A Cacti theoretical interpretation of the axioms of bialgebras and H-module algebras

We establish a dictionary between the Cacti algebra axioms on a Cacti algebra structure with underlying free associative algebra, under suitable good behavior with degrees. Using these ideas, for an associative algebra A and a bialgebra H  , we also translate Cacti algebra maps Ω(H)→C•(A)Ω(H)→C•(A) (where Ω(H)Ω(H) stands for the cobar construction on H   and C•(A)C•(A) is the Hochschild cohomology complex) with H-module algebra structures on A, and illustrate with examples of applications.