  The commutative differential graded algebra $A_{\mathrm{PL}}(X)$ of
polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy
theory. In this note, we construct an integral version $A^{\mathcal{I}}(X)$ of
$A_{\mathrm{PL}}(X)$. Our approach uses diagrams of chain complexes indexed by
the category of finite sets and injections $\mathcal{I}$ to model $E_{\infty}$
differential graded algebras by strictly commutative objects, called
commutative $\mathcal{I}$-dgas. We define a functor $A^{\mathcal{I}}$ from
simplicial sets to commutative $\mathcal{I}$-dgas and show that it is a
commutative lift of the usual cochain algebra functor. In particular, it gives
rise to a new construction of the $E_{\infty}$ dga of cochains.
  The functor $A^{\mathcal{I}}$ shares many properties of $A_{\mathrm{PL}}$,
and can be viewed as a generalization of $A_{\mathrm{PL}}$ that works over
arbitrary commutative ground rings. Working over the integers, a theorem by
Mandell implies that $A^{\mathcal{I}}(X)$ determines the homotopy type of $X$
when $X$ is a nilpotent space of finite type.
