Non-commutative groupoids obtained from the failure of 3-uniqueness in stable theories

Abstract

Given an arbitrary connected groupoid G with its vertex group G(a), if G(a) is a central subgroup of a group F, then there is a canonical extension F = G circle times F of G in the sense that Ob(G) = Ob(F), Mor(G) subset of Mor(F), and F is isomorphic to all the vertex groups of F. From the failure of 3-uniqueness of a strong type p over A = acl(eq)(A) in a stable theory T, a canonical finitary connected commutative groupoid G with the binding group G was A-type-definably constructed by John Goodrick and Alexei Kolesnikov (2012). In this paper we take a certain (possibly non-commutative) automorphism group F where G is embedded centrally (so inducing t(a) : G(a) -> Z(F)), and show that the abstract groupoid G circle times F lives A-invariantly in models of T. More precisely, we A-invariantly construct a connected groupoid T, isomorphic to G circle times F as abstract groupoids, satisfying the following: (1) Ob(F) = Ob(G), and Mor(F) and composition maps are A-invariant (i.e., described by infinite disjunctions of conjunctions of formulas over A), so that an A-automor-phism of a model of T induces a groupoid automorphism of F. (2) There is an A-invariant faithful functor I : G -> F which is the identity on the objects, and I(G(a)) = i(a) o t(a), where i(a) is a canonical group isomorphism from F onto a vertex group F-a of F. An automorphism group approximated by the vertex groups of the non-commutative groupoids is suggested as a "fundamental group" of the strong type p.