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Categorification of Hopf algebras
of rooted trees
By Joachim Kock
Centr. Eur. J. Math.
11 (2013), 401-422.
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 H0). 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.
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