Le Bois Marie
35, route de Chartres
We introduce Feynman categories and show that they naturally define bi-algebras. In good circumstances these bi-algebras have Hopf quotients. Corresponding to several levels of sophistication and decoration (both terms have technical definitions), we recover the Hopf algebras of Goncharov and Brown from number theory, a Hopf algebra of Baues used in the analysis of double loop spaces and the various Hopf algebras of Connes-Kreimer used in QFT as examples of the general theory. Co-actions also appear naturally in this context as we will explain.