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  "body": "\n## Summary\n\nA homotopy equivalence is a continuous map $f \\colon X \\to Y$ that has an inverse up to homotopy.\nThis is a map $g \\colon Y \\to X$ with $g \\circ f$ [[homotopic|F2F143KQ2M]] to $\\id_X$ and $f \\circ g$ homotopic to $\\id_Y$.\nIt is a weaker notion than a [[homeomorphism|0QT00PB5G2]].\nHomotopy invariants, such as the fundamental group and homology, cannot distinguish spaces that are related by a homotopy equivalence.\n\n## Explanations\n\n### L1 human\n\nA homotopy equivalence says that two spaces have the same shape once you allow stretching, shrinking and squashing, but no tearing.\nA thick object can be squashed to a thin one: a solid cylinder squashes to a disk, and further to a point.\nFor example, the plane without the origin is homotopy equivalent to the circle: push each point radially onto the unit circle.\nThe two spaces are not homeomorphic, but they have the same holes.\n",
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