The functor sending a smooth manifold $M$ to its de Rham algebra $\Omega^{\bullet}(M)$ does not send quotients by actions of Lie groups to invariant subalgebras. The example I have in mind is a connected Lie group $G$ acting on itself by left multiplication. The quotient is a point, which has de Rham algebra $\mathbb{R}$ concentrated in degree $0$, but the $G$-invariant subalgebra of $\Omega^{\bullet}(G)$ is much more interesting: it's the Chevalley-Eilenberg algebra $\Lambda^{\bullet}(\mathfrak{g}^{\ast})$ of the Lie algebra of $G$. 


If I wanted to fix this, it seems like I ought to derive something, maybe the operation of taking quotients. Is there a nice category of derived manifolds in which the derived quotient $G/G$ has de Rham algebra the Chevalley-Eilenberg algebra (and vector fields given by $\mathfrak{g}$, and differential operators given by $U(\mathfrak{g})$, and so forth)? Or should I be thinking in terms of Lie algebroids, or what? 


(I *think* I know what the category should be based on a talk I attended recently, but I know very little about it and would appreciate references. It should be the opposite of a category whose objects are something like commutative dg-algebras with degree $0$ part a smooth algebra.) 


**Edit:** If what I said about quotients above is silly, let me ask a slightly different question: what I really want to know is what kind of smooth object, in some category of generalized smooth spaces, has de Rham algebra the Chevalley-Eilenberg algebra, vector fields $\mathfrak{g}$, differential operators $U(\mathfrak{g})$, etc. 



For G a Lie group, can I make sense of G/G as a derived manifold in a nice way?
I fully agree with A non's answer and comments. However, here is another perspective which you might find useful. 


Recall that the de Rham stack $M\_{dR}$ of a manifold $M$ is obtained from $M$ by identifying infinitesimally close points. In general the space of (derived) functions on the de Rham stack is the de Rham complex, and the category of quasi-coherent sheaves on $M\_{dR}$ is modules for the ring of differential operators on $M$. Thus the de Rham stack is a way of encoding he kinds of invariants you are interested in.


In the case $M=G$, we have $G\_{dR} = G/\widehat{G}$, where $\widehat{G}$ is the formal group associated to $G$ (a formal neighbourhood of the identity in $G$). As you have noticed, the quotient $(G/G)\_{dR} = G\_{dR}/G\_{dR} = pt\_{dR} = pt$. So the de Rham complex of $pt = G/G$ is not the $G$-invariants of the de Rham complex of $G$.


I think what you want to look at is the stack $G\_{dR}/G$. The space of functions on $G\_{dR}/G$ is indeed $G$-invariant forms on $G$, i.e. the CE complex. And quasi-coherent sheaves on $G/G\_{dR}$ are modules for $G$-invariant differential operators, i.e. $\mathcal U(\mathfrak{g})$-modules.


Another name for $G/G\_{dR}$ would be the classifying stack $B\widehat{G}$ or $B\mathfrak{g}$.


Sorry if this is a little vague - I am thinking in an algebraic setting, but it should be possible to carry these things over to a differential geometric setting.


