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?



Polynomial differential forms on $BG$
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:


* 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
* ...


Edit: let me give you a conceptual proof


* 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 $K(\mathbb{Q},n)$.


