Representations of categories of finite relational structures and associated endomorphism monoids

TL;DR

This paper develops a unified representation theory for categories of finite relational structures, extending classical correspondences and classifying irreducible representations, with applications to topological monoids.

Contribution

It introduces a new framework for representing finite relational structures, extending Dold-Kan correspondence, and classifies irreducible modules, connecting to Artin's reconstruction theorem.

Findings

Finitely generated representations are noetherian or artinian depending on the ring.

Classified irreducible representations as either irreducible or length 2 modules.

Established a monoidal generalization of Artin's reconstruction theorem.

Abstract

We develop a unified representation theory for the categories of finite subsets and relation-preserving maps 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, we obtain a precise structural description of these representation categories. We classify irreducible representations, showing that canonical quotients of indecomposable standard modules are either irreducible (the regular case) or has length 2 (the singular case). In the case that $k$ has characteristic 0, we 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 topological transformation monoids and sheaves on the associated categories of finite subsets. In the special case where the transformation monoid is a permutation group, our result recovers Artin's theorem.