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  "body": "\n## Definition\n\nLet $\\mathcal{T}$ be an [[additive category|B2VP96NJSK]] with an [[additive|KAN07YC9T2]] automorphism $\\Sigma$.\nLet a class of distinguished triangles $X \\to Y \\to Z \\to \\Sigma X$ be given that satisfies the axioms TR1, TR2 and TR3 of a triangulated category.\nThe class satisfies the *octahedral axiom* (TR4) if the following holds.\nLet $f \\colon X \\to Y$ and $g \\colon Y \\to Z$ be morphisms, and let\n$$X \\xrightarrow{f} Y \\xrightarrow{i} Z' \\xrightarrow{i'} \\Sigma X, \\qquad Y \\xrightarrow{g} Z \\xrightarrow{j} X' \\xrightarrow{j'} \\Sigma Y, \\qquad X \\xrightarrow{g f} Z \\xrightarrow{k} Y' \\xrightarrow{k'} \\Sigma X$$\nbe distinguished triangles.\nThen there are morphisms $a \\colon Z' \\to Y'$ and $b \\colon Y' \\to X'$ such that\n\n1. $Z' \\xrightarrow{a} Y' \\xrightarrow{b} X' \\xrightarrow{(\\Sigma i) j'} \\Sigma Z'$ is a distinguished triangle;\n2. $a i = k g$ and $k' a = i'$;\n3. $b k = j$ and $j' b = (\\Sigma f) k'$.\n\n## Remarks\n\nHere $Z'$, $X'$ and $Y'$ are cones of $f$, $g$ and $g f$.\nExample: take $f = \\id_X$, so that $Z' = 0$ with the triangle $X \\xrightarrow{\\id} X \\to 0 \\to \\Sigma X$, and take the second and third triangles equal.\nThen $a = 0$ and $b = \\id_{X'}$ work: $0 \\to X' \\xrightarrow{\\id} X' \\to 0$ is distinguished, being a rotation of $X' \\xrightarrow{\\id} X' \\to 0 \\to \\Sigma X'$; $a i = 0 = k g$ because $k g$ is a composite of consecutive maps of a distinguished triangle; and $k' a = 0 = i'$, $b k = k = j$, $j' b = j' = k' = (\\Sigma f) k'$.\nIn the homotopy category of chain complexes the axiom holds, with $Y'$ and the maps $a$ and $b$ built from mapping cones.\nThe axiom is a form of the isomorphism $(Z/X)/(Y/X) \\cong Z/Y$ for $X \\subseteq Y \\subseteq Z$: read $Z'$ as $Y/X$, $Y'$ as $Z/X$ and $X'$ as $Z/Y$.\n",
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