Combinatorial Dyson-Schwinger equations and polynomial functors
I will explain what are the combinatorial Dyson-Schwinger
equations, first by briefly outlining their role in quantum
field theory, and then by showing that they can be understood
in terms of some elementary category theory. The formulation
of the equations by Bergbauer and Kreimer starts with a Hopf
algebra (of trees or graphs), and a collection of Hochschild
1-cocycles, and one of their main theorems is that the solution
spans a sub Hopf algebra isomorphic to the Faa di Bruno Hopf
algebra (the Hopf algebra dual to composition of formal power
series). The new categorical interpretation starts very
abstractly with a polynomial fixpoint equation of sets, nothing
more. I will explain how this data canonically generates
trees, Hopf algebras, Hochschild 1-cocycles, and Faa di Bruno
formula. This reveals close connections with inductive data
types in program semantics, which I can explain if time
permits.