Multisets of finite intervals and a universal category of poset representations

TL;DR

This paper explores the structure of categories formed by multisets of finite intervals over posets, introduces a universal construction for poset representations, and characterizes when these categories are abelian.

Contribution

It introduces a universal category construction for poset representations that is independent of coefficient rings and characterizes when these categories are abelian.

Findings

Multisets of finite intervals form an abelian category for finite totally ordered sets.

Counting subcategories yields familiar and new integer sequences.

The universal category is abelian iff the lattice of ideals satisfies a specific compactness condition.

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.