Is there a better name for the "Mayer-Vietoris Octahedral axiom" and has it been studied?

Question

$\newcommand{\K}{\mathcal{K}}$Say $\K$ is a triangulated category with suspension $\Sigma:\K\simeq \K$. In Iversen's book "Cohomology of Sheaves", he doesn't exactly examine triangulated categories but he implicitly demonstrates the homotopy categories of complexes are triangulated... and he shows the homotopy category satisfies something seemingly stronger. He also goes on to show this seemingly stronger version of the octahedral axiom is useful in the context of localising $\K$, in the very final theorem of the book.

I'll state the stronger version of the axiom, and then my questions are:

• Does this axiom have a standard name?
• Has it been studied at all? Do we roughly know, beyond just the homotopy categories of complexes, which triangulated categories satisfy it?

The (apparent) strengthening:

If we have an octahedron in $\K$, that is, the data of:

Arrows $X\overset{u}{\to}Y\overset{v}{\to}Z\overset{g}{\to}B$, $Y\overset{f}{\to}A\overset{x}{\to}B\overset{y}{\to}C$, $\delta:B\to\Sigma X,\partial:C\to\Sigma Y$ such that $xf=gv$, $\Sigma(u)\delta=\partial y$ and such that the following triangles are all distinguished; $(u,f,\delta x),(v,yg,\partial),(vu,g,\delta)$ and $(x,y,\Sigma(f)\partial)$

Then the induced "Mayer-Vietoris" sequences of the octahedron are also distinguished triangles: $$Y\overset{\begin{pmatrix}v\\-f\end{pmatrix}}{\longrightarrow}Z\oplus A\overset{\begin{pmatrix}g&x\end{pmatrix}}{\longrightarrow}B\overset{\partial y=\Sigma(u)\delta}{\longrightarrow}\Sigma Y\\Y\underset{xf=gv}{\longrightarrow}B\underset{\begin{pmatrix}-\delta\\y\end{pmatrix}}{\longrightarrow}\Sigma X\oplus C\underset{\begin{pmatrix}\Sigma u&\partial\end{pmatrix}}{\longrightarrow}\Sigma Y$$

Answer 1

In May's paper The additivity of traces in triangulated categories he has the following things to say about this property. (At least, I think it is an equivalent property -- his notation is different and some minus signs are in different places.)

1. Call a triangle exact if it induces long exact sequences upon application of both representable and corepresentable functors (Definition 3.2). All distinguished triangles are exact, but not conversely.
2. In any octahedron, the Mayer-Vietoris triangles are exact (Lemma 3.6, asserted to be a "lengthy but elementary diagram chase").
3. In any triangulated category arising from a well-behaved Quillen model category, given the rest of an octahedron there is some choice of $x$ and $y$ making the Mayer-Vietoris triangles distinguished (section 5). He proposes to call a triangulated category with this property strong (Definition 3.8).
4. In general, the Mayer-Vietoris triangles are not distinguished for all choices of $x$ and $y$. He sketches an argument for this that I don't completely follow, citing "Faisceaux pervers" by Beilinson, Bernstein, and Deligne (Remark 3.7).

The last point seems to contradict what you say Iversen shows. I don't have access to Iversen; does he actually show this for all choices of $x$ and $y$, or only for the particular ones constructed in his proof of the octahedral axiom?