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The homotopy category of an abelian category is triangulated
Statement
If is an abelian category, then , with its shift and distinguished cone triangles, is a triangulated category.
Facts & Assumptions
Given: An abelian category .
Proof
is additive and its shift is an additive autoequivalence.
The chosen class of cone triangles satisfies TR1, TR2 with the declared sign, TR3, and TR4 by the four preceding lemmas; these data therefore meet the definition of a triangulated category.
Depends on
The homotopy category of chain complexes
The homotopy category is additive
Shift is an additive autoequivalence of the complex and homotopy categories
Distinguished cone triangle in the homotopy category
Cone triangles satisfy TR1
Cone triangles satisfy TR2 with the declared rotation sign
Cone triangles satisfy TR3
Cone triangles satisfy the octahedral axiom