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  "body": "\n## Definition\n\nLet $\\mathcal{T}$ be an [[additive category|B2VP96NJSK]] and let $\\Sigma \\colon \\mathcal{T} \\to \\mathcal{T}$ be an [[additive functor|KAN07YC9T2]] that is an automorphism, the *shift*, also written $X \\mapsto X[1]$.\nA *triangle* is a diagram $X \\xrightarrow{u} Y \\xrightarrow{v} Z \\xrightarrow{w} \\Sigma X$.\nA morphism of triangles is a triple of morphisms $(f, g, h)$ between the first, second and third objects.\nIt must make the three squares commute, the last one with $\\Sigma f$.\nA *triangulated category* is such a pair $(\\mathcal{T}, \\Sigma)$ with a class of triangles, the *distinguished triangles*, such that:\n\n1. (TR1) for each object $X$, the triangle $X \\xrightarrow{\\id} X \\to 0 \\to \\Sigma X$ is distinguished. Each triangle isomorphic to a distinguished triangle is distinguished. Each morphism $u \\colon X \\to Y$ is the first morphism of a distinguished triangle $X \\xrightarrow{u} Y \\to Z \\to \\Sigma X$;\n2. (TR2) a triangle $X \\xrightarrow{u} Y \\xrightarrow{v} Z \\xrightarrow{w} \\Sigma X$ is distinguished if and only if $Y \\xrightarrow{v} Z \\xrightarrow{w} \\Sigma X \\xrightarrow{-\\Sigma u} \\Sigma Y$ is distinguished;\n3. (TR3) for distinguished triangles $X \\xrightarrow{u} Y \\xrightarrow{v} Z \\xrightarrow{w} \\Sigma X$ and $X' \\xrightarrow{u'} Y' \\xrightarrow{v'} Z' \\xrightarrow{w'} \\Sigma X'$ and morphisms $f \\colon X \\to X'$, $g \\colon Y \\to Y'$ with $g u = u' f$, there is $h \\colon Z \\to Z'$ such that $(f, g, h)$ is a morphism of triangles;\n4. (TR4) the [[octahedral axiom|P8PFMNCB8Y]] holds.\n\n## Remarks\n\nExample: for an additive category $\\mathcal{A}$, the homotopy category $K(\\mathcal{A})$ with the shift of complexes is triangulated, the distinguished triangles being those isomorphic to $X \\xrightarrow{f} Y \\to \\operatorname{Cone}(f) \\to X[1]$ for chain maps $f$.\nFor a distinguished triangle and an object $W$, the sequences obtained by applying $\\Hom(W, -)$ and $\\Hom(-, W)$ are long exact.\nNon-example: the category of abelian groups has no triangulated structure, for any shift and any class of triangles.\nIn a triangulated category every monomorphism $u$ is split, since in a triangle $\\Sigma^{-1} Z \\xrightarrow{t} X \\xrightarrow{u} Y \\to Z$ one has $u t = 0$, so $t = 0$, and a triangle with a zero map splits.\nBut the monomorphism $\\Z \\xrightarrow{\\cdot 2} \\Z$ does not split.\nA category satisfying TR1 to TR3 only is called *pretriangulated*.\n",
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