$\newcommand{\K}{\mathcal{K}}$Say $\K$ is a triangulated category with suspension $\Sigma:\K\simeq \K$. In Iversen's "Cohomology of Sheaves"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: $$Y\overset{\begin{pmatrix}v\\-f\end{pmatrix}}{\longrightarrow}Z\oplus A\overset{\begin{pmatrix}g&x\end{pmatrix}}{\longrightarrow}B\overset{\partial y=\Sigma(u)\delta}{\longrightarrow}\Sigma Y\\Y\underset{xf=gv}{\longrightarrow}B\underset{\begin{pmatrix}-\delta\\y\end{pmatrix}}{\longrightarrow}\Sigma X\oplus C\underset{\begin{pmatrix}\Sigma u&\partial\end{pmatrix}}{\longrightarrow}\Sigma Y$$
