  We develop an algebraic theory of flavored and pathlike co-, bi- and Hopf
algebras. This is the right framework in which to discuss antipodes for
bialgebras naturally appearing in topology, number theory and physics. In
particular, we can precisely give conditions the invertibility of characters
needed for renormalization in the formulation of Connes and Kreimer and in the
canonical examples these conditions are met. To construct the antipodes
directly, we discuss formal localization constructions and quantum
deformations. These allow to construct and naturally explain the appearance
Brown style coactions. Using previous results, we can tie all the relevant
coalgebras to categorical constructions and the bialgebra structures to Feynman
categories.
