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

TL;DR

This paper introduces a new construction of possibly non-commutative groupoids arising from the failure of 3-uniqueness in stable theories, extending previous commutative models and proposing a novel automorphism group as a fundamental group.

Contribution

It constructs non-commutative groupoids from 3-uniqueness failures and suggests a new automorphism group as a fundamental group of the strong type.

Findings

Constructed non-commutative groupoids from 3-uniqueness failure

Extended the commutative groupoid framework to non-commutative cases

Proposed an automorphism group as a fundamental group of the strong type

Abstract

We construct a possibly non-commutative groupoid from the failure of $3$-uniqueness of a strong type. The commutative groupoid constructed by John Goodrick and Alexei Kolesnikov in \cite{GK} lives in the center of the groupoid. A certain automorphism group approximated by the vertex groups of the non-commutative groupoids is suggested as a "fundamental group" of the strong type.