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  "body": "\n## Definition\n\nLet $(\\mathcal{T}, \\Sigma)$ be a [[triangulated category|T2V7GJJ2EN]].\nA *triangle* in $\\mathcal{T}$ is a diagram $X \\xrightarrow{u} Y \\xrightarrow{v} Z \\xrightarrow{w} \\Sigma X$ of objects and morphisms of $\\mathcal{T}$.\nA *distinguished triangle*, or *exact triangle*, is a triangle in the class of distinguished triangles that is part of the structure of $\\mathcal{T}$.\n\n## Remarks\n\nIn a distinguished triangle, consecutive maps compose to zero: $v u = 0$, $w v = 0$ and $(\\Sigma u) w = 0$.\nFor each object $W$, applying $\\Hom(W, -)$ gives a long exact sequence $\\cdots \\to \\Hom(W, X) \\to \\Hom(W, Y) \\to \\Hom(W, Z) \\to \\Hom(W, \\Sigma X) \\to \\cdots$.\nBy TR1 and TR3, the third object $Z$ is determined by $u$ up to a non-unique isomorphism; it is called a cone of $u$.\nExample: $X \\xrightarrow{\\id} X \\to 0 \\to \\Sigma X$ is distinguished for every $X$, by the axiom TR1.\nNon-example: if $X \\ne 0$, then $X \\to 0 \\to 0 \\to \\Sigma X$ is not distinguished.\nIndeed, the long exact sequence for $W = \\Sigma X$ would give $\\Hom(\\Sigma X, 0) \\to \\Hom(\\Sigma X, \\Sigma X) \\to \\Hom(\\Sigma X, \\Sigma 0)$ exact, which forces $\\Hom(\\Sigma X, \\Sigma X) = 0$, but $\\id_{\\Sigma X} \\ne 0$.\nIn the homotopy category of complexes, the distinguished triangles are those isomorphic to $X \\xrightarrow{f} Y \\to \\operatorname{Cone}(f) \\to X[1]$ for a chain map $f$.\n",
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