Borel Functors and Infinitary Interpretations

TL;DR

This paper establishes a deep connection between infinitary interpretations and Baire-measurable homomorphisms or functors between automorphism groups and categories of structures, showing they are essentially equivalent.

Contribution

It proves that every Baire-measurable homomorphism or functor arises from an infinitary interpretation, extending the understanding of the relationship between structures and their automorphism groups.

Findings

Baire-measurable homomorphisms are induced by infinitary interpretations.

Baire-measurable functors are induced by infinitary interpretations.

Complexity levels are preserved in the interpretation-functor correspondence.

Abstract

We introduce the notion of infinitary interpretation of structures. In general, an interpretation between structures induces a continuous homomorphism between their automorphism groups, and furthermore, it induces a functor between the categories of copies of each structure. We show that for the case of infinitary interpretation the reversals are also true: Every Baire-measurable homomorphism between the automorphism groups of two countable structures is induced by an infinitary interpretation, and every Baire-measurable functor between the set of copies of two countable structures is induced by an infinitary interpretation. Furthermore, we show the complexities are maintained in the sense that if the functor is $\mathbf{\Delta}^0_\alpha$, then the interpretation that induces it is $\Delta^{\mathtt{in}}_\alpha$ up to $\mathbf{\Delta}^0_\alpha$ equivalence.