Homotopy coherent commutative multiplications on chain complexes of modules over a commutative ring can be encoded by the action of an $E$-infinity operad. Alternatively, one can model such $E$-infinity differential graded algebras by commutative $I$-dgas, which are strictly commutative objects in diagrams of chain complexes indexed by the category of finite sets and injections $I$. In this talk $I$ will explain how the cochain algebra of a space arises as a commutative $I$-dga in a natural way. This construction can be viewed as a generalization of the functor of polynomial forms on simplicial sets from rational homotopy theory that works over arbitrary commutative ground rings. Passing to the homotopy colimit over $I$, the commutative $I$-dga model for cochain algebras gives rise to a new construction of the well known $E$-infinity structure on cochains.

This is joint work with Birgit Richter.