Incidence Categories

Given a family $\text{F}$ of posets closed under disjoint unions and the operation of taking convex subposets, we construct a category ${\text{C}}_{\text{F}}$ called the \emph{incidence category of $\text{F}$}. 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 ${\text{C}}_{\text{F}}$ is isomorphic to the incidence Hopf algebra of the collection $\mathrm{\P }(\text{F})$ of order ideals of posets in $\text{F}$. This construction generalizes the categories introduced by K. Kremnizer and the author In the case when $\text{F}$ is the collection of posets coming from rooted forests or Feynman graphs.