A strictly commutative model for the cochain algebra of a space

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.