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  "body": "\n## Summary\n\nThe octahedral axiom is the fourth axiom, TR4, in the definition of a [[triangulated category|T2V7GJJ2EN]].\nIt relates the cones of two composable morphisms $f$ and $g$ to the cone of their composite $g f$ by a further distinguished triangle.\nIt is the triangulated analogue of the third isomorphism theorem $(C/A)/(B/A) \\cong C/B$.\nIts name comes from drawing the four triangles involved as the faces of an octahedron.\n\n## Explanations\n\n### L1 human\n\nIf $X \\subseteq Y \\subseteq Z$ are abelian groups, then $Y/X$ sits inside $Z/X$ with quotient $Z/Y$.\nThe octahedral axiom says the same thing for cones in a triangulated category.\nThe cone of $X \\to Y$ maps to the cone of $X \\to Z$, and the \"quotient\" of that map is the cone of $Y \\to Z$.\nThe four triangles that appear can be drawn as four of the eight faces of an octahedron.\n",
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