Faa di Bruno for operads and internal algebras

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Faa di Bruno for operads and internal algebras. / Kock, Joachim; Weber, Mark.

I: Journal of the London Mathematical Society, Bind 99, Nr. 3, 06.2019, s. 919-944.

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt

Harvard

Kock, J & Weber, M 2019, 'Faa di Bruno for operads and internal algebras', Journal of the London Mathematical Society, bind 99, nr. 3, s. 919-944. https://doi.org/10.1112/jlms.12201

APA

Kock, J., & Weber, M. (2019). Faa di Bruno for operads and internal algebras. Journal of the London Mathematical Society, 99(3), 919-944. https://doi.org/10.1112/jlms.12201

Vancouver

Kock J, Weber M. Faa di Bruno for operads and internal algebras. Journal of the London Mathematical Society. 2019 jun.;99(3):919-944. https://doi.org/10.1112/jlms.12201

Author

Kock, Joachim ; Weber, Mark. / Faa di Bruno for operads and internal algebras. I: Journal of the London Mathematical Society. 2019 ; Bind 99, Nr. 3. s. 919-944.

Bibtex

@article{6c5e3341ed0f47da84773997abfb26de,
title = "Faa di Bruno for operads and internal algebras",
abstract = "For any coloured operad R, we prove a Faa di Bruno formula for the 'connected Green function' in the incidence bialgebra of R. This generalises on one hand the classical Faa 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 Faa di Bruno formula for P-trees of Galvez-Kock-Tonks (P a finitary polynomial endofunctor), which corresponds to the case where R is the free operad on P. Following Galvez-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 Faa di Bruno formula of Brouder-Frabetti-Krattenthaler).",
keywords = "QUANTUM-FIELD THEORY, HOPF ALGEBRA, POLYNOMIAL FUNCTORS, RENORMALIZATION, BIALGEBRAS, PARTITIONS, CATEGORIES, FORMULA, GRAPHS, TREES",
author = "Joachim Kock and Mark Weber",
year = "2019",
month = jun,
doi = "10.1112/jlms.12201",
language = "English",
volume = "99",
pages = "919--944",
journal = "Journal of the London Mathematical Society",
issn = "0024-6107",
publisher = "Oxford University Press",
number = "3",

}

RIS

TY - JOUR

T1 - Faa di Bruno for operads and internal algebras

AU - Kock, Joachim

AU - Weber, Mark

PY - 2019/6

Y1 - 2019/6

N2 - For any coloured operad R, we prove a Faa di Bruno formula for the 'connected Green function' in the incidence bialgebra of R. This generalises on one hand the classical Faa 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 Faa di Bruno formula for P-trees of Galvez-Kock-Tonks (P a finitary polynomial endofunctor), which corresponds to the case where R is the free operad on P. Following Galvez-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 Faa di Bruno formula of Brouder-Frabetti-Krattenthaler).

AB - For any coloured operad R, we prove a Faa di Bruno formula for the 'connected Green function' in the incidence bialgebra of R. This generalises on one hand the classical Faa 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 Faa di Bruno formula for P-trees of Galvez-Kock-Tonks (P a finitary polynomial endofunctor), which corresponds to the case where R is the free operad on P. Following Galvez-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 Faa di Bruno formula of Brouder-Frabetti-Krattenthaler).

KW - QUANTUM-FIELD THEORY

KW - HOPF ALGEBRA

KW - POLYNOMIAL FUNCTORS

KW - RENORMALIZATION

KW - BIALGEBRAS

KW - PARTITIONS

KW - CATEGORIES

KW - FORMULA

KW - GRAPHS

KW - TREES

U2 - 10.1112/jlms.12201

DO - 10.1112/jlms.12201

M3 - Journal article

VL - 99

SP - 919

EP - 944

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 3

ER -

ID: 331497852