Twisted bialgebras, cofreeness and cointeraction

We study twisted bialgebras and double twisted bialgebras, that is to say bialgebras in the category of linear species, or in the category of species in the category of coalgebras. We define the notion of cofree twisted coalgebra and generalize Hoffman's quasi-shuffle product, obtaining in particular a twisted bialgebra of set compositions Comp. Given a special character , this twisted bialgebra satisfies a terminal property, generalizing the one of the Hopf algebra of quasisymmetric functions proved by Aguiar, Bergeron and Sottile. We give Comp a second coproduct, making it a double twisted bialgebra, and prove that it is a terminal object in the category of double twisted bialgebras. Actions of characters on morphisms allow to obtain every twisted bialgebra morphisms from a connected double twisted bialgebra to Comp. These results are applied to examples based on graphs and on finite topologies, obtaining species versions of the chromatic symmetric series and chromatic polynomials, or of the Ehrhart polynomials. Moreover, through actions of monoids of characters, we obtain a Twisted bialgebraic interpretation of the duality principle.