  We develop an algebraic theory of colored, semigrouplike-flavored and
pathlike co-, bi- and Hopf algebras. This is the right framework in which to
discuss antipodes for bialgebras naturally appearing in combinatorics,
topology, number theory and physics. In particular, we can precisely give
conditions for the invertibility of characters that is needed for
renormalization in the formulation of Connes and Kreimer. These are met in the
relevant examples. In order to construct antipodes, we discuss formal
localization constructions and quantum deformations. These allow to define and
explain the appearance of Brown style coactions. Using previous results, we can
interpret all the relevant coalgebras as stemming from a categorical
construction, tie the bialgebra structures to Feynman categories, and apply the
developed theory in this setting.
