On representations of permutation groups and orbit categories

Abstract

Given an infinite set $\Omega$ and a ring $R$ as well as a group $G$ acting on them, we show that $G$ and a subgroup $H$ share the same canonical relational structure on $\Omega$ if and only if the restriction functor gives an equivalence from the category of discrete representations of $G$ to that of $H$. Moreover, the age of this relational structure satisfies the strong amalgamation property if and only if there is a canonical isomorphism from the category of finite substructures of $\Omega$ and embeddings to the opposite category of the orbit category of $G$. As an application, we prove that finitely generated discrete representations of highly homogeneous groups over the polynomial ring $k[\Omega]$ are Noetherian.