TY - JOUR

T1 - The incidence comodule bialgebra of the Baez–Dolan construction

AU - Kock, Joachim

N1 - Funding Information:
This work was presented at the Workshop on comodule bialgebras (GDR Renormalisation) in Clermont-Ferrand, November 2018. I wish to thank Dominique Manchon for a wonderful conference, and for the perfect opportunity for me to expose this material. I have benefitted much from related collaborations with Imma Gálvez, Andy Tonks, Mark Weber, Louis Carlier, Kurusch Ebrahimi-Fard, Loïc Foissy, and Frédéric Patras, all of whom I thank for their influence on various parts of this work. Thanks are due also to Marcelo Fiore, Ander Murua, Pierre-Louis Curien, Paul-André Melliès, André Joyal, Gabriella Böhm, and Birgit Richter, for input and feedback, and to the anonymous referees for catching a couple of small mistakes and for other pertinent remarks. Support from grants MTM2016-80439-P ( AEI/FEDER, UE ) of Spain and 2017-SGR-1725 of Catalonia is gratefully acknowledged.
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/6/4

Y1 - 2021/6/4

N2 - Starting from any operad P, one can consider on one hand the free operad on P, and on the other hand the Baez–Dolan construction on P. These two new operads have the same space of operations, but with very different notions of arity and substitution. The main result of this paper is that the incidence bialgebras of the two-sided bar constructions of the two operads constitute together a comodule bialgebra. The result is objective: it concerns comodule-bialgebra structures on groupoid slices, and the proof is given in terms of equivalences of groupoids and homotopy pullbacks. Comodule bialgebras in the usual sense are obtained by taking homotopy cardinality. The simplest instances of the construction cover several comodule bialgebras of current interest in analysis. If P is the identity monad, then the result is the Faà di Bruno comodule bialgebra (dual to multiplication and substitution of power series). If P is any monoid Ω (considered as a one-coloured operad with only unary operations), the resulting comodule bialgebra is the dual of the near-semiring of Ω-moulds under product and composition, as employed in Écalle's theory of resurgent functions in local dynamical systems. If P is the terminal operad, then the result is essentially the Calaque–Ebrahimi-Fard–Manchon comodule bialgebra of rooted trees, dual to composition and substitution of B-series in numerical analysis (Chartier–Hairer–Vilmart). The full generality is of interest in category theory. As it holds for any operad, the result is actually about the Baez–Dolan construction itself, providing it with a new algebraic perspective.

AB - Starting from any operad P, one can consider on one hand the free operad on P, and on the other hand the Baez–Dolan construction on P. These two new operads have the same space of operations, but with very different notions of arity and substitution. The main result of this paper is that the incidence bialgebras of the two-sided bar constructions of the two operads constitute together a comodule bialgebra. The result is objective: it concerns comodule-bialgebra structures on groupoid slices, and the proof is given in terms of equivalences of groupoids and homotopy pullbacks. Comodule bialgebras in the usual sense are obtained by taking homotopy cardinality. The simplest instances of the construction cover several comodule bialgebras of current interest in analysis. If P is the identity monad, then the result is the Faà di Bruno comodule bialgebra (dual to multiplication and substitution of power series). If P is any monoid Ω (considered as a one-coloured operad with only unary operations), the resulting comodule bialgebra is the dual of the near-semiring of Ω-moulds under product and composition, as employed in Écalle's theory of resurgent functions in local dynamical systems. If P is the terminal operad, then the result is essentially the Calaque–Ebrahimi-Fard–Manchon comodule bialgebra of rooted trees, dual to composition and substitution of B-series in numerical analysis (Chartier–Hairer–Vilmart). The full generality is of interest in category theory. As it holds for any operad, the result is actually about the Baez–Dolan construction itself, providing it with a new algebraic perspective.

KW - Comodule bialgebra

KW - Decomposition space

KW - Incidence bialgebra

KW - Operad

KW - Tree

KW - Two-sided bar construction

UR - http://www.scopus.com/inward/record.url?scp=85103068178&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2021.107693

DO - 10.1016/j.aim.2021.107693

M3 - Article

AN - SCOPUS:85103068178

SN - 0001-8708

VL - 383

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 107693

ER -