  Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral
equations have inspired the study of algebraic structures with replicating
operations. To understand their algebraic and combinatorial nature, we first
use rooted forests with multiple decoration sets to construct free Hopf
algebras with multiple Hochschild 1-cocycle conditions. Applying the universal
property of the underlying operated algebras and the method of
Gr\"obner-Shirshov bases, we then construct free objects in the category of
matching Rota-Baxter algebras which is a generalization of Rota-Baxter algebras
to allow multiple Rota-Baxter operators. Finally the free matching Rota-Baxter
algebras are equipped with a cocycle Hopf algebra structure.
