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  "body": "\n## Summary\n\nThe homotopy fiber of a map $f \\colon X \\to Y$ over $y_0$ is the space of pairs of a point $x \\in X$ and a path from $f(x)$ to $y_0$.\nIt replaces the ordinary fiber $f^{-1}(y_0)$ by a version that depends only on the homotopy class of $f$.\nFor a [[Serre fibration|PB9JXQ2KC3]] the two fibers have the same homotopy groups.\nSo homotopy fibers extend the [[long exact sequence of homotopy groups|E0RV5D1GQ0]] to arbitrary maps.\nThe homotopy fiber of the inclusion of a point is the [[loop space|2JFF5F3EGF]].\n\n## Explanations\n\n### L1 human\n\nThe ordinary fiber of $f$ over $y_0$ collects the points that land exactly on $y_0$.\nThe homotopy fiber is more forgiving: a point $x$ counts if it lands anywhere, as long as one also says how to walk from $f(x)$ to $y_0$.\nSo its points are pairs, a point $x$ and a path from $f(x)$ to $y_0$.\nIf $X$ is a single point mapped to $y_0$, the pairs are just the loops at $y_0$.\n",
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