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Ordnung muss sein

Henning Krause

TL;DR

The paper addresses when a $\\mathbb{K}$-linear length category is the representation category of a finite poset, i.e., $\\operatorname{rep}(P,\\mathbb{K})$ for some finite $P$. It presents a bidirectional criterion based on the ext-quiver, endomorphism rings of simples, and the existence of a distributive-generator object $M$, yielding an explicit reconstruction of the poset from Ext-data and an isomorphism with the incidence algebra $\\mathbb{K}P$. The main contribution is a precise characterization (Theorem) linking length categories to poset representations, together with the observation that distributive generators are not unique and can be classified by 2-cocycles in $H^*(\\Sigma(P),\\mathbb{K}^{\times})$. This work integrates length-category theory, poset representations, and incidence-algebra perspectives, providing a concrete structural criterion for when a category is determined by its ext-quiver and suggesting avenues for generalization to skew-field settings.

Abstract

For any length category, we establish a set of rules (necessary and sufficient) that ensure a partial order on the isomorphism classes of simple objects such that the category is equivalent to the category of finite dimensional representations of this partially ordered set.

Ordnung muss sein

TL;DR

The paper addresses when a -linear length category is the representation category of a finite poset, i.e., for some finite . It presents a bidirectional criterion based on the ext-quiver, endomorphism rings of simples, and the existence of a distributive-generator object , yielding an explicit reconstruction of the poset from Ext-data and an isomorphism with the incidence algebra . The main contribution is a precise characterization (Theorem) linking length categories to poset representations, together with the observation that distributive generators are not unique and can be classified by 2-cocycles in . This work integrates length-category theory, poset representations, and incidence-algebra perspectives, providing a concrete structural criterion for when a category is determined by its ext-quiver and suggesting avenues for generalization to skew-field settings.

Abstract

For any length category, we establish a set of rules (necessary and sufficient) that ensure a partial order on the isomorphism classes of simple objects such that the category is equivalent to the category of finite dimensional representations of this partially ordered set.
Paper Structure (2 sections, 6 theorems, 17 equations)

This paper contains 2 sections, 6 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. The proof

Key Result

Theorem 1

Let $\mathbb K$ be a field. A $\mathbb K$-linear length category is equivalent to the category of finite dimensional representations of a finite poset if and only if the following holds: In this case the poset is given by the isomorphism classes of simple objects, with $[S]\le [T]$ if there is a path $[S]\to\cdots\to [T]$ in the ext-quiver. In particular, the Hasse diagram of the poset identifies

Theorems & Definitions (16)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 6 more