Incidence categories

Given a family F of posets closed under disjoint unions and the operation of taking convex subposets, we construct a category CF called the 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 CF 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.