It is shown in Remark 7.1.12 of (a newer version of) Mark Hovey's book *Model Categories* that, in a stable model category, homotopy pullback squares coincide with homotopy pushout squares. The argument goes as follows: given a square $S$ of the form
$$
\begin{matrix}
W & \to & X\\
\downarrow\_h & & \downarrow\_g\\
Z & \to & Y 
\end{matrix}
$$
one shows that $S$ is a pullback square if and only if the induced map $\mathrm{HoFibre}(h)\to\mathrm{HoFibre}(g)$ is an isomorphism in the homotopy category. Similarly, $S$ is a pushout square if and only if the induced map $\mathrm{HoCofibre}(h)\to\mathrm{HoCofibre}(g)$ is an isomorphism in the homotopy category. One concludes by observing that $\mathrm{HoCofibre}(h)$ is the suspension of $\mathrm{HoFibre}(h)$ and similarly for $g$.


**Question:** *Is it possible to generalize this statement to say something like as follows?*


"In a diagram of the form
$$
\begin{matrix}
W & & \to && X\\
 &\searrow & \\
\downarrow &&Y&&\downarrow\\
&&&\searrow\\
Z & &\to && V, 
\end{matrix}
$$
$V$ is a homotopy colimit of
$$
\begin{matrix}
W & & \to && X\\
 &\searrow & \\
\downarrow &&Y&&\\
&&&\\\
Z & & && 
\end{matrix}
$$
if and only if 
$W$ is a homotopy limit of
$$
\begin{matrix}
 & & && X\\
 & & \\
 &&Y&&\downarrow\\
&&&\searrow\\
Z & &\to && V." 
\end{matrix}
$$



Homotopy limit-colimit diagrams in stable model categories
This is not true. For example, take $W = X = Y = 0$, and the map $Z \rightarrow V$ to be an isomorphism. Then you've got a homotopy colimit diagram, but it is only a homotopy limit diagram if $Z$ is weakly contractible.


