nLab definable groupoid



Model theory

Internal categories



In a model MM of a first-order theory TT a definable set GG 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 TT, it is interpreted in every model of TT, and is hence an invariant of the syntactic category (walking model) Def(T)\mathbf{Def}(T) of TT.


To make this precise:

  • A definable group is just a group object in Def(T)\mathbf{Def}(T).
  • A definable groupoid is just a groupoid object, i.e. an internal groupoid in Def(T)\mathbf{Def}(T).
  • A definable ring is just a semigroup object in the category of abelian group objects of Def(T)\mathbf{Def}(T).
  • A definable category is just a category object, i.e. an internal category in Def(T)\mathbf{Def}(T).

Since groups, groupoids, rings, and categories can be given by algebraic theories, a definition (modulo having EI) of one of these in TT is just an interpretation of one of those theories in TT is just a logical functor from the walking models of one of these to Def(T)\mathbf{Def}(T).


  • Evaluating the unit of the Makkai duality adjunction at TT yields Def(T)Hom Ult(Mod(T),Set)\mathbf{Def}(T) \simeq \operatorname{Hom}_{\mathbf{Ult}}(\mathbf{Mod}(T), \mathbf{Set}) the category of ultrafunctors? from the category of models (logical functors Def(T)Set\mathbf{Def}(T) \to \mathbf{Set}) to Set\mathbf{Set} viewed as ultracategories, so that one may identify a definable set (and definable sets with extra definable structure on them) XDef(T)X \in \mathbf{Def}(T) with a functor of points MX(M)M \mapsto X(M) on the category of models.

  • The fact that the external axiom of choice (all epimorphisms split) holds in Set\mathbf{Set} 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 Def(T)\mathbf{Def}(T). This means: if the theory has two constants, then TT 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 Def(T)\mathbf{Def}(T).

  • 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 TT and definable concrete groupoids with a single isomorphism class, up to bi-interpretability over TT for the internal imaginary sorts and Hrushovski-equivalence for the definable concrete groupoids.

This is (Hrushovski 2006, Th.3.2).


  • Y. Peterzil, A. Pillay, Generic sets in definably compact groups, Fundamenta Mathematicae 193 (2007), pp. 153–170, MR2282713, doi
  • Y. Peterzil, A. Pillay, S. Starchenko, Linear groups definable in o-minimal structures, J. Algebra 247 (2002), no. 1, pp. 1–23, MR1873380, doi

  • 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

  • Ehud Hrushovski, Groupoids, imaginaries and internal covers Turk. J. Math 36 (2012) 173 – 198 doi arxiv/math.LO/0603413 (2006)

  • Ehud Hrushovski, On finite imaginaries, arxiv/0902.0842

Hrushovski’s correspondence between definable connected groupoids in a theory TT and internal generalised imaginary sorts of TT is extended to an equivalence of categories in

  • Levon Haykazyan, Rahim Moosa, Functoriality and uniformity in Hrushovski’s groupoid-cover correspondence, Annals of Pure and Applied Logic 169:8 (2018) 705– 730 arXiv/1711.03531 doi

On higher categorical analogues of definable groupoids and internal covers:


Last revised on April 19, 2023 at 19:06:18. See the history of this page for a list of all contributions to it.