Type-amalgamation properties and polygroupoids in stable theories

TL;DR

This paper explores how failures of higher-dimensional type amalgamation in stable theories can be characterized by algebraic structures called n-ary polygroupoids, extending previous results about definable groupoids.

Contribution

It introduces n-ary polygroupoids as witnesses for higher-dimensional amalgamation failures in stable theories, generalizing earlier work on 4-amalgamation.

Findings

Failures of higher-dimensional amalgamation are witnessed by n-ary polygroupoids.

n-ary polygroupoids are definable in a mild expansion of the language.

The work generalizes Hrushovski's result on 4-amalgamation and groupoids.

Abstract

We show that in a stable first-order theory, the failure of higher-dimensional type amalgamation can always be witnessed by algebraic structures which we call n-ary polygroupoids. This generalizes a result of Hrushovski that failures of 4-amalgamation in stable theories are witnessed by definable groupoids (which are 2-ary polygroupoids in our terminology). The n-ary polygroupoids are definable in a mild expansion of the language (adding a unary predicate for an infinite Morley sequence).