TY - JOUR

T1 - Faà di Bruno for operads and internal algebras

AU - Kock, Joachim

AU - Weber, Mark

N1 - Funding Information:
Received 27 September 2016; revised 26 August 2018; published online 12 December 2018. 2010 Mathematics Subject Classification 16T10, 05A19, 18D50 (primary), 18C15, 18B40, 32A05, 57T30, 18G30 (secondary). J. Kock was supported by grant number MTM2013-42293-P of Spain. M. Weber acknowledges the support of the Australian Research Council grant no. DP130101172, and grant no. GA CR P201/12/G028 from the Czech Science Foundation.
Publisher Copyright:
© 2018 London Mathematical Society

PY - 2019/6

Y1 - 2019/6

N2 - For any coloured operad R, we prove a Faà di Bruno formula for the ‘connected Green function’ in the incidence bialgebra of R. This generalises on one hand the classical Faà di Bruno formula (dual to composition of power series), corresponding to the case where R is the terminal reduced operad, and on the other hand the Faà di Bruno formula for R -trees of Gálvez–Kock–Tonks (P a finitary polynomial endofunctor), which corresponds to the case where R is the free operad on P. Following Gálvez–Kock–Tonks, we work at the objective level of groupoid slices, hence all proofs are ‘bijective’: the formula is established as the homotopy cardinality of an explicit equivalence of groupoids, in turn derived from a certain two-sided bar construction. In fact we establish the formula more generally in a relative situation, for algebras of one polynomial monad internal to another. This covers in particular nonsymmetric operads (for which the terminal reduced case yields the noncommutative Faà di Bruno formula of Brouder–Frabetti–Krattenthaler).

AB - For any coloured operad R, we prove a Faà di Bruno formula for the ‘connected Green function’ in the incidence bialgebra of R. This generalises on one hand the classical Faà di Bruno formula (dual to composition of power series), corresponding to the case where R is the terminal reduced operad, and on the other hand the Faà di Bruno formula for R -trees of Gálvez–Kock–Tonks (P a finitary polynomial endofunctor), which corresponds to the case where R is the free operad on P. Following Gálvez–Kock–Tonks, we work at the objective level of groupoid slices, hence all proofs are ‘bijective’: the formula is established as the homotopy cardinality of an explicit equivalence of groupoids, in turn derived from a certain two-sided bar construction. In fact we establish the formula more generally in a relative situation, for algebras of one polynomial monad internal to another. This covers in particular nonsymmetric operads (for which the terminal reduced case yields the noncommutative Faà di Bruno formula of Brouder–Frabetti–Krattenthaler).

UR - http://www.scopus.com/inward/record.url?scp=85058369555&partnerID=8YFLogxK

U2 - 10.1112/jlms.12201

DO - 10.1112/jlms.12201

M3 - Article

AN - SCOPUS:85058369555

SN - 0024-6107

VL - 99

SP - 919

EP - 944

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 3

ER -