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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.

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up vote 12 down vote accepted

Shulman's paper ``Homotopy limits and colimits and enriched homotopy theory'' (on arXiv) gives a thorough study of homotopy colimits in enriched contexts, and his enriching category $V$ can be taken to be the category of $G$-spaces for a topological group $G$. It specializes to answer your questions, in greater generality than you ask. The domain category for your diagrams, which I will denote $D$ as in Shulman, is implicitly a classical category (discrete hom sets), whereas Shulman allows hom objects in $V$. He defines "corrected'' weighted homotopy colimits in terms of two-sided bar construction in sections 13 and 20 and shows in his Theorem 13.7 and Corollary 13.12 that this gives the appropriate left derived functors of weighted colimits. (This becomes more general and explicit in his Theorems 20.4 and 21.1) The upshot is that anything you can prove for spaces will generalize appropriately to G-spaces, with the usual kind of CW replacements. Making $D$ explicit helps answer your last quandary. If your $D$ is just a plain category, a functor from $D$ to $G$-spaces is just a diagram of $G$-spaces and you just see things in spaces but with induced $G$-actions. The story is more interesting when $D$ has $G$-spaces of morphisms between objects and you consider enriched diagrams ($G$-maps of $G$-spaces on Hom's).

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