Incidence Categories

TL;DR

This paper introduces the incidence category of a family of posets, demonstrating its nearly abelian structure and linking its Ringel-Hall algebra to the incidence Hopf algebra of order ideals, generalizing previous constructions.

Contribution

It constructs a new category called the incidence category for families of posets, establishing its algebraic properties and connecting it to incidence Hopf algebras, extending prior work on rooted forests and Feynman graphs.

Findings

The incidence category is nearly abelian with kernels and cokernels.

The Ringel-Hall algebra of the category is isomorphic to the incidence Hopf algebra.

Generalizes previous categories for rooted forests and Feynman graphs.

Abstract

Given a family $\F$ of posets closed under disjoint unions and the operation of taking convex subposets, we construct a category $\C_{\F}$ called the \emph{incidence category of $\F$}. This category is "nearly abelian" in the sense that all morphisms have kernels/cokernels, and possesses a symmetric monoidal structure akin to direct sum. The Ringel-Hall algebra of $\C_{\F}$ is isomorphic to the incidence Hopf algebra of the collection $\P(\F)$ of order ideals of posets in $\F$. This construction generalizes the categories introduced by K. Kremnizer and the author In the case when $\F$ is the collection of posets coming from rooted forests or Feynman graphs.