The Axioms for Triangulated Categories

This is an edited extract from my paper [11]. We define triangulated categories and discuss homotopy pushouts and pullbacks in such categories in §1 and §2. We focus on Verdier’s octahedral axiom, since the axiom that is usually regarded as the most substantive one is redundant: it is implied by Verdier’s axiom and the remaining, less substantial, axioms. Strangely, since triangulated categories have been in common use for over thirty years, this observation seemed to be new in [11]. We explain intuitively what is involved in the verification of the axioms in §3.