Let G be some nice group. Let X be a G-space (topological space, simplicial set, spectra - adjusting G accordingly). Can $X^{hG}$ be stated in terms of a holim?
For example, if G is seen as a groupoid with one point, then we have a map $X \colon G \rightarrow Spaces$. Is it true that $X^{hG} \cong holim_G X$? When would such a thing be true? If this is true, is it also true that the homotopy orbit space is the homotopy colimit of the same diagram?
Note that given a diagram $X:I\to \mathbf{Top}$, a model for $\mathrm{holim}\;X$ is as the equalizer of $\displaystyle\prod_i X_i^{|N(I/i)|} \rightrightarrows \prod_{i\to j}X_j^{|N(I/i)|}$ where $N(C)$ denotes the nerve (simplicial set) of the category $C$.
Now if you look at $I=G$ a discrete group, then $N(G/\bullet)$ (where $\bullet$ is the only object of $G$ seen as a category) has $EG$ as its nerve, and taking a look at the arrows in the above diagram shows that, seeing your $G$-space as a functor $G\to \mathbf{Top}$, $\mathrm{holim}_G \; X = \mathrm{Map}(EG,X)^G = \mathrm{Map}^G(EG,X) = X^{hG}$.
I don't know about more general versions of "$G$-space" (spectra and so on), or what happens if you let $G$ be non-discrete. This probably works for simplicial sets too ($G$ still being discete)
