Categorification of Hopf algebras of rooted trees

TL;DR

This paper constructs a monoidal category structure on P-trees that categorifies the Connes--Kreimer Hopf algebra of rooted trees, linking algebraic and categorical frameworks.

Contribution

It introduces a monoidal structure on categories of P-trees that lifts the algebraic structure of the Connes--Kreimer Hopf algebra into a categorical setting.

Findings

Categorifies the Connes--Kreimer Hopf algebra using P-trees

Defines a monoidal structure via polynomial functors

Connects the structure to the free monad on P

Abstract

We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec N) whose semiring of functions is (a P-version of) the Connes--Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to Z and collapsing H_0). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring in the polynomial representation of the free monad on P.