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?