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- W2782043977 abstract "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 (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." @default.
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- W2782043977 date "2020-08-01" @default.
- W2782043977 modified "2023-10-18" @default.
- W2782043977 title "A strictly commutative model for the cochain algebra of a space" @default.
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- W2782043977 doi "https://doi.org/10.1112/s0010437x20007319" @default.
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