nLab interpretation

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 toposes

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 $\mathcal{E}(T)$.

Related entries

• formula

• model theory, structure (model theory)

• geometric morphism

References

• wikipedia interpretation (model theory)

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

• Olivia Caramello, Topos-theoretic preliminaries.