Thanks to the work of Cisinski-Deglise: http://arxiv.org/abs/0912.2110, we now have a triangulated category of `motivic sheaves' available that admits the standard yoga of the six functors.
Is there any literature out there, or does anyone know how to use/adapt this to obtain a formalism of `equivariant motivic sheaves'. Slightly more precisely, I am interested in having a motivic version of the Bernstein-Lunts equivariant derived category $D_G(X)$ (say $G$ linear algebraic acting on $X$). See no. 51 at http://www.math.tau.ac.il/~bernstei/Publication_list/Publication_list.html for the Bernstein-Lunts construction in the topological setting.
Let me just briefly indicate why (I think) there are problems with just trying to run through the Bernstein-Lunts construction in the motivic setting. Very roughly what one wants is that $D_G(X)$ to be the same as $D(EG\times_G X)$ (a la Borel construction for equivariant cohomology). The space $EG\times_G X$ is a bit problematic (infinite dimensional, not a variety, etc.). Bernstein-Lunts solution is to recover "chunks" of $D(EG\times_G X)$ using sufficiently acyclic approximations to the universal bundle $EG \to BG$. The problem with adapting this naively to the motivic setting is that "chunks" means complexes of a certain length. I.e., you need some t-structure to start making sense of "chunks". And well, motivic t-structures haven't yet been constructed in any sort of generality that would make this issue moot. Does anyone know how to get around this?
Added later: since there seems to be some discussion in the comments about defining this equivariant motivic category (let me denote it $DM_G(X)$) via simplicial varieties, let me ask for two related properties that one would like (in order to justify calling it an equivariant derived category):
1) if the quotient $X/G$ exists, then $DM_G(X) \simeq DM(X/G)$;
2) a realization functor $DM_G(X) \to D_G(X)$, where the right hand side is the Bernstein-Lunts construction in the non-motivic setting.
(I am being deliberately sloppy about what I mean by quotient, and as to what base field I am working over, make whatever assumptions about these things as you see fit.)
Perhaps I should also point out that I am a bit skeptical about getting a suitable formalism using just simplicial varieties, since even in the ordinary complex algebraic setting I am not aware of nor have any idea how to get a functor yoga going using just simplicial varieties. I would assume that if one could do such a thing in the motivic setting, then one should also be able to do it (perhaps even in a simpler way) in the topological setting (which would be quite interesting in my opinion).