As should be evident from the title this question has a similar flavor to:

Formality of classifying spaces

However, unlike Geordie's question, I will be working with torsion free coefficients (say the complex numbers). The `torsion' in the current question is going to be from a different source.

Let $G$ be a reductive linear algebraic group (over $\mathbb{C}$) say. Relatively general formal considerations show that the $G$-equivariant derived category of a point is equivalent to the dg-derived category of a dg-algebra (I am going to ignore finiteness issues).

For instance, if $G$ is connected (so that $BG$ is simply connected), the dg-algebra is the algebra associated to the De Rham complex of $BG$ (let's be friendly and ignore finiteness issues again). In this case the $G$-equivariant derived category of a point is the subcategory of $D(BG)$ generated by the constant sheaf. The latter is equivalent to the derived category of the De Rham complex. Further, since $H^*(BG)$ is a polynomial ring (I am still in the $G$ is connected case) generated in even degree, it is easy to see that this dg-algebra is formal (the point is that the De Rham complex is (super)commutative before even passing to cohomology, so you can very naively construct your quasi-isomorphism by hand).

Ok, that's pretty nice. Now let's look at $G$ not connected. Then the $G$-equivariant derived category of a point is still governed by a dg-algebra, but it's a bit nastier. Essentially we want the subcategory of $D(BG)$ generated by local systems, and since $BG$ isn't simply connected anymore we are going to have non-trivial local systems. The dg-algebra we are after is the $Ext$-algebra of the sum of all the irreducible local systems. Now we can certainly compute the cohomology of this dg-algebra: it is the $H^*(BG^0)$-twisted group algebra of $G/G^0 = \pi_1(BG)$. **Added later: **to see this, loop the fibration $BG^0\hookrightarrow BG \twoheadrightarrow B(G/G^0)$.

**The question:** is the dg-algebra governing $D(BG)$ (for $G$ not necessarily connected) formal? If so, is it possible to deduce this formality from the connected case?

I am quite happy assuming that $G/G^0$ is abelian.

Proc. Kon. Ned. Akad. v. Wet.7 (1976) 296-302. Can this be linked with the other material you give? There are related ideas in my paper with O. Mucuk, "Covering groups of non-connected topological groups revisited",Math. Proc. Camb. Phil. Soc, 115 (1994) 97-110. – Ronnie Brown Oct 6 '13 at 9:41`DG-modules and sheaves of topological spaces') explains why one might care. The`

natural subcategory' of $D(X)$ alluded to in the middle of the page is the subcategory generated by the constant sheaf. If you wish I can add additional comments about how I think about the equivariant derived category of a point. But I'd be surprised if any of it were news to you! – Reladenine Vakalwe Oct 6 '13 at 15:03