---
tag: P8PFMNCB8Y
kind: concept
name: Octahedral axiom
msc: [18G80]
sources:
  - url: https://en.wikipedia.org/wiki/Triangulated_category
---

## Summary

The octahedral axiom is the fourth axiom, TR4, in the definition of a [[triangulated category|T2V7GJJ2EN]].
It relates the cones of two composable morphisms $f$ and $g$ to the cone of their composite $g f$ by a further distinguished triangle.
It is the triangulated analogue of the third isomorphism theorem $(C/A)/(B/A) \cong C/B$.
Its name comes from drawing the four triangles involved as the faces of an octahedron.

## Explanations

### L1 human

If $X \subseteq Y \subseteq Z$ are abelian groups, then $Y/X$ sits inside $Z/X$ with quotient $Z/Y$.
The octahedral axiom says the same thing for cones in a triangulated category.
The cone of $X \to Y$ maps to the cone of $X \to Z$, and the "quotient" of that map is the cone of $Y \to Z$.
The four triangles that appear can be drawn as four of the eight faces of an octahedron.