theorem 3.13, we see that T and S are definitionally equivalent.

Thus, we see that for sufficiently strong theories, the distinction between Morita equivalence and definitional equivalence collapses.

6.1.1 Some remaining questions

While we’ve come to understand a lot of the relationships between equivalences in our table a number of questions remain. We highlight them in the following table.