Co-theory of sorted profinite groups for PAC structures

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 pseudo-algebraically closed (PAC) substructures of an ambient structure with no finite cover property (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.