# Co-theory of sorted profinite groups for PAC structures

## TL;DR

This paper extends Galois theory to many-sorted structures, providing coding methods, independence theorems, and characterizations of algebraic closure in PAC structures within model theory.

## Contribution

It introduces a new framework for Galois groups in many-sorted structures and develops tools for analyzing PAC substructures and their properties.

## Key findings

- Developed a variant of Galois theory for many-sorted structures
- Provided a coding method for Galois groups in monster models
- Proved the Weak Independence Theorem for PAC substructures

## Abstract

We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove the "Weak Independence Theorem" for PAC substructures of an ambient structure with nfcp and the property B(3). Fourth, we describe Kim-dividing in these PAC substructures and show several results related to the SOPn hierarchy. Fifth, we characterize the algebraic closure in PAC structures