Categorification of Hopf algebras of rooted trees

Joachim Kock

Open Mathematics (2013)

  • Volume: 11, Issue: 3, page 401-422
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Joachim Kock. "Categorification of Hopf algebras of rooted trees." Open Mathematics 11.3 (2013): 401-422. <http://eudml.org/doc/269593>.

@article{JoachimKock2013,
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.},
author = {Joachim Kock},
journal = {Open Mathematics},
keywords = {Rooted trees; Hopf algebras; Categorification; Monoidal categories; Polynomial functors; Finite sets; rooted trees; Connes-Kreimer bialgebras; categorification; monoidal categories; polynomial functors; operadic trees},
language = {eng},
number = {3},
pages = {401-422},
title = {Categorification of Hopf algebras of rooted trees},
url = {http://eudml.org/doc/269593},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Joachim Kock
TI - Categorification of Hopf algebras of rooted trees
JO - Open Mathematics
PY - 2013
VL - 11
IS - 3
SP - 401
EP - 422
AB - 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.
LA - eng
KW - Rooted trees; Hopf algebras; Categorification; Monoidal categories; Polynomial functors; Finite sets; rooted trees; Connes-Kreimer bialgebras; categorification; monoidal categories; polynomial functors; operadic trees
UR - http://eudml.org/doc/269593
ER -

References

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