TITLE: idea and intuition behind triangulated category
QUESTION [6 upvotes]: I have some trouble in understanding the significance of some axiom of triangulated category.
If someone could explain to me each axiom with some intuition, and explain to me the intuition behind the translation functor
I would be
very grateful !!
Thank you in advance

REPLY [8 votes]: TR3 is redundant: it is  implied by the other axioms.  TR4 is confused by the initial octahedral shape: it starts with a square in which one side is the identity.  If  one instead starts with a triangle given by a pair of composable arrows and their composition and writes it as a braid, one readily sees that it is just describing the behavior of triangles under composition.  The description becomes obvious in the topological framework that was Verdier's starting point (he credited work of Puppe).   For the redundancy claim, the braid description, and the behavior of distinguished triangles under products, see https://www.sciencedirect.com/science/article/pii/S0001870801919954 or https://www.math.uchicago.edu/~may/PAPERS/AddJan01.pdf.