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.