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  "body": "\n## Summary\n\nA distinguished triangle is a sequence $X \\to Y \\to Z \\to X[1]$ of morphisms in a [[triangulated category|T2V7GJJ2EN]] that belongs to the chosen class of exact triangles.\nIt is the analogue of a short exact sequence: $Z$ plays the role of a cokernel of $X \\to Y$, and $X[1]$ records the connecting map.\nIn the [[derived category|5ST7NVJFSM]], every short exact sequence of complexes gives a distinguished triangle.\nIn the homotopy category of chain complexes the standard example is built from the [[mapping cone|6CXPGB13WD]].\n\n## Explanations\n\n### L1 human\n\nA distinguished triangle is three objects and three maps that go around in a circle, with a shift on the way back.\nIt says that $Z$ is what is left of $Y$ after removing the image of $X$, in a homotopy-invariant sense.\nUnlike a cokernel, $Z$ also remembers what $X \\to Y$ kills, shifted by one degree.\nFor complexes, the triangle $X \\to Y \\to \\operatorname{Cone}(f) \\to X[1]$ of a chain map $f$ is the basic example.\n",
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