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?
Where can I learn more about categories with interpretations as morphisms?
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).