Quasishuffle double bialgebras

Quasishuffle Hopf algebras, usually defined on a commutative monoid, can be more generally defined on any associative algebra V. If V is a commutative and cocommutative bialgebra, the associated quasishuffle bialgebra QSh(V) inherits a second coproduct $\delta$ of contraction and extraction of words, cointeracting with the deconcatenation coproduct $\Delta$, making QSh(V) a double bialgebra. In order to generalize the universal property of the Hopf algebra of quasisymmetric functions QSym (a particular case of quasishuffle Hopf algebra) as exposed by Aguiar, Bergeron and Sottile, we introduce the notion of double bialgebra over V. A bialgebra over V is a bialgebra in the category of right V-comodules and an extra condition is required on the second coproduct for double bialgebras over V. We prove that the quasishuffle bialgebra QSh(V) is a double bialgebra over V , and that it satisfies a universal property: for any bialgebra B over V and for any character $\lambda$ of B, under a connectedness condition, there exists a unique morphism $\phi$ of bialgebras over V from B to QSh(V) such that $\epsilon_\delta \circ \phi=\lambda$. When V is a double bialgebra over V , we obtain a unique morphism of double bialgebras over V from B to QSh(V), and show that this morphism $\phi_1$ allows to obtain any morphism of bialgebra over V from B to QSh(V) thanks to an action of a monoid of characters. This formalism is applied to a double bialgebra of V-decorated graphs