There are lots of similar questions that have been answered on this topic (particularly [Homotopy limit-colimit diagrams in stable model categories](https://mathoverflow.net/questions/135462/homotopy-limit-colimit-diagrams-in-stable-model-categories)), but I have a specific question that I do not believe has been answered. I suspect there is a simple answer that I'm just not seeing, so apologize in advance if this is the case. Given a commutative square
$\require{AMScd}$
\begin{CD}
A @>{}>> X\\
@V{f}VV @V{g}VV \\
B @>{}>> Y
\end{CD}
we say it is a **homotopy pullback square** if the canonical map $A \to holim(B \to Y \leftarrow X)$ is a weak equivalence and that it is a **homotopy pushout square** if the canonical map $hocolim(B \leftarrow A \rightarrow X) \to Y$ is a weak equivalence. 


I'm hoping to understand the proof that in a stable category, the two notions coincide. As pointed out in the reference above, in the 2007 version of Hovey's ``Model Categories,'' an argument is given in Remark 7.1.12 that goes roughly as follows. One can show that the square above is a homotopy pullback square if and only if 
$$
hofib(f) \overset{\sim}{\to} hofib(g) 
$$
is a weak equivalence. Similarly, it is a homotopy pushout square if and only if 
$$
hocofib(f) \overset{\sim}{\to} hocofib(g)
$$
is a weak equivalence. One then concludes the result by claiming that 
$$
\Sigma hofib(f) \simeq hocofib(f) \text{ and } \Sigma hofib(g) \simeq hocofib(g)
$$
This is the part on which I am stuck. Hovey claims that it follows from Thereom 7.1.11 (which is the same in both the 1999 and 2007 version) but I do not see this. It seems to me (and here is where I must be going wrong) that Hovey is claiming that the fiber in the homotopy category is the homotopy fiber -- because from what I understand, it is the homotopy category which is triangulated (to which 7.1.11 would apply), not the original stable model category. Since the fiber in the homotopy category is *not* (usually) the homotopy fiber, I must be misinterpreting what Hovey is saying.


So my question is: how do we know that the suspension of the homotopy fiber is weakly equivalent to the homotopy cofiber (or, equivalently, that loops of the homotopy cofiber is weakly equivalent to the homotopy fiber). Am I misunderstanding what Hovey means by "fiber sequence?" Are the distinguished triangles actually *homotopy* cofiber sequences in the original model category? As is probably clear by now, I do not know much about triangulated categories. Any help/reference would be appreciated!



Homotopy pullbacks and pushouts in stable model categories
Many thanks and much credit to Dmitri Pavlov for clearing up my confusions. In the interest of having a definitive answer, I've organized his comments and my understanding into the following answer.


Let $\mathcal{C}$ be a stable model category; then (by definition, for Hovey) $Ho(\mathcal{C})$ is a triangulated category.


**Note:** By definition 6.2.6 (same in both the 1999 and 2007 version of Hovey's "Model Categories") when we say $A \to B \to C$ in $Ho(\mathcal{C})$ is a cofiber sequence, we mean that there is a natural zig-zag of weak equivalences *in $\mathcal{C}$* to a cofibration sequence of cofibrant objects $A' \hookrightarrow B' \to C'$. Dually, saying that $X \to Y \to Z$ in $Ho(\mathcal{C})$ is a fiber sequence means that there is a natural zig-zag of weak equivalences *in $\mathcal{C}$* to a fibration sequence of fibrant objects $X' \to Y' \twoheadrightarrow Z'$. 


**Claim:** Given a map $X \overset{g}{\to} Y$ in $\mathcal{C}$, we have that $\Sigma hofib(g) \simeq hocofib(g)$, where $\Sigma$ is derived suspension (i.e. the suspension of a cofibrant replacement). 


*Proof:* Without loss of generality (replacing if needed) we may assume $g$ is a fibration between fibrant objects. Take the (strict) fiber of $g$ (which will also be the homotopy fiber) to obtain
$$
F \to X \overset{g}{\twoheadrightarrow} Y
$$
Then replace the map $F \to X$ with a fibration and again take the fiber to obtain the following fibration sequence of fibrant objects in $\mathcal{C}$
$$
\Omega Y \to G \twoheadrightarrow X \twoheadrightarrow Y
$$
By definition, this descends to a fiber sequence 
$$
\Omega Y \to G \to X \to Y
$$
in $Ho(\mathcal{C})$. By Remark 7.1.9 (same in both versions) the following is also then a fiber sequence in $Ho(\mathcal{C})$ (note our notation clashes with Hovey's here, i.e. our $G$ is his $X$, etc.).
$$
\Omega\Sigma G \to X \to Y \to \Sigma G
$$
which means (by 6.2.6) that $hofib(X \to Y) \simeq \Omega\Sigma G$.
Now apply 7.1.11 (same in both versions) in $Ho(\mathcal{C})$ (taking $Z = \Sigma G$) to obtain a cofiber sequence
$$
\Omega \Sigma G \to X \to Y \to \Sigma(\Omega\Sigma G)
$$
in $Ho(\mathcal{C})$ which means (by 6.2.6) that $hocofib(X \overset{g}{\to} Y) \simeq \Sigma(\Omega\Sigma G)$. Since $\Sigma(\Omega\Sigma G) \simeq \Sigma hofib(X \overset{g}{\to} Y)$, this completes the proof.


