Using a Day convolution product on diagrams of chain complexes indexed by the category of finite sets and injections I allows one to model E-infinity differential graded algebras by strictly commutative objects, called commutative I-dgas. In this note we introduce a functor A^I from simplicial sets to commutative I-dgas that is a commutative lift of the usual cochain algebra functor. In particular, A^I gives rise a new construction of the E-infinity dga of cochains. Our approach is motivated by the functor A_{PL} of polynomial forms on simplicial sets used in rational homotopy theory. The functor A^I shares many properties of A_{PL}, and can be viewed as a generalization of A_{PL} that works over arbitrary commutative ground rings.

中文翻译：

---

空间链代数的严格交换模型

在由有限集和注入 I 的类别索引的链复合图上使用 Day 卷积乘积，可以通过严格交换对象（称为交换 I-dgas）对 E-无穷大微分分级代数进行建模。在这篇笔记中，我们介绍了一个从单纯集合到交换 I-dgas 的函子 A^I，它是通常的 cochain 代数函子的交换提升。特别是，A^I 产生了 Cochain 的 E-infinity dga 的新构造。我们的方法受到有理同伦理论中使用的单纯集上多项式形式的函子 A_{PL} 的启发。函子 A^I 共享 A_{PL} 的许多性质，可以看作是 A_{PL} 的泛化，适用于任意交换接地环。