Categorification of Hopf algebras of rooted trees
arXiv:1109.5785
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.
29 pages. Does not compile with pdflatex due to dependency on the texdraw package. v2: expository improvements, following suggestions from the referees; final version to appear in Centr. Eur. J. Math