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  "body": "\n## Definition\n\nLet $X$ be a topological space, let $(Y, y_0)$ be a [[pointed space|T9DG8DDE75]], and let $f \\colon X \\to Y$ be a [[continuous map|0X3WGX95FE]].\nLet $I = [0, 1]$ and let $Y^I$ be the space of [[paths|03NAMV1DHJ]] $\\gamma \\colon I \\to Y$ with the compact-open topology.\nThe *homotopy fiber* of $f$ over $y_0$ is\n$$F_f = \\set{(x, \\gamma) \\in X \\times Y^I : \\gamma(0) = f(x), \\ \\gamma(1) = y_0},$$\nwith the [[subspace topology|0WS9P12FQR]] of the [[product|0GBG8B49AA]] $X \\times Y^I$.\n\n## Remarks\n\nThe map $F_f \\to X$, $(x, \\gamma) \\mapsto x$, is a fibration, and its fiber over $x$ is the space of paths from $f(x)$ to $y_0$.\nThe ordinary fiber $f^{-1}(y_0)$ embeds in $F_f$ by $x \\mapsto (x, c_{y_0})$, where $c_{y_0}$ is the constant path.\nIf $f$ is a [[Serre fibration|PB9JXQ2KC3]], this embedding is a weak homotopy equivalence.\n\nExample: if $X = \\set{x_0}$ and $f(x_0) = y_0$, then $F_f$ is the space of paths from $y_0$ to $y_0$, which is the [[loop space|2JFF5F3EGF]] $\\Omega Y$.\n\nNon-example: the ordinary fiber need not be a homotopy fiber.\nLet $X = \\set{x_0}$, $Y = \\R$, $f(x_0) = 0$ and $y_0 = 1$.\nThen $f^{-1}(1)$ is empty, while $F_f$ is the space of paths in $\\R$ from $0$ to $1$, which is nonempty and contractible.\n\nThe notations $\\operatorname{hofib}(f)$ and $\\operatorname{hofib}_{y_0}(f)$ are also used.\n",
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