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  "body": "\n## Summary\n\nA spectral sequence is a sequence of pages $E_r$ of objects indexed by two integers.\nEach page is the [[homology|2J4JJJYP22]] of the previous one for a differential $d_r$.\nIt is a bookkeeping tool that computes the homology of a filtered complex or a [[double complex|N80ABTQHEQ]] by successive approximations.\nSpectral sequences connect derived functors of composites, the homology of fibrations, and many other computations in algebra and topology.\n\n## Explanations\n\n### L1 human\n\nThink of a grid of groups, one at each point of the plane.\nArrows of a given length and direction connect the groups, and taking homology gives a new, smaller grid.\nThen one repeats with longer arrows.\nIn good cases the grid stops changing after a few steps, and the final grid gives the pieces of the answer, like the layers of a cake.\nIf all the groups lie on one row, then from the second page on every arrow leaves the row, so the grid stops changing at once.\n",
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