Call a diagram $E$ in a model category a *homotopy colimit diagram* if the morphism $$\mathrm{hocolim}~E\to \mathrm{colim}~ E$$ is a weak equivalence. A *homotopy colimit* is defined as the categorical colimit of a cofibrant replacement of the diagram in the projective model structure and this is where the morphism comes from.

Let $F:C\rightleftarrows D:G$ be a Quillen equivalence between model categories $C$ and $D$. The (**Edit**: derived!) left adjoint $F$ *preserves homotopy colimits*, i.e. if $E$ is a homotopy colimit diagram in $C$, then $F\circ Q\circ E$ is a homotopy colimit diagram in $D$ where $Q$ denotes a cofibrant replacement.

Does the (

Edit: derived!) right adjoint $G$ preserve homotopy colimits if the adjunction is a Quillen equivalence?To be more precise, if $E$ is a homotopy colimit diagram in $D$, is $G\circ R\circ E$ is a homotopy colimit diagram in $C$ where $R$ denotes a fibrant replacement?

I suppose that this is true since the notion of homotopy colimit should depend only on the homotopy category and not on the model, I guess, but I cannot think of an argument.

derivedleft Quillen functor preserves hocolims. (Think of $B\otimes_A: Ch(A)\to Ch(B)$, where $A\to B$ isn't flat.) $\endgroup$ – Charles Rezk Dec 6 '11 at 20:47