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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 (dga) 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.
