Take the 2-minute tour ×  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?  share|improve this