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  "body": "\n## Summary\n\nA homotopy limit is the version of a [[limit|CHS5AFJGWE]] that respects weak equivalences.\nOrdinary limits of diagrams of spaces can change drastically when the diagram is replaced by a weakly equivalent one; homotopy limits do not.\nIn a [[model category|SDTZVZ6B4Q]] the homotopy limit is the right derived functor of the limit functor.\nHomotopy pullbacks, homotopy fixed points and the space of maps out of a homotopy colimit are typical instances.\n\n## Explanations\n\n### L1 human\n\nThe pullback of two maps to a space $Z$ consists of pairs of points with the same image.\nIn homotopy theory equality is too strict; one should allow a path in $Z$ between the two images.\nThe homotopy pullback of $X \\to Z \\leftarrow Y$ is therefore the space of triples $(x, \\omega, y)$ with $\\omega$ a path from the image of $x$ to the image of $y$.\nFor two points of the real line, the ordinary pullback is empty if the points differ, while the homotopy pullback is the space of paths between them, which is contractible.\nHomotopy limits extend this idea from pullbacks to all diagrams.\n",
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