TITLE: Do homotopy pullbacks commute with homotopy orbits (in spaces)?
QUESTION [6 upvotes]: Suppose we are given a diagram $X \to Z \gets Y$ of $G$-spaces ($G$ a discrete group). Let $(- \times^h -)$ denote homotopy pullback. Is $(X \times^h_Z Y)_{hG}$ weakly equivalent to $X_{hG} \times^h_{Z_{hG}} Y_{hG}$?

REPLY [7 votes]: Yes.  A sketch:
Taking products with the free $G$-space $EG$ commutes with the pullback diagram (because product is also a limit) and so you can assume they're free, and one of the maps is a fibration.  
Having done this, there is a natural long exact sequence of homotopy groups
$\to \pi_* (U) \to \pi_*(U_{hG}) \to \pi_*(BG) \to \dots$
and applying this to the pullback diagram you can deduce (from the 5-lemma) that the natural map from the orbit of pullbacks to the pullback of the orbits is a weak equivalence.