In a model of a first-order theory a definable set might have additional algebraic stucture (e.g. that of a group, ring, groupoid, category, etc.) also given by definable functions and predicates. Since this set with extra structure is given by some collection of formulas in the language of , it is interpreted in every model of , and is hence an invariant of the syntactic category (walking model) of .
To make this precise:
Since groups, groupoids, rings, and categories can be given by algebraic theories, a definition (modulo having EI) of one of these in is just an interpretation of one of those theories in is just a logical functor from the walking models of one of these to .
Evaluating the unit of the Makkai duality adjunction at yields the category of ultrafunctors? from the category of models (logical functors ) to viewed as ultracategories?, so that one may identify a definable set (and definable sets with extra definable structure on them) with a functor of points on the category of models.
The fact that the external axiom of choice? (all epimorphisms split) holds in if and only if every fully faithful essentially surjective functor between small? categories is a true equivalence of categories can be word-for-word internalized to . This means: if the theory has two constants, then has definable Skolem functions if and only if every fully faithful essentially surjective definable functor between definable categories admits a pseudoinverse?.
With a suitable amount of choice (definable Skolem functions, for example) Freyd’s general adjoint functor theorem also carries over word-for-word to the setting of internal categories in .
In particular, there is much studied case of definable groups, cf. e.g. (Peterzil-Pillay)
There is a bijective correspondence between internal imaginary sorts of and definable concrete groupoids with a single isomorphism class, up to bi-interpretability over for the internal imaginary sorts and Hrushovski-equivalence for the definable concrete groupoids.
This is (Hrushovski 2006, Th.3.2).
Alessandro Berarducci, Definable groups in o-minimal structures, pdf; Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup, J. Symbolic Logic 74:3 (2009), 891-900, MR2548466, euclid, doi, O-minimal spectra, infinitesimal subgroups and cohomology, J. Symbolic Logic 72 (2007), no. 4, pp. 1177–1193, MR2371198, euclid, doi
Margarita Otero, A survey on groups definable in o-minimal structures, in: Model theory with applications to algebra and analysis. Vol. 2, 177–206, London Math. Soc. Lecture Note Ser. 350, Cambridge Univ. Press 2008, MR2010b:03042, doi