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In their book, Bernstein an Lunts define the equivariant derived category in several ways. One can be expressed as follows:

Let $X$ be a say complex variety with an action by an algebraic group $G$.

Consider the fibration $\pi:D^b(- , \mathbb Z) \rightarrow Var$ which assigns to every variety $T$ the bounded derived category of sheaves of say abelian groups $D^b(T, \mathbb Z)$. Then the equivariant derived category $D_G^b(X,\mathbb Z)$ is defined to be the category of cartesian sections of $\pi$ over the quotient stack $X/G \rightarrow Var$. Informally one can read this as:

"An equivariant sheaf is on a space $X$ is just a map from $X/G$ to the classifying space of sheaves"

Now my questions are:

In what situation is this the correct definition? What properties of $X/G$ and $D^b(T,\mathbb Z)$ make this definition reasonable?

For example:

If we replace $X/G$ by an artin stack $\mathfrak{X}$ does this give a reasonable definition of $D^b(\mathfrak{X}, \mathbb Z)$?

If we replace $D^b(T, \mathbb Z)$ by some other "derived category of sheaves" do we still get a reasonable definition?

For example we could drop the boundedness assumption in one or both directions/allow non finitely generated cohomology sheaves, replace sheaves of abelian groups by quasi-coherent sheaves etc...

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