A category with translations is a category equipped with a rudimentary notion of suspension objects. Categories with translation underly triangulated categories where the “translation” becomes a genuine suspension as in homotopy fiber sequences.
called the shift functor or translation functor or suspension functor.
If , are additive and is additive is a “morphism of additive categories with translation”.
In any additive category with translation a triangle is a sequence of morphisms of the form