For questions about triangulated categories. A triangulated category is an additive category with an additive auto-equivalence called a translation (or shift) functor, and a class of distinguished triangles satisfying various axioms.

Let $\mathcal{A}$ be an additive category and $\Sigma:\mathcal{A}\to\mathcal{A}$ be an additive auto-equivalence. A *triangle* in $(\mathcal{A},\Sigma)$ consists of three objects $A,B,C$ and three morphisms $u:A\to B$, $v:B\to C$ and $w:C\to\Sigma(A)$.

A *triangulated category* is an additive category $\mathcal{A}$ equipped with an additive auto-equivalence $\Sigma:\mathcal{A}\to\mathcal{A}$ which we call its *translation functor* (or shift functor), and a class of distinguished triangles satisfying various axioms.

Important examples include the homotopy category of complexes of an additive category, the derived category of an abelian category and stable categories of Frobenius algebras.

**Reference:** *Triangulated Categories*, edited by Holm T., Jorgensen P. & Rouquier R., London Math. Soc., 2010