NOTES ON THE OCTAHEDRAL AXIOM 1. Pre-Triangulated ...

6 ANDREW HUBERYthere exists a commutative diagram with exact rowsfgX −−−−→ Y −−−−→ Z −−−−→ X[1]⏐ ⏐ ⏐↓∥ ↓ yz ∥Xf−−−−→ ′Y ′ g−−−−→ ′Z ′ h−−−−→ ′X[1]such that δ = f[1]h. Moreover, we may fix beforehand either (f, g, h) or (f ′ , g ′ , h ′ ).Axiom D. Given three exact triangles (f, g, h), (f ′ , g ′ , h ′ ) and (u, v, w) such thatf ′ = uf, there exists a fourth exact triangle (u ′ , v ′ , w ′ ) fitting into a commutativediagramX∥Xf−−−−→Y⏐↓ug−−−−→f−−−−→ ′Y ′ g ′⏐↓vhhZ −−−−→ X[1]∥⏐∥∥↓u ′−−−−→ Z ′ h−−−−→ ′X[1]⏐↓v ′W⏐↓wW⏐↓w ′and such that f[1]h ′ = wv ′ .Y [1]g[1]−−−−→ Z[1]Axiom D’. Under the same conditions as Axiom D, we may further assume thatthe triangleY“ y−g”is exact, where δ := f[1]h ′ = wv ′ .Axiom E. Given an exact triangle−−−−→ Y ′ ⊕ Z ( g′ z )−−−−→ Z ′ δ−→ Y [1]Y ( u −g )−−−→ Y ′ ⊕ Z ( g′ u ′ )−−−−→ Z ′ δ−→ Y [1]there exists a commutative diagram with exact rows and columnsX∥Xf−−−−→Y⏐↓ug−−−−→f−−−−→ ′Y ′ g ′⏐↓vhZ −−−−→ X[1]∥⏐∥∥↓u ′−−−−→ Z ′ h−−−−→ ′X[1]⏐↓v ′W⏐↓wW⏐↓w ′Y [1]g[1]−−−−→ Z[1]such that δ = f[1]h ′ = wv ′ . Moreover, we may fix beforehand either (f, g, h) or(f ′ , g ′ , h ′ ) and either (u, v, w) or (u ′ , v ′ , w ′ ).