Representations of categories of finite relational structures and associated endomorphism monoids

Abstract: We develop a unified representation theory for the categories of finite substructures of highly homogeneous relational structures classified by Cameron. For any commutative coefficient ring $k$, we extend the classical Dold-Kan correspondence to this setting-with the sole exception of the category $\mathrm{FA}$-and prove that finitely generated representations are noetherian (resp., artinian) when $k$ is noetherian (resp., artinian). When $k$ is a field of characteristic $0$, we obtain a precise structural description of these representation categories. We classify irreducible representations, show that every indecomposable standard modules either is irreducible (the regular case) or has length 2 (the singular case), and establish a (direct sum or triangular) decomposition into a singular component governed by classical Dold-Kan theory and a regular component exhibiting semisimplicity or representation stability. Finally, we establish a monoidal generalization of Artin's reconstruction theorem for topological groups, proving an equivalence between uniformly continuous representations of infinite endomorphism monoids and sheaves on the associated finite substructure categories. In the special case where the endomorphism monoid is a permutation group, our result recovers Artin's theorem.