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My question is about the octahedral axiom (OA) in the definition of a triangulated category. For what I can understand so far (cf. Huybrechts, Fourier-Mukai in algebraic gometry, Definition 1.32), this axiom wants to roughly generalise the "double quotient" situation in the category of abelian groups, i.e. if $A\subset B\subset C$ are abelian groups then $C/B\cong(C/A)/(B/A)$.

I would like to know why people think that this axiom is superfluous.

I reported the "double quotient" situation because one may say that if it wants to generalise a situation which is natural in the non-generalised case, then one would expect this situation to be natural too. But it seems to me that this argument is too weak and probably there are better arguments...

Moreover, is it true that everyone is convinced about that?

As a motivation to this question I would like to say that: $1)$ last summer a paper by Maciocia appeared in which he proved the OA is a consequence of the previous ones. But unfortunately there was an error which is still not fixed. $2)$ In the Huybrechts book, I have cited before, he doesn't state the OA because he "will never use it explicitely and only once implicitely". $3)$ I am very curious about that.

Thank you all!

P.S.: I have tried to find something in the literature or in StackExchange as well but I was apparently unable. Sorry if it is a duplicate.

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    $\begingroup$ What convinced be of the axiom is this observation: let's have composable morphisms $f$ and $g$. Then you can form distinguished triangles over $f$, $g$ and $g\circ f$, and obviously they should be somehow related. The axiom then says that there is a distinguished triangle such that everything commutes. $\endgroup$
    – Ennar
    Commented Dec 28, 2015 at 19:35
  • $\begingroup$ @Ennar, thank you for your comment. Sorry but I cannot see your point: it seems to me that you are just paraphrasing the definition. And actually this is my point: why this assumption (which seems strong) should be implied by the other ones? (Of course I don't want an answer, but just a "feeling"). $\endgroup$
    – User3773
    Commented Dec 28, 2015 at 22:23
  • $\begingroup$ Sorry, I misunderstood your question (I should have clicked on the link before commenting). I'm afraid I cannot offer much help in this regard, but am rather interested in this as well. $\endgroup$
    – Ennar
    Commented Dec 28, 2015 at 22:55

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The octahedral axiom is probably not superfluous. That is, Maciocia's error will probably never be repaired. People were very surprised by his claim and unsurprised by the discovery of the flaw.

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  • $\begingroup$ I see, thank you. Actually the common feeling is that it is necessaire. By the way, do you maybe know what leaded Maciocia to think it was superfluous? $\endgroup$
    – User3773
    Commented Dec 29, 2015 at 12:37
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    $\begingroup$ Well, he thought he had a proof! He started looking because of phenomena you mention: it's an ugly and relatively little used axiom, so it would be nice if it were redundant. $\endgroup$ Commented Dec 29, 2015 at 15:15
  • $\begingroup$ I thought (or maybe hoped) there was a cooler thing after that.. :) why the easiest is always the better!? Thank you very much! $\endgroup$
    – User3773
    Commented Dec 29, 2015 at 15:45
  • $\begingroup$ Similarly, people tried getting rid of the 5th postulate by proving it from the others for centuries. Luckily, situations get resolved faster nowadays. However, I wonder whether we can reject the octahedral axiom and discover new structures in the same way non-Euclidean geometry was discovered. $\endgroup$ Commented Oct 25, 2022 at 16:36

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