A strictly commutative model for the cochain algebra of a space

Abstract

AbstractThe 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.