# Interpretations as morphisms

Model-theoretic interpretations seem to give rise to categories in which morphisms are not functions. To name just two small examples:

• the category of finite graphs with interpretations between them
• the category of finite relational structures of arbitrary signature with interpretations between them

In the category of finite graphs with graph homomorphisms one is primarily interested in the finitely many in-going morphisms of an object (which tell all there is to know about a given object). In an interpretational category there are infinitely many in-going morphisms (= interpretations), but only finitely many out-going ones: to those objects (= structures) a given object is able to interpret. So maybe the opposite category might be better suited for comparison's sake?

Maybe I am blind, but I didn't manage to find any thorough analysis of such a category with interpretations as morphisms. Maybe I just missed the right keywords to search with?

Addendum: I should have mentioned the reminiscence of homotopy when thinking of interpretations this way:

• Homotopy categories are standard examples of categories with morphisms that are not functions.

• In the Wikipedia article on interpretations we read: "[Bi-interpretability] permits to define an equivalence relation among structures, reminiscent of the homotopy equivalence among topological spaces."

• In chapter 5.4 Shapes and sizes of interpretations of his Model Theory, Wilfrid Hodges uses the term "homotopy" in the context of (c) Homotopies and bi-interpretations (but the connection with the standard definition of a homotopy isn't clear to me, to be honest).

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You might be interested in this paper by Albert Visser, thought it treats interpretations of theories rather than models: igitur-archive.library.uu.nl/lg/2008-0403-201345/… . –  Emil Jeřábek Aug 17 '12 at 10:24
Thanks, Emil. I stumbled over this paper in the course of my search, and certainly there will be connections... –  Hans Stricker Aug 17 '12 at 10:34
One of the deepest problems in a branch of model theory called "geometric stability theory" is classification of theories up to biinterpretability. The philosophy is that one first needs to classify "minimal" theories or definable sets where "minimal" can be given precise formal meaning in several ways. A good progress has been made for so-called stable theories and their generalisations (simple theories). One can define certain properties that determine the "geometric complexity" of theories. For example, "one-based" theories cannot interpret a field. –  Dima Sustretov Aug 17 '12 at 17:44
..and any group interpretable in a one-based theory is Abelian-by-finite. There are also "converse" results: if theory is complex enough then it must interpret a field or division ring. A comment really is not enough to give a substantial account of the theory, so I refer to some survey articles: Anand Pillay "Aspects of geometric model theory", David Marker "Strongly minimal sets and geometry", . All these are freely accesible, just google them. –  Dima Sustretov Aug 17 '12 at 17:53
By the way I suppose things work in this way: we have two interpretations $\Delta,\Gamma \colon \mathbf W \to \mathbf V$, where $\mathbf W$ and $\mathbf V$ are respectively a $L$ and a $K$ family of structures. An homotopy $\chi$ associate to every $A \in \mathbf V$ an isomorphism $\chi_A \colon \Delta(A) \to \Gamma(A)$, like an homotopy $h$ between to continuous function $f,g \colon X \to Y$ associate to every $a \in X$ a path $h_a \colon f(a) \to f(b)$. Hope this seems like a possible clarification. :) –  Giorgio Mossa Aug 21 '12 at 20:51