Generic algebras and iterated Hochschild homology

We investigate the properties of the iterated bar construction and generalize to it well-known results of Quillen, Gerstenhaber and Schack on the bar construction. In particular, we show that the iterated bar construction and the associated iterated Hochschild homology theories split canonically over any field, refining and extending previous results of Pirashvili. For that purpose, we introduce and study algebraic structures such as algebras, coalgebras or Hopf algebras in the Eilenberg–MacLane “category of constructions”. These results on constructions should be of interest on their own.