Suppose we are given a diagram of topological spaces. We can restrict ourselves to the diagrams over finite partially ordered sets and let all spaces be good enough (e.g. CW-complexes). One can take the colimit and homotopy colimit of this diagram and there is a set of rules to manipulate these objects, I mean first of all:
1) If $D_1\to D_2$ is coherent map of diagrams and $D_1(p)\to D_2(p)$ is homotopy equivalence for each $p$, then there is a homotopy equivalence $hocolim D_1\to hocolim D_2$.
2) If diagram $D$ is cofibrant in the Reedy category of diagrams (or some other equivalent conditions on $D$), then the canonical map $hocolim D\to colim D$ is a homotopy equivalence.
The question is about the equivariant version of these statements. Suppose topological group G acts (e.g. at the left) on each space of the diagram and all the maps in the diagram are equivariant. Then colim D and hocolim D seems to carry the induced G-action.
1) Is it true that: If $D_1\to D_2$ is coherent G-equivariant map of diagrams of G-spaces and $D_1(p)\to D_2(p)$ is equivariant homotopy equivalence for each $p$, then there is an equivariant homotopy equivalence $hocolim D_1\to hocolim D_2$?
2) What conditions should be imposed on the diagram $D$ to guarantee the equivariant homotopy equivalence $hocolim D\to colim D$?
To be strict, here by hocolim I mean the specific topological space (not the homotopy type) given by the bar-construction.
What are good references for this subject? I had found some references on hocolims and colims in the categories of $G$-spaces. But I still don't understand, what is the connection between these objects and usual colims and hocolims with induced $G$-action.
