TITLE: Is this a definition of equivariant derived category?
QUESTION [7 upvotes]: Let $X$ be a topological space and $G$ be a topological group acting on $X$, both locally compact Hausdorff. Denote by $D^b(X)$ the derived category of sheaves (say of abelian groups) on $X$. We define a new category in the following way:
An object is an object $M$ of $D^b(X)$, together with an isomorphism 
$$\pi^*M\leftrightarrow \rho^*M$$
that satisfies the cocycle condition, where $\pi,\rho\colon G\times X\to X$ are the projection and the action maps. A morphism is a morphism of objects in $D^b(X)$ which is compatible with the action isomorphisms.
Note that this category is a special case of the category of equivariant objects:
http://ncatlab.org/nlab/show/equivariant+object

Is this category equivalent to the equivariant derived category? If not, why is the equivariant derived category a better construction in this case?

PS: Bernstein and Lunts give among others the following definition of the equivariant derived category in their book "Equivariant sheaves and functors" (p.32):
Denote by $[X/G]$ the "action simplex" $\Delta(n)=G^{n-1}\times X$.
By a simplicial sheaf we mean a collection of sheaves $F^n$ on $\Delta(n)$ together with maps
$\alpha_h: h^*F^m \rightarrow F^n$ for each map $h$ in $[X/G]$, which satisfy the cocycle condition:
$$\alpha_{h' h}=\alpha_h \circ h^* \alpha_{h'}$$ 
Denote by $Sh([X/G])$ the category of simplicial sheaves and by $Sh_{eq}([X/G])$ the full subcategory where all $\alpha_h$ are isomorphisms.
The derived equivariant category $D^b_G(X)$ is then the full subcategory of $D^b(Sh([X/G]))$ consisting of objects, which have cohomology in $Sh_{eq}([X/G])$.

REPLY [7 votes]: There's a related discussion here. The idea that the equivariant version of the derived category consists simply of complexes with equivariance structure is correct, once you work in a refined enough setting such as differential graded, $A_\infty$ or stable $\infty$-categories. As was explained it's certainly not true just at the homotopy level. But at this enhanced level, the equivariant derived category is indeed just the derived invariants of the group acting on the derived category of X.  The Bernstein-Lunts definition is the same, just writing down invariants using a standard resolution (ie the simplicial model for BG). A good reference for such things is chapter 7 of the manuscript of Beilinson-Drinfeld on Quantization of Hitchin's Hamiltonians.