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

Given an arbitrary connected groupoid G with its vertex group Ga, if G_a is a central subgroup of a group F, then there is a canonical extension F = G ⊗ F of G in the sense that Ob(G) = Ob(F), Mor(G) ⊆ 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 = acleq(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 ι_a: G_a → Z(F)), and show that the abstract groupoid G ⊗ F lives A-invariantly in models of T. More precisely, we A-invariantly construct a connected groupoid F, isomorphic to G ⊗ 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-automorphism 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(Ga) = i_a o ι_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.