A strictly commutative model for the cochain algebra of a space

Abstract

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.