  We generalize the notion of a coloring complex of a graph to linearized
combinatorial Hopf monoids. We determine when a linearized combinatorial Hopf
monoid has such a construction, and discover some inequalities that are
satisfied by the quasisymmetric function invariants associated to the
combinatorial Hopf monoid. We show that the collection of all such coloring
complexes forms a linearized combinatorial Hopf monoid, which is the terminal
object in the category of combinatorial Hopf monoids with convex characters. We
also study several examples of combinatorial Hopf monoids.
