Graphical calculus underlying Grassmann, Clifford, and symmetric function Hopf algebras

Graphical calculus is a categorical method to describe abstract properties of algebraic structures in a dual way to the use of commutative diagrams. I will show that behind a rather astonishing wealth of algebraic structures there is a rather simple ribbon graph category which governs these structures in a unified way. Consequent usage of this tool led to new mathematical results, like new branching rules for non-classical groups, a cohomological classification of algebraic structures in QFT etc. Perhaps more interesting is that this line of thought shows up the need to weaken the notion of a Hopf algebra to deal with subjects like Dirichlet convolution rings in number theory or renormalization in quantum field theory.