Groupoids and Relative Internality

TL;DR

This paper explores how families of internal types in stable theories vary uniformly, introducing a type-definable groupoid structure that links properties of types to their internality and symmetry groups.

Contribution

It establishes the uniform variation of binding groups in relative internality and introduces a type-definable groupoid framework connecting internality properties to groupoid structures.

Findings

Binding groups vary uniformly in families of internal types.

Type-definable groupoids encode properties of internal types.

Criteria for internality of 2-analysable types are derived.

Abstract

In a stable theory, a stationary type $q \in S(A)$ internal to a family of partial types $\mathcal{P}$ over $A$ gives rise to a type-definable group, called its binding group. This group is isomorphic to the group $\mathrm{Aut}(q/\mathcal{P},A)$ of permutations of the set of realizations of $q$, induced by automorphisms of the monster model, fixing $\mathcal{P} \cup A$ pointwise. In this paper, we investigate families of internal types varying uniformly, what we will call relative internality. We prove that the binding groups also vary uniformly, and are the isotropy groups of a natural type-definable groupoid (and even more). We then investigate how properties of this groupoid are related to properties of the type. In particular, we obtain internality criteria for certain 2-analysable types, and a sufficient condition for a type to preserve internality.