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...

We continue the study started recently by Agore, Bontea and Militaru in “Classifying bicrossed products of Hopf algebras” (2014), by describing and classifying all Hopf algebras $E$ that factorize through two Sweedler’s Hopf algebras. Equivalently, we classify all bicrossed products $H_4\bowtie H_4$. There are three steps in our approach. First, we explicitly describe the set of all matched pairs $(H_4,H_4,▹,◃)$ by proving that, with the exception of the trivial pair, this set is parameterized by the ground field $k$. Then, for...

Let (A,α) and (B,β) be two Hom-Hopf algebras. We construct a new class of Hom-Hopf algebras: R-smash products $(A{♮}_{R}B,\alpha \otimes \beta )$. Moreover, necessary and sufficient conditions for $(A{♮}_{R}B,\alpha \otimes \beta )$ to be a cobraided Hom-Hopf algebra are given.