Title: Polynomial functors and combinatorial Dyson--Schwinger equations
Abstract: I'll 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. Many standard 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, the starting point is an
abstract fixpoint equation in groupoids, whose solution spans its own
bialgebra. 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 Faˆ di Bruno formula, and hence generates a
sub-bialgebra isomorphic to the Faˆ 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 Green
functions as inductive datatypes in the sense of Martin-Lšf Type
Theory, but this might not fit into the talk.