As far as I know, one way to take a homotopy colimit in a model category is to replace (up to acyclic fibration) all arrows in the diagram with cofibrations, and take the strict colimit of the resulting diagram.

In Top with the model structure given by Serre fibrations, cofibrations, and weak equivalences, if one wants to obtain a homotopy pushout of the diagram $X \leftarrow A \rightarrow Y$, it is "enough" to replace only one of these arrows with a cofibration: that is, there is a natural map (by the universal property of the pushout) $Cyl(X)\cup_A Cyl(Y) \to Cyl(X) \cup_A Y$ that is a homotopy equivalence of spaces.

Question 1: What conditions on the model category $\mathcal{C}$ (or objects $X,Y,A$) will guarantee that the natural map $Cyl(X) \cup_A Cyl(Y) \to Cyl(X) \cup_A Y$ is a weak equivalence?

Question 2: This question is less precise, but if the map above is a weak equivalence, does that mean $Cyl(X) \cup_A Y$ is a good model for the homotopy pushout?

constant(Delta^op is generated by arrows which have either left or right inverses, so such an F has to take all these to isomorphisms.) $\endgroup$ – Charles Rezk Jan 8 '10 at 21:22