  We endow the category of bialgebras over a pair of operads in distribution
with a cofibrantly generated model category structure. We work in the category
of chain complexes over a field of characteristic zero. We split our
construction in two steps. In the first step, we equip coalgebras over an
operad with a cofibrantly generated model category structure. In the second one
we use the adjunction between bialgebras and coalgebras via the free algebra
functor. This result allows us to do classical homotopical algebra in various
categories such as associative bialgebras, Lie bialgebras or Poisson bialgebras
in chain complexes.
