I would like to know if there is a "moral" reason why in the definition of [triangulated categories](https://en.wikipedia.org/wiki/Triangulated_category#TR_2) the "rotation axiom" TR2 requires that we have to add a negative sign to an arrow when we rotate the triangles? What was (presumably Verdier's) initial motivation to pose this negative sign in the arrow $−u[1]$ as stated in Wikipedia or standard literature? Does it come only from a convention or is there a deeper reason for it?



Moral reason for negative sign in rotation axiom for triangulated categories
One deeper reason is that this is what you get (no convention) when you have a triangulated category that comes from a stable $\infty$-category.


Ultimately, this boils down to the fact that when you take a loop in some space and reverse its direction, you get minus that loop in $\pi\_1$. The connection to this becomes clear when you then examine what the rotation of triangles corresponds to in stable $\infty$-categories. It is worked out in detail in lemma 1.1.2.13 of Lurie's *Higher algebra*.


More historically, this comes from the example of the triangulated category of chain complexes, which was the motivating example. In this case, if I understand correctly, if you are careful about sign conventions, it is also forced on you. This is worked out in Lawson's note on sign conventions ([https://www-users.cse.umn.edu/~tlawson/papers/signs.pdf](https://www-users.cse.umn.edu/%7Etlawson/papers/signs.pdf)), in paragraph 11.


