Polynomial functors and combinatorial Dyson-Schwinger equations

Abstract

We present a general abstract framework for combinatorial Dyson-Schwinger equations, in which combinatorial identities are lifted to explicit bijections of sets, and more generally equivalences of groupoids. Key features of combinatorial Dyson-Schwinger equations are revealed to follow from general categorical constructions and universal properties. Rather than beginning with an equation inside a given Hopf algebra and referring to given Hochschild $1$-cocycles, our starting point is an abstract fixpoint equation in groupoids, shown canonically to generate all the algebraic structure. Precisely, for any finitary polynomial endofunctor $P$ defined over groupoids, the system of combinatorial Dyson-Schwinger equations $X=1+P(X)$ has a universal solution, namely the groupoid of $P$-trees. The isoclasses of $P$-trees generate naturally a Connes-Kreimer-like bialgebra, in which the abstract Dyson-Schwinger equation can be internalised in terms of canonical $B_{+}$-operators. The solution to this equation is a series (the Green function) which always enjoys a Faà di Bruno formula, and hence generates a sub-bialgebra isomorphic to the Faà di Bruno bialgebra. Varying $P$ yields different bialgebras, and cartesian natural transformations between various $P$ yield bialgebra homomorphisms and sub-bialgebras, corresponding for example to truncation of Dyson-Schwinger equations. Finally, all constructions can be pushed inside the classical Connes-Kreimer Hopf algebra of trees by the operation of taking core of $P$-trees. A byproduct of the theory is an interpretation of combinatorial Green functions as inductive data types in the sense of Martin-Löf Type Theory (expounded elsewhere).

v4: minor adjustments, 49pp, final version to appear in J. Math. Phys

References in corpus (6)

• An Étude in non-linear Dyson--Schwinger Equations
• Hopf algebras, from basics to applications to renormalization
• Decomposition spaces, incidence algebras and Möbius inversion I: basic theory
• Exponential renormalization
• Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion
• Decomposition spaces, incidence algebras and Möbius inversion II: completeness, length filtration, and finiteness

Cited by in corpus (6)

• Decomposition spaces, incidence algebras and Möbius inversion I: basic theory
• Decomposition spaces, incidence algebras and Möbius inversion II: completeness, length filtration, and finiteness
• Decomposition spaces and restriction species
• Faà di Bruno for operads and internal algebras
• The incidence comodule bialgebra of the Baez-Dolan construction
• Differential 2-rigs