Seminar

In simple theory, n-amalgamation is the property that any coherent, independent system of types indexed by proper subsets of {1, ..., n} has a consistent extension. For stable theories, there is a three-way connection (first discovered by Hrushovski) between 4-amalgamation of types, definable groupoids, and the splitting of certain finite covers.

The present talk reports on joint work with Byunghan Kim and Alexei Kolesnikov which generalizes the equivalence of 4-amalgamation and eliminability of definable groupoids: in a stable theory, if n is minimal such that n-amalgamation fails, then in a mild expansion (adding a predicate for a Morley sequence) the theory interprets a homogeneous, locally finite structure which we call an (n-2)-ary polygroupoid that witnesses the failure of amalgamation. A 2-ary polygroupoid is an ordinary groupoid, and a k-ary polygroupoid has a k-ary "composition" operation which satisfies a generalized associativity law.