  We introduce the notion of a quadri-bialgebra, which gives a bialgebra theory
for the quadri-algebra introduced by Aguiar and Loday. We show that a
quadri-bialgebra is equivalent to a Manin triple of dendriform algebras
associated to a nondegenerate 2-cocycle, and to a Manin triple of
quadri-algebras associated to a nondegenerate invariant bilinear form.
Quadri-bialgebras also come from a variation of the classical Yang-Baxter
equation, called the $Q$-equations. Moreover, quadri-bialgebras fit into the
framework of construction of Rota-Baxter operators and Nijenhuis operators on
the double spaces of quadri-algebras.
