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  "body": "\n## Summary\n\nThe Lusternik–Schnirelmann category of a space $X$ is the least number of open sets, minus one, that cover $X$ and are each contractible inside $X$.\nIt is a homotopy invariant that measures how far a space is from being contractible.\nLusternik and Schnirelmann introduced it to bound from below the number of critical points of a smooth function on a closed manifold.\nIt is itself bounded below by the cup-length of the cohomology ring.\n\n## Explanations\n\n### L1 human\n\nTry to cover a space by as few open patches as possible, where each patch can be shrunk to a point inside the whole space.\nA contractible space needs one patch, so its category is $0$.\nA sphere cannot be shrunk in one piece.\nBut the sphere minus the north pole and the sphere minus the south pole each shrink to a point, so the sphere has category $1$.\nThe more patches a space needs, the more complicated its shape.\n",
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