# nLab interpretation

Contents

model theory

## Dimension, ranks, forking

• forking and dividing?

• Morley rank?

• Shelah 2-rank?

• Lascar U-rank?

• Vapnik–Chervonenkis dimension?

# Contents

## Idea

In formal logic and model theory, interpretation refers to equipping the syntax of some theory with a semantics. Naïvely this means finding a model (in the category of sets) for the theory. This is subsumed by treating interpretations as functors out of syntactic categories.

## Definition

In the language of categorical logic, interpretations are representations of theories inside some category $\mathbf{C}$. Depending on the kind of theory (cartesian, regular, coherent, first-order, geometric), an interpretation of $T$ in $\mathbf{C}$ is just a (cartesian, regular, coherent, first-order, geometric) functor to $\mathbf{C}$.

When $\mathbf{C}$ is Set, $T$ is first-order, and the functor (say $M$) is logical (in the sense of Makkai-Reyes, equivalently coherent if we take the Morleyization of $T$), we get models in the sense usually studied in model theory.

## Interpretations of theories in each other

Let $T_1$ and $T_2$ be (cartesian, regular, coherent, first-order, geometric) theories. A (cartesian, regular, coherent, first-order, geometric) interpretation $T_1 \to T_2$ is just a functor between the syntactic categories $\mathbf{Def}(T_1) \to \mathbf{Def}(T_2)$.

Elsewhere, interpretations have been defined as assignments of symbols in the language $\mathcal{L}_1$ of $T_1$ to definable sets of $T_2$ satisfying various coherence conditions (usually at least product-preserving) which amount to functoriality.

Note that via the duality between taking syntactic categories and internal logics, a model of $T$ in $\mathbf{Set}$ is just an interpretation of $T$ in the theory $\mathsf{Lang}(\mathbf{Set})$.

### Bi-interpretations of theories

We say that $T_1$ and $T_2$ are bi-interpretable if there are functors (of appropriate logical strength) $\mathbf{Def}(T_1) \leftrightarrows \mathbf{Def}(T_2)$ forming an equivalence of categories. One direction of conceptual completeness is that bi-interpretable theories have equivalent categories of models. This follows from the fact that (cartesian, regular, coherent, first-order, geometric) functors are closed under composition, and that equivalent categories induce equivalences of functor categories.

Many notions from geometric stability theory and classification theory? are invariant under bi-interpretability, e.g. stability, quantifier elimination, elimination of imaginaries, etc.

Since bi-interpretations induce equivalences of categories of models, monsters of bi-interpretable first-order theories $T_1, T_2$ will have isomorphic automorphism groups, with the isomorphism induced by restrictions along reducts. This indicates the bi-interpretation extends to the imaginaries of $T_1$ and $T_2$ also, so that $T_1^{\operatorname{eq}} \simeq T_2^{\operatorname{eq}}$, in fact uniquely—which is the universal property of the pretopos completion.

## Interpretations of models in each other

This notion has appeared in the model-theoretic literature, and is what some model theorists mean when they say “interpretation.” Specialize to first-order logic and models in Set, and fix models $M_1 \models T_1, M_2 \models T_2$. An interpretation of $M_1$ in $M_2$ is a surjection $U \overset{f}{\twoheadrightarrow} M_1$ for $U$ some subset of $M_2^k$, some $k \in \mathbb{N}$, such that the pullback of $f^*X$ of any definable (with parameters) set $X$ of $M_1$ along $f$ is again definable in $Y$. This is enough to induce a logical functor $\mathbf{Def}(T_1) \to \mathbf{Def}(T_2)$ (surjectivity implies witnessed existentials continue to be witnessed, and the functor being induced by pullback implies logicalness), in fact a logical functor $\mathbf{Def}(T_1(M_1)) \to \mathbf{Def}(T_2(M_2))$ of $T_1$ and $T_2$ enriched with the elementary diagrams of $M_1$ and $M_2$.

Facts:

• Every interpretation between theories can be realized as being induced by a concrete interpretation between sufficiently saturated models of those theories.

• Given an $f : U \twoheadrightarrow M_1$, any other $g : U \twoheadrightarrow M_1$ which is also an interpretation $M_1 \to M_2$ is of the form $\sigma \circ f$ for some $\sigma \in \operatorname{Aut}(M_1)$.

### Examples

• The construction of the Grothendieck group of a commutative monoid implements a canonical interpretation of the ring $\mathbb{Z}$ in the semiring $\mathbb{N}$ (in fact inside $\mathbb{N}$ viewed as the standard model of Peano arithmetic).

• The complex field $\mathbb{C}$ is canonically interpreted in the real field $\mathbb{R}$ by identifying $\mathbb{C}$ with $\mathbb{R}^2$ and noting that the multiplication of complex numbers is definable from multiplication on the reals.

## Classifying topses

An interpretation $T \to T'$ is precisely the inverse image part of a geometric morphism of classifying toposes $\mathcal{E}(T') \to \mathcal{E}(T)$. In particular, models of $T$ are points of Set in $\mathcal{E}(T)$.

• wikipedia interpretation (model theory)

• Wilfrid Hodges, A shorter model theory, Cambridge Univ. Press 1997

• Olivia Caramello, Topos-theoretic preliminaries.