Categories of finite relational structures — graphs, partial orders, hypergraphs, metric spaces — have a rich algebraic life that mirrors classical representation theory. Each finite substructure of a homogeneous relational structure defines an object in a category, and the morphisms between objects are the embeddings and extensions that the relational structure permits.

The unified representation theory for these categories extends the Dold-Kan correspondence — a classical result connecting chain complexes with simplicial abelian groups — to the setting of finite relational structures. The extension isn’t straightforward because relational structures lack the symmetry and linearity that simplicial objects enjoy. The category of finite graphs, for instance, has a much wilder morphism structure than the category of finite totally ordered sets.

The classification of irreducible representations proceeds through a filtration by size: representations built on substructures of size at most n form a subcategory, and the simple objects in each layer are classified by the combinatorics of the relational structure’s automorphism groups.

The connection to sheaves on topological spaces associated to the relational structure provides geometric intuition. The representation category “looks like” sheaves on a space whose points are the substructures and whose topology encodes the extension relations. This isn’t metaphor — there’s a precise functor establishing the connection.

The result unifies representation-theoretic studies that were previously done case-by-case: representations of graph categories, representations of poset categories, representations of matroids. The common framework reveals that the same structural theorems hold across all homogeneous relational structures, with the differences confined to the combinatorics of the specific structure’s automorphism groups.

One framework, many categories. The representation theory was always the same; only the examples varied.