Multisets of finite intervals and a universal category of poset representations

TL;DR

The article develops a universal, coefficient-free framework for poset representations by constructing a category $\operatorname{rep} P$ from any poset $P$, which is finitely cocomplete (and abelian when $P$ is finite) and enjoys a universal property with respect to functors into finitely cocomplete categories. It connects this framework to a concrete interval-based model for quivers by identifying $\mathcal{A}_n$ with $\operatorname{rep}(A_n, \mathbb{F}_2)$, and it analyzes a broad spectrum of subcategory closures, revealing rich combinatorial structures (Catalan numbers, Tamari lattices, Dyck paths) and explicit closure criteria. The universal construction further satisfies a natural, ring-agnostic equivalence $\operatorname{Fun}(\operatorname{rep} P, \mathcal{C}) \simeq \operatorname{Fun}(P, \mathcal{C})$ for any finitely cocomplete category $\mathcal{C}$, highlighting its broad applicability. The final part characterizes when $\operatorname{rep}(P, \mathbb{F})$ is abelian via the notion of coherent posets, linking categorical abelianness to lattice-theoretic compactness properties of ideals. Together, these results fuse poset combinatorics with abelian category theory and yield a universal, coefficient-free lens for poset representations.

Abstract

For any finite totally ordered set, the multisets of intervals form an abelian category. Various classes of subcategories admit natural combinatorial descriptions, and counting them yields familiar integer sequences. Surprisingly, in some cases new integer sequences arise. The formulation of this counting problem leads to a universal construction which assigns to any poset a finitely cocomplete additive category; it is abelian when the poset is finite and does not depend on the choice of any ring of coefficients. For a general poset the universal category of representations is abelian if and only if for the lattice of ideals the meet of two compact elements is again compact.

Theorems & Definitions (25)

• Definition 1
• Definition 2
• Lemma 3
• proof
• Remark 4
• Lemma 5
• Lemma 6
• proof
• Lemma 7
• proof