Polynomial differential forms on $BG$ Let $\Omega^{*}_{\text{poly}}\: : \: sSet\to dg_{\geq 0}Comm_{+}$ be the polynomial De Rahm functor on simplicial sets, where the codomain is the category of commutative differential graded algebras over a field of charachteristic zero. Let $G$ be a group, what is $\Omega^{*}_{\text{poly}}(BG)$? Its cohomology? Do you have some references for that?
 A: Let $\mathbb{F}$ be a field of characteristic zero.
For any simplicial set $Y$ denote by $RY$ its topological realization then you have an isomorphism of graded algebras:
$$H^*(\Omega_{poly}^*(Y,\mathbb{F}))\cong H^*_{Sing}(RY,\mathbb{F}).$$
Any book about rational homotopy theory will be a good reference:


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*A. K. Bousfield and V. K. A. M. Gugenheim, On $\mathrm{PL}$ de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179

*Griffiths, P.; Morgan, J. (1981), Rational homotopy theory and differential forms, Progress in Mathematics, 16, Birkhäuser

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Edit: let me give you a conceptual proof


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*1) you have a first adjunction
$$\Omega^*_{Poly}:SSets^{op}\leftrightarrows cdga_{\mathbb{F}}:\mathfrak{R}$$
this is a Quillen adjunction we get on homotopy classes:
$$[Y,\mathfrak{R}A^*]_{SSets}\cong [A^*,\Omega^*_{Poly}(Y)]_{cdga_{\mathbb{F}}}.$$

*2) let $\Lambda_{\mathbb{F}}(x_n)$ be the free graded commutative algebra on a generator $x_n$ of degree $n$ we have:
$$[\Lambda_{\mathbb{F}}(x_n),B^*]_{cdga_{\mathbb{F}}}\cong H^n(B^*)$$
putting 1)+2) together we have that $H^n(\Omega^*_{poly}(Y))\cong [Y,\mathfrak{R}\Lambda_{\mathbb{F}}(x_n)]_{SSets}$.

*3) We use the fact that $\mathfrak{R}B^*$ is fibrant (it is always a Kan simplicial set) and the Quillen equivalence between SSets and Top (given by the realization and the singular set of a topologica space) to get that
$$[Y,\mathfrak{R}B^*]_{Ssets}\cong [RY,R\mathfrak{R}B^*]_{Top}$$
this gives us: $H^n(\Omega_{Poly}^*(Y))\cong [RY,R\mathfrak{R}\Lambda(x_n)].$

*4) The last step is the identification of $R\mathfrak{R}\Lambda(x_n)$ together with the Eilenberg-MacLane space $K(\mathbb{Q},n)$.
