nLab
definable groupoid

Contents

Idea

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.

Definition

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).

Remarks

  • 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)

Properties

Theorem

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).

References

  • 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 (2006), arxiv/math.LO/0603413

Revised on September 18, 2016 14:43:57 by jesse (130.113.105.198)