  Bialgebras and Hopf (bi)modules are typical algebraic structures with several
interacting operations. Their structural and homological study is therefore
quite involved. We develop the machinery of braided systems, tailored for
handling such multi-operation situations. Our construction covers the above
examples (as well as Poisson algebras, Yetter--Drinfel$'$d modules, and several
other structures, treated in separate publications). In spite of this
generality, graphical tools allow an efficient study of braided systems, in
particular of their representation and homology theories. These latter
naturally recover, generalize, and unify standard homology theories for
bialgebras and Hopf (bi)modules (due to Gerstenhaber--Schack,
Panaite--{\c{S}}tefan, Ospel, Taillefer); and the algebras encoding their
representation theories (Heisenberg double, algebras~$\mathscr X$, $\mathscr
Y$, $\mathscr Z$ of Cibils--Rosso and Panaite). Our approach yields simplified
and conceptual proofs of the properties of these objects.
