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A triangulated category is an additive category equipped with the additional structure of an autoequivalence (called the translation functor) and a class of of triangles satisfying certain axioms.
3
votes
when does a "triangulated" functor factor over the homotopy category?
Yes, that is true.
I will try to give an elementary proof.
First of all the category $K$ is obtained as a quotient category of $C$ in the sense of [MacLane, Categories for the Working Mathematician …