We endow the category of coalgebras over an operad in connective non-negatively graded chain complexes with a co fibrantly generated model category structure. Then, we prove our main result : the category of bialgebras over a pair of operads (P,Q) in distribution admits a co fibrantly generated model category structure inherited from those of the P-algebras and the Q -coalgebras. This is the fi rst result of this type for a bialgebras category. It allows to do classical homotopical algebra in various categories such as associative bialgebras, Lie bialgebras or Poisson bialgebras in connective non-negatively graded chain complexes.