Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Representations of categories of finite relational structures and associated endomorphism monoids

Liping Li

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.

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 , we extend the classical Dold-Kan correspondence to this setting-with the sole exception of the category -and prove that finitely generated representations are noetherian (resp., artinian) when is noetherian (resp., artinian). When is a field of characteristic , 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.
Paper Structure (20 sections, 55 theorems, 238 equations)

This paper contains 20 sections, 55 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Notations
  4. Main results
  5. Organization and conventions
  6. Reedy Structure
  7. Highly homogeneous structures and mutation graphs
  8. Normalization
  9. Standard modules
  10. Further results for ${\mathrm{CA}}$, ${\mathrm{SA}}$ and ${\mathrm{FA}}$
  11. ${\mathrm{CA}}$ and ${\mathrm{SA}}$
  12. ${\mathrm{FA}}$
  13. Uniformly continuous representations of endomorphism monoids
  14. Continuous and uniformly continuous representations
  15. Realization and its right adjoint
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $k$ be a commutative ring and let ${\mathscr{C}} \in \{{\mathrm{OA}},{\mathrm{CA}},{\mathrm{BA}},{\mathrm{SA}}\}$. Then ${\mathscr{C}} {\textrm{-}\mathrm{Mod}}$ is equivalent to the representation category of the $k$-linear category $\mathscr{K}$ generated by: The automorphism groups $G_n$ form a crossed simplicial group in the sense of Fiedorowicz--Loday FL91: elements of $G_n$ permute the f

Theorems & Definitions (120)

  • Theorem 1.1: Theorem \ref{['generalized D-K correspondence']}
  • Remark 1.2
  • Corollary 1.3: Corollary \ref{['finite length 1']}
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6: Theorem \ref{['hom spaces']}
  • Theorem 1.7: Propositions \ref{['regular standard is simple']} and \ref{['length 2']}
  • Corollary 1.8: Theorem \ref{['classification of irreducibles']}
  • Remark 1.9
  • Theorem 1.10: Theorem \ref{['equivalence 2']} and Corollary \ref{['equivalence 3']}
  • ...and 110 more